From sohtani@math.sci.hokudai.ac.jp Thu Sep 7 23:54:41 2000 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.9.3/8.9.1) with ESMTP id XAA12486 for ; Thu, 7 Sep 2000 23:55:36 -0400 (EDT) Received: from mgate2.sys.hokudai.ac.jp (mgate2.sys.hokudai.ac.jp [133.87.1.132]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id UAA16262 for ; Thu, 7 Sep 2000 20:54:46 -0700 (PDT) Received: from wwwcs2.sys.hokudai.ac.jp (localhost [127.0.0.1]) by mgate2.sys.hokudai.ac.jp (8.8.7/3.6Wbeta5) with ESMTP id MAA01979 for ; Fri, 8 Sep 2000 12:54:43 +0900 (JST) Received: from math.sci.hokudai.ac.jp by wwwcs2.sys.hokudai.ac.jp (8.9.3/3.7W) id MAA18108; Fri, 8 Sep 2000 12:52:28 +0900 (JST) Received: (qmail 12964 invoked by uid 30409); 8 Sep 2000 03:54:41 -0000 Date: 8 Sep 2000 03:54:41 -0000 Message-ID: <20000908035441.12963.qmail@math.sci.hokudai.ac.jp> From: OHTANI Sachiko To: was@gold.math.berkeley.edu Cc: sohtani@math.sci.hokudai.ac.jp Subject: Hi ! Content-Type: text Content-Length: 576 Status: RO X-Status: O Hi William, This is Sachiko Ohtani in Japan, who see you in MSRI and UCB 2 weeks ago. Do you remenber me ? I arrived at Japan on Sepember 2 safely. I went to also Boston last week and enjoyed staying there. If you also came back to Boston during my staying. I would meet you in Boston. As I told you, I am sending .tex source of my thesis in next message. If you read it, please give me your advice. When I met Prof. Ribet, he suggested another way of proof, so my thesis will be better than now. I look forward to seeing you again. Sincerely yours, Sachiko Ohtani From sohtani@math.sci.hokudai.ac.jp Thu Sep 7 23:57:38 2000 Received: from math.berkeley.edu (gold-slow.Math.Berkeley.EDU [128.32.183.94]) by abel.math.harvard.edu (8.9.3/8.9.1) with ESMTP id XAA12503 for ; Thu, 7 Sep 2000 23:58:34 -0400 (EDT) Received: from mgate2.sys.hokudai.ac.jp (mgate2.sys.hokudai.ac.jp [133.87.1.132]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id UAA16407 for ; Thu, 7 Sep 2000 20:57:43 -0700 (PDT) Received: from wwwcs2.sys.hokudai.ac.jp (localhost [127.0.0.1]) by mgate2.sys.hokudai.ac.jp (8.8.7/3.6Wbeta5) with ESMTP id MAA02308 for ; Fri, 8 Sep 2000 12:57:41 +0900 (JST) Received: from math.sci.hokudai.ac.jp by wwwcs2.sys.hokudai.ac.jp (8.9.3/3.7W) id MAA18410; Fri, 8 Sep 2000 12:55:25 +0900 (JST) Received: (qmail 12975 invoked by uid 30409); 8 Sep 2000 03:57:38 -0000 Date: 8 Sep 2000 03:57:38 -0000 Message-ID: <20000908035738.12974.qmail@math.sci.hokudai.ac.jp> From: OHTANI Sachiko To: was@gold.math.berkeley.edu Subject: Thesis by S. Ohtani Content-Type: text Content-Length: 45242 Status: RO X-Status: O %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%% Construction of unramified Galois extensions over maximal abelian %%%%% %%%%%%%%%%%%%%%% extensions of algebraic number fields %%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt,a4paper]{amsart} \usepackage[mathscr]{eucal} \usepackage{amssymb} \usepackage{latexsym} \usepackage{amsthm} \usepackage{amscd} \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bF}{\mathbb{F}} \newcommand{\Gm}{\mathbb{G}_m} \newcommand{\ab}{\mathrm{ab}} \newcommand{\nr}{\mathrm{nr}} \newcommand{\tors}{\mathrm{tors}} \newcommand{\Aut}{\mathrm{Aut}\,} \newcommand{\Hom}{\mathrm{Hom}\,} \newcommand{\End}{\mathrm{End}\,} \newcommand{\Gal}{\mathrm{Gal}\,} \newcommand{\Char}{\mathrm{char}\,} \newcommand{\Dim}{\mathrm{dim}\,} \newcommand{\Spec}{\mathrm{Spec}\,} \newcommand{\for}{\mathrm{for}} \newcommand{\Rank}{\mathrm{rank}\,} \newcommand{\Ker}{\mathrm{Ker}\,} \newcommand{\ord}{\mathrm{ord}\,} \newcommand{\Det}{\mathrm{det}\,} \newcommand{\Tr}{\mathrm{Tr}\,} \newcommand{\fp}{\mathfrak{p}} \newcommand{\fP}{\mathfrak{P}} \newcommand{\fl}{\mathfrak{l}} \newcommand{\fL}{\mathfrak{L}} \newcommand{\fm}{\mathfrak{m}} \newcommand{\cA}{\mathcal{A}} \newcommand{\cX}{\mathcal{X}} \newcommand{\cO}{\mathcal{O}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\cE}{\mathcal{E}} \newcommand{\Ie}{\textit{i.e.}} \newcommand{\Eg}{\textit{e.g.}} \newcommand{\Cf}{\textit{cf.}\;} \newcommand{\lr}{\longrightarrow} \newcommand{\hr}{\hookrightarrow} \newcommand{\Llr}{\Longleftrightarrow} \newcommand{\lmt}{\longmapsto} \newcommand{\Lim}{\varinjlim\,} \newcommand{\plim}{\varprojlim\,} \newcommand{\Kvz}{K_v(\zeta_{p^r})} \newcommand{\Kz}{K(\zeta_{p^r})} \newcommand{\Kzi}{K(\zeta_{p^{\infty}})} \newcommand{\Qzi}{\mathbb{Q}(\zeta_{p^{\infty}})} \newcommand{\Qlz}{\mathbb{Q}_l(\zeta_{p^r})} \newcommand{\lmd}{\lambda} \newcommand{\Lmd}{\Lambda} \newcommand{\dlt}{\varDelta} \newcommand{\Gtld}{\tilde{G}} \newcommand{\Xbar}{\underline{X}} \newcommand{\Ybar}{\underline{Y}} \title{Construction of unramified Galois extensions over maximal abelian extensions of algebraic number fields} \author{Sachiko Ohtani} \date{} \address{Department of Mathematics, Hokkaido University, Sapporo 060-0810, Japan} \email{sohtani@math.sci.hokudai.ac.jp} \pagestyle{myheadings} \footskip = 32pt \topmargin=0pt \oddsidemargin=1truecm \evensidemargin=1truecm \textheight=22cm \textwidth=14cm \begin{document} \maketitle \markboth{SACHIKO OHTANI}{UNRAMIFIED GALOIS EXTENSIONS} \setcounter{section}{-1} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% abstruct %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \begin{abstract} We construct unramified Galois extensions over maximal abelian extensions of algebraic number fields by using division points of abelian varieties which have everywhere semistable reduction. Further, by using division points of elliptic curves, we construct infinitely many linearly independent unramified Galois extensions of $\Qzi^{\ab}$ having $SL_2(\bZ_p)$ as the Galois group over $\Qzi^{\ab}$. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%% Introduction %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Introduction} Our purpose in this paper is to construct unramified Galois extensions over maximal abelian extensions of algebraic number fields by using division points of abelian varieties. Let $A$ be an abelian variety over an algebraic number field $K$ and $p \geq 5$ a prime number. Suppose that the pair $(A, p)$ satisfies the following conditions: \begin{quote} \begin{itemize} \item[(A0)] $A$ has everywhere semistable reduction over $\cO_K$, where $\cO_K$ is the ring of integers of $K$. \item[(A1)] $A$ has bad reduction at all extensions of $p$ to $K$. \end{itemize} \end{quote} Note that, if we replace $K$ by a suitable finite extension $L$ of $K$, then the abelian variety $A\otimes_{K} L$ over $L$ satisfies the condition $(\mathrm{A0})$ by the \textit{semistable reduction theorem} (\Cf Grothendieck \cite{SGA7}, Proposition 3.6). For a place $v$ of $K$, we denote by $\cA_v^0$ the identity component of the special fibre of the N\'eron model of $A$ at $v$. For an algebraic number field $k$, we denote by $k^{\ab}$ the maximal abelian extension of $k$. Our main result is the following \begin{theorem}\label{thA} Let $A, K$ and $p$ be as above, and $F$ the field obtained by adjoining to $K$ the coordinates of all $p$-power division points on $A$.\\ $(a)$ If $\cA_v^0$ is a split torus for every place $v$ of $K$ such that $A$ has bad reduction at $v$, then $F\Kzi^{\ab}$ is an unramified Galois extension of $\Kzi^{\ab}$, where $\Kzi$ is the field obtained by adjoining to $K$ the $p$-power roots of unity.\\ $(b)$ If $\cA_v^0$ is a split torus for every place $v$ of $K$ lying above $p$, then $F(K\bQ^{\ab})^{\ab}$ is an unramified Galois extension of $(K\bQ^{\ab})^{\ab}$. \end{theorem} Further, by using division points of elliptic curves (\Ie, abelian varieties of dimension one), we obtain the following \begin{theorem}\label{thE} Let $p \geq 5$ be a prime number. Then there exist infinitely many linearly independent unramified Galois extensions of $\Qzi^{\ab}$ having $SL_2 (\bZ_p)$ as the Galois group over $\Qzi^{\ab}$. \end{theorem} Here, for a sequence of extensions $K_1, K_2, \cdots$ of an algebraic number field $k$, if $K_{n+1} \cap K_1K_2 \cdots K_n = k$ for any $n \geq 1$, then we say that the extensions are \textit{linearly independent} over $k$. \\ We shall now explain the background of these results. Unramified abelian extensions over maximal abelian extensions of algebraic number fields have been investigated by many people, \Eg, Cornell \cite{gC79}, Brumer \cite{aB81} and Kurihara \cite{mK99}. Uchida \cite{kU82} and Horie \cite{kH90} determined the structure of the Galois group of the maximal unramified solvable extension over maximal abelian extensions of algebraic number fields. For unramified non-solvable extensions, Asada \cite{mA85}, \cite{mA85'} gave some results. First, Asada \cite{mA85} considered elliptic curves over $\bQ$ whose modular invariants are integral and which have good reduction at a supersingular prime $p$, and then constructed unramified Galois extensions over maximal abelian extensions of algebraic number fields by using their $p$-power division points. Second, Asada \cite{mA85'} constructed an infinite family of elliptic curves over $\bQ$ which have multiplicative reduction at a prime $p$ and of which the order at $p$ of the modular invariants are divisible by $p$, and then constructed unramified Galois extensions over $\bQ^{\ab}$ having $PSL_2 (\bZ/p^r\bZ)$ as the Galois group over $\bQ^{\ab}$ by using their $p^r$-division points. Further this paper contains an example of unramified Galois extensions having $SL_2(\bZ_p)$ as the Galois group using results of \cite{mA85}. Asada's family in \cite{mA85'} is chosen so carefully that it appears very special. Also his elliptic curves in \cite{mA85} have everywhere potential good reduction. So we would like to find more general phenomena. In this paper, we give a general construction, at the expense of enlarging the base field, using abelian varieties which have everywhere semistable reduction. Next we shall explain the contents of this paper. In Section 1, we give the proof of Theorem \ref{thA}. In Section 2, we give examples of abelian varieties and algebraic number fields which satisfy the conditions of Theorem \ref{thA}. In Section 3, by using division points of elliptic curves over $\bQ$ which satisfy the conditions of Theorem \ref{thA} and some additional assumptions, we construct infinitely many unramified Galois extensions over $\Qzi^{\ab}$ having $SL_2(\bZ_p)$ as the Galois group over $\Qzi^{\ab}$, and we give the proof of Theorem \ref{thE}. In Section 4, we consider elliptic curves of prime conductor $p$ of Setzer \cite{bS75}, and construct unramified Galois extensions over finite extensions over $\bQ$ having $SL_2 (\bZ/p^r\bZ)$ as the Galois group. The author would like to express her sincere gratitude to Professor Yuichiro Taguchi who suggested her to consider this problem and gave her many valuable suggestions. The author also thanks Professor Iku Nakamura for enjoyable discussions on degeneration of abelian varieties.\\ In this paper we use the following notation: \begin{description} \item[$\bQ$] the field of rational numbers. \item[$\bZ$] the ring of rational integers. \end{description} For a rational prime number $p$, \begin{description} \item[$\bQ_p$] the field of $p$-adic numbers, \item[$\bZ_p$] the ring of $p$-adic integers, \item[$\bF_p$] the prime field of $p$ elements. \end{description} For an algebraic number field $K$, \begin{description} \item[$K^{\ab}$] the maximal abelian extension of $K$, \item[$K_v$] the completion of $K$ at a place $v$ of $K$, \item[$\cO_K$] the ring of integers of $K$, \item[$G_K$] the absolute Galois group over $K$. \end{description} For a positive integer $m$, \begin{description} \item[$\zeta_m$] a primitive $m$-th root of unity, \item[{$A[m]$}] the group of $m$-division points of an abelian variety $A$.\end{description} \bigskip %%%%%%%%%%%%%%%%%%%%%%%%% Key Lemmas %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Division Points of Abelian Varieties} In this section, we shall give the proof of Theorem \ref{thA}. \subsection{Key lemmas} In our construction of unramified Galois extensions over maximal abelian extensions of algebraic number fields, a key is the following lemma due to Y. Ihara. \begin{lemma}[Asada \cite{mA85}, Proposition\;1]\label{ihara} Let $k$ be an algebraic number field, $k^{\ab}$ the maximal abelian extension of $k$, and $K$ a finite Galois extension of $k$. Then $Kk^{\ab}/k^{\ab}$ is unramified if and only if its decomposition group in $K/k$ is commutative for any prime divisor of $K$. \end{lemma} Next we shall give an application of \textit{Mumford's construction} of degenerate abelian varieties (\Cf Faltings-Chai \cite{FC90}). Let $A$ be an abelian variety of dimension $g$ over an algebraic number field $K$ and $p \geq 5$ a prime number. Suppose that the pair $(A, p)$ satisfies the following conditions: \begin{quote} \begin{itemize} \item[(A0)] $A$ has everywhere semistable reduction over $\cO_K$. \item[(A1)] $A$ has bad reduction at all extensions of $p$ to $K$. \end{itemize} \end{quote} Let $v$ be a place of $K$ such that $A$ has bad reduction at $v$, and $G$ the identity component of the N\'eron model of $A$ at $v$. By the condition (A0), $G$ is a semi-abelian group scheme over $\Spec(\cO_{K_v})$. If the special fibre $\cA_v^0$ of $G$ is a split torus, then there exists a group isomorphism $$ \phi : (K_v^{\times})^g / \Lmd \overset{\sim}{\lr} G(K_v) \simeq A(K_v), \quad \Lmd \simeq \bZ^g, $$ by Faltings-Chai \cite{FC90}, Ch.III, Proposition 8.1. Hence we obtain the following \begin{lemma}\label{dp} $(a)$ Let $m \geq 2$ be an integer and $\Lmd^{1/m} = \{ x \in (K_v^{\times})^g \;|\; x^m \in \Lmd \}$. Then the group isomorphism $\phi$ induces an isomorphism of $G_{K_v}$-modules $$ \Lmd^{1/m}/\Lmd \simeq A[m]. $$ $(b)$ Further, $\Lmd^{1/m}/\Lmd$ fits into the following exact sequence of $G_{K_v}$-modules: \begin{equation*} \begin{CD} 1 @>>> {{\boldsymbol{\mu}}_m}^g @>>> \Lmd^{1/m}/\Lmd @>>> (\bZ/m\bZ)^g @>>> 0 \end{CD} \end{equation*} where $\boldsymbol{\mu}_m$ is the group of m-th roots of unity.\\ $(c)$ Let $(q_i)_{1 \leq i \leq g }$ be a basis of $\Lmd$, and put $q_i = (q_{i1}, q_{i2}, \cdots, q_{ig}) \in (K_v^{\times})^g$ for each $1 \leq i \leq g$. Then $$ K_v(A[m]) = K_v\!\left(\zeta_m, \{ q_{ij}^{1/m} \}_{1 \leq i,j \leq g} \right), $$ where $K_v(A[m])$ is the field obtained by adjoining to $K$ the coordinates of m-division points on $A$. \end{lemma} \begin{proof} The group isomorphism $\phi$ above induces a homomorphism $$ \tilde{\phi} : \Lmd^{1/m} \lr \; A[m], $$ and the kernel is equal to $\Lmd$. Let $(q_i)_{1 \leq i \leq g }$ be a basis of $\Lmd$, and put $q_i = (q_{i1}, q_{i2}, \cdots, q_{ig}) \in (K_v^{\times})^g$ for each $1 \leq i \leq g$. If $x \in \Lmd^{1/m}$, then $x$ may be written as follows: \begin{equation} \tag{*} x = \left(\zeta_m^{r_1} \prod_{i=1}^g {q_{i1}^{s_i/m}}, \cdots, \zeta_m^{r_g} \prod_{i=1}^g {q_{ig}^{s_i/m}} \right), \end{equation} where $r_1, \cdots, r_g \in \bZ/m\bZ, s_1, \cdots, s_g \in \bZ$. Since the order of the group $\Lmd^{1/m}/\Lmd$ is equal to the order of the group $A[m]$, we see that $\tilde{\phi}$ is surjective. Further if $x \in \Lmd^{1/m}$, then we see that $\tilde{\phi} (\sigma(x)) = \tilde{\phi}(x) ^{\sigma}$ for any $\sigma \in G_{K_v}$. Hence we obtain an isomorphism of $G_{K_v}$-modules $\Lmd^{1/m}/\Lmd \lr A[m]$. >From the equation (*), we see that $\Lmd^{1/m}/\Lmd$ fits into the exact sequence of the statement of (b), and we obtain the following isomorphisms: $$ K_v(A[m]) = K_v(\Lmd^{1/m}/\Lmd) = K_v\!\left(\zeta_m, \{ q_{ij}^{1/m} \}_{1 \leq i,j \leq g} \right). $$ \end{proof} \smallskip %%%%%%%%%%%%%%%%%%%%%%%%% Proof of Theorem 0.1. %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Proof of Theorem 0.1.} Let $A$ be an abelian variety of dimension $g$ over an algebraic number field $K$ and $p \geq 5$ a prime number. Suppose that the pair $(A, p)$ satisfies the conditions (A0) and (A1) in 1.1. Let $r$ be a positive integer, and $F_r$ the field obtained by adjoining to $K$ the coordinates of $p^r$-division points on $A$. Theorem \ref{thA} $(a)$ follows from the following \begin{theorem}\label{unrA} Suppose that $\cA_v^0$ is a split torus for every place $v$ of $K$ such that $A$ has bad reduction at $v$. Then $F_r\Kz^{\ab}$ is an unramified Galois extension of $\Kz^{\ab}$. \end{theorem} \begin{proof} By Lemma \ref{ihara}, it suffices to show that $F_rK_v$ is an abelian extension of $\Kvz$ for every place $v$ of $K$. If $v$ is unramified in $F_r$, there is nothing to prove. Hence we only have to verify it for the places which are ramified in $F_r$. By the \textit{criterion of N\'eron-Ogg-Shafarevich} (\Cf Serre-Tate \cite{ST68}, Theorem 1), such places are the bad places of $A$. (Recall that we assumed that $A$ has bad reduction at all extensions of $p$ to $K$.) Let $v$ be a place of $K$ such that $A$ has bad reduction at $v$. By Lemma \ref{dp} (c), we have $$ F_rK_v = K_v\!\left(\zeta_{p^r}, \{ q_{ij}^{1/{p^r}} \}_{1 \leq i,j \leq g} \right).$$ Hence $F_rK_v$ is a Kummer extension over $K_v(\zeta_{p^r})$. In particular, it is abelian. \end{proof} Theorem \ref{thA} $(b)$ follows from the following \begin{theorem}\label{unrAb} If $\cA_v^0$ is a split torus for every place $v$ of $K$ lying above $p$, then $F_r(K\bQ^{\ab})^{\ab}$ is an unramified Galois extension of $(K\bQ^{\ab})^{\ab}$. \end{theorem} \begin{proof} It suffices to show that $F_rK_v\bQ^{\ab}$ is an abelian extension of $K_v\bQ^{\ab}$ for the places $v$ of $K$ prime to $p$ such that $A$ has bad reduction at $v$, using Lemma \ref{ihara} again. Let $v$ be a place of $K$ prime to $p$ such that $A$ has bad reduction at $v$, and $T_p(A)$ the $p$-adic Tate module associated to $A$. We have the following representation $$ \rho_p : G_{K_v} \lr \Aut(T_p(A)) \simeq GL_{2g}(\bZ_p). $$ Since $A$ has semistable reduction at $v$, we see that the image of the absolute inertia subgroup of $G_{K_v}$ by $\rho_p$ is isomorphic to $\bZ_p(1)$ by Grothendieck \cite{SGA7}, Proposition\;3.5, Corollary\;3.5.2. Hence $F_rK_v^{\nr}$ is abelian over $K_v^{\nr}$, where $K_v^{\nr}$ is the maximal unramified extension of $K_v$. Since $K_v^{\nr}$ is cyclotomic over $K_v$, we see that $F_rK_v\bQ^{\ab}$ is an abelian extension of $K_v\bQ^{\ab}$. \end{proof} \medskip %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% Example 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Examples} In this section, we shall give examples of abelian varieties and algebraic number fields which satisfy the conditions of Theorem \ref{thA}. \subsection{The case of dim $A$ = 1} Let $E$ be an elliptic curve over an algebraic number field $k$ with modular invariant $j = j(E)$ and $p \geq 5$ a prime number. Suppose that $v(j)< 0$ for any places $v$ of $k$ lying above $p$. Then the pair $(E, p)$ satisfies the condition (A1) of Theorem \ref{thA}. Further there exists a finite Galois extension $K$ of $k$ such that the elliptic curve $E \otimes_k K$ over $K$ satisfies the condition (A0) of Theorem \ref{thA} by the semistable reduction theorem (\Cf \cite{SGA7}, Proposition 3.6) and the assumption of Theorem \ref{thA} (a) by \textit{Tate's theory} (\Cf Lang \cite{sL70}, \S 15). In particular, if $k = \bQ$ then we can see a field $K$ explicitly such that $F\Kzi^{\ab}$ is an unramified over $\Kzi^{\ab}$. We shall prove that we can take a solvable extension of $\bQ$ as such a field $K$, and if $K/\bQ$ is solvable and $E$ satisfies an assumption, then we can also determine the Galois group of $F\Kzi^{\ab}$ over $\Kzi^{\ab}$ which is unramified Galois extension of Theorem \ref{thA}. \begin{proposition}\label{ex1} $(a)$ Let $E$ be an elliptic curve over $\bQ$, $p \geq 5$ a prime number. Suppose that the modular invariant $j(E)$ is non-integral. Let $r$ be a positive integer and $F_r$ the field obtained by adjoining to $\bQ$ the coordinates of $p^r$-division points on $E$. Then there exists a finite solvable extension $K$ over $\bQ$ depending only on $E$ such that $F_r\Kz^{\ab}$ is an unramified Galois extension of $\Kz^{\ab}$.\\ $(b)$ Let $F_1$ be the field obtained by adjoining to $\bQ$ the coordinates of $p$-division points on $E$. If $\Gal(F_1/\bQ)$ is isomorphic to $GL_2(\bF_p)$, then $$ \Gal(F_r\Kz^{\ab}/\Kz^{\ab}) \simeq SL_2(\bZ/p^r\bZ). $$ \end{proposition} Note that it is well-known that $\Gal(F_1/\bQ) \simeq GL_2(\bF_p)$ for almost all primes $p$ in this case by Serre \cite{jS72}, Th\'eor\`eme 2. \begin{proof}\:$(a)$\: First we shall find a finite extension $K$ of $\bQ$ such that $E \otimes_{\bQ} K$ has everywhere semistable reduction over $\cO_{K}$. Suppose that $E$ is defined by the following equation: $$ E \;:\; y^2 = (x - e_1)(x - e_2)(x - e_3), \qquad e_1, e_2, e_3 \in \Bar{\bQ}, $$ where $e_1, e_2, e_3$ are distinct. Then the change of coordinates $$ x = (e_2 - e_1)x' + e_1, \; y = (e_2 - e_1)^{3/2}y' $$ gives the following form: $$ E_{\lmd} \;:\; (y')^2 = x'(x' - 1)(x' - \lmd), \qquad \lmd = (e_3 - e_1)/(e_2 - e_1) \in \Bar{\bQ}. $$ Let $L$ be the Galois closure of $\bQ(\sqrt{e_2 - e_1},e_1)$. Then $L$ is solvable over $\bQ$. Since $E$ is isomorphic to $E_{\lmd}$ over $\bQ(\sqrt{e_2 - e_1},e_1)$, so is over $L$. Hence we may regard $E$ as defined over $L$ and given by the following equation: $$ E \;:\; y^2 = x(x - 1)(x - \lmd). $$ Note that $\lmd \in L.$ Now we shall use the notation of Silverman \cite{AEC}. Then we see the quantity $c_4 = 16(\lmd^2 - \lmd +1)$ and the discriminant $\dlt = 16\lmd^2(\lmd - 1)^2$ by simple calculations. Now we put $S = \{ v : \text{a place of $L$}\,|\, v(\lmd) < 0 \}$. If $S = \varnothing$, then $E$ has everywhere semistable reduction over $\cO_L$. In this case we can take $L$ as the field $K$ of the statement of $(a)$. Next suppose that $S \ne \varnothing$. Let $H$ be the Hilbert class field of $L$ and $\fp_v$ a prime divisor of $L$ corresponding to $v \in S$. Since the extensions of all prime divisors of $L$ to $H$ are principal, there exists an element $\pi_v$ in $\cO_H$ such that $\fp_v \cO_H = (\pi_v)$. Further, for any $v$ in $S$, there exist positive integers $r_v$ such that $v(\lmd \prod_{v \in S} \pi_v^{r_v}) = 0$ for any $v \in S$. We put $\pi = \prod_{v \in S} \pi_v^{r_v}$ and $u = \lmd\pi$. Then the change of coordinates $$ x = \pi^{-1}x', \; y = \pi^{-3/2}y' $$ gives the following form: $$ E' \;:\; (y')^2 = x'(x' - u)(x' - \pi). $$ Let $w$ be the extension of $v \in S$ to $H(\sqrt{\pi})$. Then we see that the equation of $E'$ is minimal and $E'$ has multiplicative reduction over $H(\sqrt{\pi})_w$ by Silverman \cite{AEC}, Ch.VII, Remark 1.1, Proposition 3.6. Since $E$ is isomorphic to $E'$ over $H(\sqrt{\pi})$, $E$ also has multiplicative reduction over $H(\sqrt{\pi})_w$. In this case we can take $H(\sqrt{\pi})$ as the field $K$ of the statement of $(a)$. Thus we see that there exists the solvable extension $K$ of $\bQ$ such that $E \otimes_{\bQ} K$ has everywhere semistable reduction over $\cO_{K}$. Next we shall show that $F_r\Kz^{\ab}$ is an unramified Galois extension of $\Kz^{\ab}$. Let $v$ be a place of $K$ at which $E \otimes_K K_v$ has bad reduction. By Tate's theory (\Cf Serre \cite{jS68}, Appendix, A.1.1), over the unramified quadratic extension $K_v'$ of $K_v$, we see that $E \otimes_K K_v$ is isomorphic to a Tate curve $E(q)$ over $K_v$ with modular invariant $j(E)$. Let $F(q)_r$ be the field obtained by adjoining to $K_v$ the coordinates of $p^r$-division points on $E(q)$. Then $F(q)_r$ is equal to $K_v(\zeta_{p^r}, q^{1/{p^r}})$, where $q$ is the element of $K_v^{\times}$ which corresponds to $j(E)$ by Tate's theory (\Cf Lang \cite{sL70}, Ch.15, \S 2). Hence $F(q)_rK_v$ is a Kummer extension over $K_v(\zeta_{p^r})$, in particular, it is abelian. Then $F_rK_v'$ is also an abelian extension of $K_v(\zeta_{p^r})$. Hence we see that $F_rK_v$ is also abelian over $K_v(\zeta_{p^r})$. $(b)$ By the assumption, we have $$ \Gal(F_1/\bQ(\zeta_p)) \cong SL_2 (\bF_p). $$ Since $SL_2 (\bF_p)$ has no non-trivial abelian quotients, we see that $$ \Kz^{\ab} \cap F_1 = \bQ(\zeta_p). $$ Hence we see $$ \Gal(F_1\Kz^{\ab}/\Kz^{\ab}) \cong SL_2 (\bF_p). $$ The proposition follows from the following \begin{lemma}[Serre \cite{jS68}, Ch.IV, 3.4, Lemma\;3]\label{srr} Let $X$ be a closed subgroup of $SL_2 (\bZ_p)$ whose image in $SL_2 (\bF_p)$ is $SL_2 (\bF_p)$. If $p \geq 5$, then $X = SL_2 (\bZ_p)$. \end{lemma} \end{proof} >From Proposition \ref{ex1}, we obtain the following \begin{corollary}\label{cor-ex1} $(a)$ Let $E$ and $p$ be as above, and $F$ the field obtained by adjoining to $\bQ$ the coordinates of all $p$-power division points on $E$. Then there exists a finite solvable extension $K$ over $\bQ$ depending only on $E$ such that $F\Kzi^{\ab}$ is an unramified Galois extension of $\Kzi^{\ab}$.\\ $(b)$ Let $F_1$ be the field obtained by adjoining to $\bQ$ the coordinates of $p$-division points on $E$. If $\Gal(F_1/\bQ)$ is isomorphic to $GL_2(\bF_p)$, then $$ \Gal(F\Kzi^{\ab}/\Kzi^{\ab}) \simeq SL_2(\bZ_p). $$ \end{corollary} \medskip %%%%%%%%%%%%%% Example 2 jacobian variety of level $p$ %%%%%%%%%%%%%%%%%%%%%% \subsection{The jacobian variety of the modular curve of level $p$} For any integer $m$, and a subgroup $H$ of $GL_2(\bZ/m\bZ)$, we have an algebraic stack proper over $\Spec\bZ$, which may be interpreted over $\Spec\bZ[1/m]$ as the fine moduli stack classifying \textit{generalized} elliptic curves with level $H$-structure (\Cf \cite{DL73}, IV, (3.1)). Its associated \textit{coarse} moduli stack (\Cf \cite{DL73}, I, (8.1)) may be denoted by $M_H$. If $H= \Gamma_0(N) = \left\{ \left( \begin{smallmatrix} a& b \\ c& d \\ \end{smallmatrix} \right) \in SL_2(\bZ) \;|\; c \equiv 0 \pmod N \right\}$, then we write $M_H = M_0(N)$. \begin{lemma}[Mazur \cite{bM77}, Appendix, Theorem (A.1)]\label{jacob} Let $p \geq 5$ be a prime number, $J$ the jacobian variety of dimension $g$ of the modular curve $M_0(p)$, $\cJ$ the N\'eron model of $J$ at $p$. The identity component $\cJ_p^0$ of the fibre at $\bF_p$ of $\cJ$ is a group scheme of multiplicative type, in other words, if we consider it over $\Bar{\bF}_p$ then $\cJ_p^0 \otimes_{\bF_p}\Bar{\bF}_p \simeq \Gm^g$. \end{lemma} Note that $J$ has good reduction outside $p$. From this lemma, we see that there exists a finite extension $k$ over $\bF_p$ such that $\cJ_p^0 \otimes_{\bF_p} \! k \simeq \Gm^g$. Hence there exists an finite Galois extension $K$ over $\bQ$ such that the abelian variety $J$ over $K$ satisfies the condition (A0), (A1) and the assumption of Theorem \ref{thA} $(a)$.\\ Now we shall describe $k$ and $K$ explicitly. We put $X(\cJ_p^0) = \Hom_{\Bar{\bF}_p}(\cJ_p^0, \Gm)$. We see that $X(\cJ_p^0) = \Hom(\Gm^g, \Gm) \simeq \bZ^g$, since we have $\cJ_p^0 \otimes_{\bF_p}\Bar{\bF}_p \simeq \Gm^g$ by Lemma \ref{jacob}. Further $G_{\bF_p}$ acts on $\Gm$. Hence we obtain a continuous representation $$ \rho \;:\; G_{\bF_p} \lr \Aut(X(\cJ_p^0)) \simeq GL_{2g}(\bZ). $$ Note that the image of $\rho$ is finite. Hence we see that there exists a positive integer $N$ such that $\rho$ factors through a finite quotient $\Gal(\bF_{p^N}/\bF_p) \simeq \bZ/N\bZ$. Thus $\cJ_p^0 \otimes_{\bF_p}\bF_{p^N}$ is a split torus. We can take as $k$ the field $\bF_{p^N} = \bF_p(\zeta_{p^N - 1})$. Further we may take as $K$ a field such that the residue field of $v$ contains $\bF_{p^N}$ for every place $v$ lying above $p$, for example, $\bQ(\zeta_{p^N - 1})$. Thus we obtain the following \begin{proposition} Let $J$ and $p$ be as above, and $F$ the field obtained by adjoining to $\bQ$ the coordinates of all $p$-power division points on $J$. Then there exists a finite solvable extension $K$ over $\bQ$ such that $F\Kzi^{\ab}$ is an unramified Galois extension of $\Kzi^{\ab}$. \end{proposition} \bigskip %%%%%%%%%%%%%%%%%% Division Points of Elliptic Curves %%%%%%%%%%%%%%%%%%%%%%% \section{Division Points of Elliptic Curves} In this section, we shall give the proof of Theorem \ref{thE}. Throughout this section, let $k$ be the field obtained by adjoining to $\bQ$ the coodinates of all $p$-power roots of unity, $k_r$ the field obtained by adjoining to $\bQ$ the coodinates of $p^r$-th roots of unity. \subsection{Unramifiedness over $\Qzi^{\ab}$} In this subsection, by using division points of elliptic curves over $\bQ$, we construct unramified Galois extensions over $k^{\ab}$. Let $E$ be an elliptic curve over $\bQ$, $p \geq 5$ a prime number. Suppose that the pair $(E, p)$ satisfies the following conditions: \begin{quote} \begin{itemize} \item[(E0)] $E$ has everywhere semistable reduction over $\bZ$. \item[(E1)] $E$ has bad reduction at $p$. \end{itemize} \end{quote} \begin{proposition}\label{unrE} Let $E$ and $p$ be as above. Let $r$ be a positive integer and $F_r$ the field obtained by adjoining to $\bQ$ the coordinates of $p^r$-th power division points on $E$.Then $F_rk_r^{\ab}$ is an unramified Galois extension of $k_r^{\ab}$. \end{proposition} \begin{proof} We can prove this proposition in a way similar to the proof of Proposition \ref{ex1}. In this case, we can take $\bQ$ itself as the field $K$ in 2.1. \end{proof} >From Proposition \ref{unrE}, we obtain the following \begin{corollary}\label{corunr} Let $E,p,k$ and $F$ be as above. Let $F$ be the field obtained by adjoining to $\bQ$ the coordinates of all $p$-power division points on $E$. Then $Fk^{\ab}$ is an unramified Galois extension of $k^{\ab}$. \end{corollary} \medskip %%%%%%%%%%%%%%%%%%%% Determination of the Galois group %%%%%%%%%%%%%%%% \subsection{Determination of the Galois group} Let $E$ be an elliptic curve over $\bQ$, $p \geq 5$ a prime number. Suppose that the pair $(E, p)$ satisfies the condisions (E0), (E1) of 3.1 and the following conditions: \begin{quote} \begin{itemize} \item[(E2)] $E$ has three rational $2$-division points. \item[(E3)] $p$ does not divide the valuation at $p$ of the modular invariant $j(E)$ of $E$. \end{itemize} \end{quote} In this subsection, for such $E$ and $p$, we shall prove that the unramified extention $Fk^{\ab}$ over $k^{\ab}$ which we constructed in Corollary \ref{corunr} has $SL_2(\bZ_p)$ as the Galois group over $k^{\ab}$. By Corollary \ref{cor-ex1} $(b)$, it suffices to prove the following \begin{proposition}\label{p-gal} Let $E$ and $p$ be as above, and $F_1$ the field obtained by adjoining to $\bQ$ the coordinates of the $p$-division points on $E$. Then $$ \Gal(F_1/\bQ) \cong GL_2 (\bF_p). $$ \end{proposition} \begin{proof} In general, let $E$ be an arbitrary elliptic curve defined over $\bQ$. Let $l \geq 5$ be a prime number, and $E[l]$ the group of the $l$-division points of $E$. Then we obtain a representation $$ \rho_l\; : \; G_{\bQ} \longrightarrow \Aut(E[l]) \cong GL_2 (\bF_l). $$ Now we denote $G_l = \mathrm{Im}\rho_l$. To show this proposition, we shall verify the conditions of the following lemma (\Cf Serre \cite{jS68}, Ch.IV, 3.2, Lemma\;2). \begin{lemma}\label{gl2} If $G_l$ satisfies the next three conditions $(a), (b), (c),$ then $G_l$ is equal to $GL_2 (\bZ/l\bZ)$.\\ $(a)$\;$\mathrm{det}\;G_l = \bF_l^{\times}.$\\ $(b)$\; $G_l$ contains the element\; $ \begin{pmatrix} 1& 1 \\ 0& 1 \\ \end{pmatrix} $ \;with respect to a suitable basis of $E[l]$.\\ $(c)$\;$E[l]$ is irreducible as a $G_l$-module. \end{lemma} Apply this to our $E$ and $l = p$. The condition $(a)$ is satisfied for $l = p$ since $F_1$ contains $\zeta_p$. The condition $(b)$ is also satisfied for $l = p$, by the condition (E3) and Lemma\;1 of Serre \cite{jS68}, Ch.IV, 3.2. Hence we shall show that the condition $(c)$ is satisfied for $l = p$. Assume that this is not the case, \Ie, $E[p]$ is reducible as a $G_p$-module. Then $E$ contains a subgroup $X$ whose order is $p$ and which is stable under the action of $G_p$. Since $E$ has everywhere semistable reduction and multiplicative reduction at $p$, by Serre \cite{jS72}, p.307, the action of $G_{\bQ})$ on $X$ is described by two characters $\chi'$ and $\chi''$ as follows: $\chi' = 1,\;\chi'' = \chi$ or $\chi' = \chi,\; \chi'' = 1$, where $\chi : G_{\bQ} \lr \bF_p^{\times}$ is the cyclotomic character. In the first case, $E$ has a $\bQ$-rational point of order $p$. Since $E$ has three rational $2$-division points, $$ |\, E(\bQ)_{\tors} \,| \geq 4p \geq 20. $$ Here $E(\bQ)_{\tors}$ is the torsion subgroup of the group $E(\bQ)$ of $\bQ$-rational points of $E$. But this is impossible by \textit{Mazur's Theorem} (\Cf Mazur \cite{bM77}, Theorem 8). In the second case, the quotient curve $E' = E/X$ has $\bQ$-rational points of order $p$. Further, since $p \ne 2$, the curve $E'$ also has three rational $2$-division points. Hence we can apply the argument in the first case to $E'$. This completes the proof of Proposition \ref{p-gal} \end{proof} \medskip %%%%%%%%%%%%%%%%%%%%%%%%% Infinite existence %%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Infinite existence} In this subsection, we shall give the proof of Theorem \ref{thE}. Let $p$, $p_i \geq 5$ $(i = 1,2, \cdots)$ be distinct prime numbers, and $E^{(i)}$ the elliptic curve over $\bQ$ defined by the following equation $$ E^{(i)} : y^2 = x(x - p)(x + p_i). $$ By Serre \cite{jS87}, 4.1, the elliptic curve $E^{(i)}$ has everywhere semistable reduction over $\bZ$, and multiplicative reduction at $p$, $p_i$ and the prime divisors of $p+p_i$. Let $j(E^{(i)})$ be the modular invariant of $E^{(i)}$. Then, by simple calculations, we see that $$ j(E^{(i)}) = 2^8\frac{(p^2 + pp_i + p_i^2)^3}{p^2p_i^2(p+p_i)^2}. $$ Hence $E^{(i)}$ satisfies the condition (E3). Thus we obtain an infinite family of elliptic curves over $\bQ$ which satisfies the conditions (E0), (E1), (E2) and (E3). Let $F^{(i)}$ be the field obtained by adjoining to $\bQ$ the coordinates of all $p$-power division points on $E^{(i)}$. Then, by Proposition \ref{unrE} and \ref{p-gal}, we see that the extensions $F^{(i)}k^{\ab}$ over $k^{\ab}$ are unramified Galois extensions having $SL_2(\bZ_p)$ as the Galois group over $k^{\ab}$ for any $i \geq 1$. Let $F_1^{(i)}$ be the field obtained by adjoining to $\bQ$ the coordinates of all $p$-division points on $E^{(i)}$. \begin{lemma}\label{pindep} Let $p$, $p_1 \geq 5$ be distinct prime numbers, and $E^{(1)}$ as above. If we take a prime number $p_2$ such that $E^{(2)}$ has good reduction at $p_1$, then $$ F_1^{(1)} \cap F_1^{(2)} = \bQ(\zeta_p). $$ \end{lemma} Note that $E^{(2)}$ has good reduction at $p_1$ if and only if $p_2 \not\equiv - p \pmod {p_1}$. By \textit{Dirichlet's theorem}, there exist infinitely many prime numbers satisfying this condition. \begin{proof} By the assumption and the criterion of N\'eron-Ogg-Shafarevich (\cite{ST68}, Theorem 1), we see that $p_1$ is unramified in $F_1^{(2)}$. Next we consider the ramification of $p_1$ in $F_1^{(1)}$. $E^{(1)} \otimes_{\bQ} \bQ_{p_1}$ is isomorphic to a Tate curve over $\bQ_{p_1}$ with modular invariant $j(E^{(1)})$ over the unramified quadratic extension $L$ of $\bQ_{p_1}$. Then we have $$ F_1^{(1)}L \simeq L(\zeta_p, q^{1/p}), $$ where $q$ is the element of $L^{\times}$ which corresponds to $j(E^{(1)})$ by Tate's theory. Since $p_1$ is unramified in $L$, if $v_{p_1}$ is an extension of the normalized $p_1$-adic valuation of $\bQ$ to $L$, then we see that $$ v_{p_1}(q) = - v_{p_1}(j(E^{(1)})) = - \,\ord_{p_1}(j(E^{(1)})) = 2. $$ Further since $p_1$ is unramified in $\bQ(\zeta_p)$, if $v'_{p_1}$ is an extension of $v_{p_1}$ to $\bQ(\zeta_p)$, then $v'_{p_1}(q) = 2$. Let $w_{p_1}$ be an extension of $v'_{p_1}$ to $F_1^{(1)}$ and $e_{p_1}$ the ramification index of $w_{p_1}$ over $v'_{p_1}$. Then $$ w_{p_1}(q^{1/p}) = \frac{1}{p}\,w_{p_1}(q) = \frac{1}{p}\,e_{p_1} v'_{p_1}(q) = \frac{2}{p}\,e_{p_1}. $$ Since $p \geq 5$ and $w_{p_1}(q^{1/p})$ is in $\bZ$, we see that $e_{p_1} \gneq 2$. In particular, $p_1$ is ramified in $F_1^{(1)}$. So we obtain $F_1^{(1)}\ne F_1^{(2)}$. Let ${F'}_1^{(i)}$ be the subextension of $F_1^{(i)}$ of $\bQ(\zeta_p)$ corresponding to the normal subgroup $\{ \pm 1 \}$ of $SL_2(\bF_p) \;(i = 1, 2)$. Suppose that ${F'}_1^{(1)} = {F'}_1^{(2)}$. Then the prime of ${F'}_1^{(1)}$ lying above $p_1$ must be ramified in $F_1^{(1)}$ and the ramification index is equal to $e_{p_1}$ because $p_1$ is unramified in ${F'}_1^{(1)}$ and ramified in $F_1^{(1)}$. However $$ [F_1^{(1)} : {F'}_1^{(1)}] = 2 \lneq e_{p_1} \leq [F_1^{(1)} : {F'}_1^{(1)}]. $$ It is impossible. Hence we see that ${F'}_1^{(1)} \ne {F'}_1^{(2)}$. Since $$ \Gal({F'}_1^{(i)}/\bQ(\zeta_p)) \simeq PSL_2(\bF_p), $$ which is a simple group, we have $$ F_1^{(1)} \cap F_1^{(2)} = \bQ(\zeta_p). $$ \end{proof} \begin{proposition}\label{indep} Let $k$ and $F^{(i)}$ be as in Lemma $\ref{pindep}$. Then $$ F^{(1)}k^{\ab} \cap F^{(2)}k^{\ab} = k^{\ab}. $$ \end{proposition} \begin{proof} By Lemma \ref{pindep} and Proposition \ref{p-gal}, we obtain \begin{align*} \Gal(F_1^{(1)}F_1^{(2)}/\bQ(\zeta_p)) &\simeq \Gal(F_1^{(1)}/\bQ(\zeta_p)) \times \Gal(F_1^{(2)}/\bQ(\zeta_p)) \\ &\simeq SL_2(\bF_p) \times SL_2(\bF_p). \end{align*} Since there are no non-trivial abelian quotients of $SL_2(\bF_p) \times SL_2(\bF_p)$, we see $$ \Gal(F_1^{(1)}F_1^{(2)}k^{\ab}/k^{\ab}) \simeq SL_2(\bF_p) \times SL_2(\bF_p). $$ By Lemma \ref{srr}, we obtain $$ F^{(1)}k^{\ab} \cap F^{(2)}k^{\ab} = k^{\ab}. $$ \end{proof} Now we can choose an infinite family of elliptic curves over $\bQ$ inductively. Suppose that we have taken $p_1, \cdots, p_n$ such that $F^{(1)}k^{\ab}, \cdots, F^{(n)}k^{\ab}$ are linearly independent over $k^{\ab}$ having $SL_2(\bZ_p)$ as the Galois group over $k^{\ab}$. Let $p_{n+1}$ be a prime number different from $p_1, \cdots, p_n$ such that $p_{n+1} \not\equiv - p \pmod {p_i}$ and $p_i \not\equiv - p \pmod {p_{n+1}}$ for all $1 \leq i \leq n$. By using Dirichlet's theorem again, we can take such a prime number. Then we see that $E^{(n+1)}$ has good reduction at $p_{i}\;(1 \leq i \leq n)$ from the first condition for $p_{n+1}$. In a way similar to the above argument, we obtain $$ F^{(n+1)}k^{\ab} \cap F^{(i)}k^{\ab} = k^{\ab}. $$ Further we see that $F_1^{(n+1)} \cap \:F_1^{(1)}F_1^{(2)} \cdots F_1^{(n)} = \bQ(\zeta_p)$. Indeed, assume that $M = F_1^{(n+1)} \cap \:F_1^{(1)} F_1^{(2)} \cdots F_1^{(n)}$ is a non-trivial extension of $\bQ(\zeta_p)$. Then the group $\Gal(F_1^{(n+1)}/M)$ must be isomorphic to the group $\{ \pm 1 \}$ or $\{ 1 \}$ because $\Gal(F_1^{(n+1)}/\bQ(\zeta_p)) \simeq SL_2(\bF_p)$. By the same argument as in the proof of Lemma \ref{pindep}, the prime of $\bQ(\zeta_p)$ lying above $p_{n+1}$ is ramified in $M$. However $p_{n+1}$ is unramified in $F_1^{(1)}F_1^{(2)} \cdots F_1^{(n)}$ by the latter condition for $p_{n+1}$. Base cases are impossible. Hence we obtain $$ F^{(n+1)}k^{\ab} \cap \:F^{(1)}F^{(2)} \cdots F^{(n)}k^{\ab} = k^{\ab}. $$ This completes the proof of Theorem \ref{thE}. \begin{remark} In Theorem \ref{thE} we have constructed unramified Galois extensions over algebraic number fields of infinite degree having $SL_2(\bZ_p)$ as the Galois group. Then can we construct unramified Galois extensions over \textit{finite} algebraic number fields having $SL_2(\bZ_p)$ as the Galois group ? If the \textit{Fontaine-Mazur conjecture} (\Cf Fontaine-Mazur \cite{FM93}, Conjecture 5a) is true, then we cannot do this. Now we try to construct unramified Galois extensions having $SL_2(\bZ/p^r\bZ)$ as the Galois group over \textit{smaller} algebraic number fields. In the next section, we shall give an example of this problem. \end{remark} \bigskip %%%%%%%%%%%%%%%%%%%%%%%%% Supplement %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Supplement} In this section, we shall construct unramified Galois extension over a finite extension of $\bQ$ having $SL_2(\bZ/p^r\bZ)$ as the Galois group by using a family of elliptic curves of conductor $p$ over $\bQ$. Let $p$ be a rational prime such that $p =u^2 +64$ for some rational integer $u$, and $E$ an elliptic curve of conductor $p$ over $\bQ$ with a non-trivial rational 2-division point. By Theorem\;2 of Setzer \cite{bS75},\;if $p$ is of the form $u^2 +64$, where $u$ is a rational integer, and the sign of $u$ is chosen so that $u \equiv 1 \; \pmod{4}$, then there exist, up to isomorphism, just two such curves. The two curves are connected by a 2-isogeny, and one of them is given by $$ y^2 = x^3 + ux^2 - 16x. $$ Now we assume that $E$ is given by this equation. The modurlar invariant $j(E)$ is $p^{-1}(p-16)^3.$ Then we see that $E$ has no complex multiplication. \begin{proposition}\label{p-cond} Let $E$ and $p$ be as above, and $F_r$ the field obtained by adjoining to $\bQ$ the coordinates of $p^r$-th power division points on $E$. Then there exists a finite solvable extension $K_r$ over $\bQ$ of degree at most $2\varphi(p^r)p^r$ such that $F_rK_r$ is an unramified Galois extension of $K_r$ having $SL_2 (\bZ/p^r\bZ)$ as the Galois group over $K_r$, where $\varphi$ is the Euler funtion.\\ \end{proposition} \begin{proof} By the assumption, the extension $F_r$ is unramified outside $p$ over $\bQ$. Since $E$ has multiplicative reductiuon at $p$, we have $F_rL = L(\zeta_{p^r}, q^{1/{p^r}})$ in a way similar to the proof of Proposition \ref{ex1} (a), where $L$ is the unramified quadratic extension of $\bQ_p$ and $q$ is the element of $L^{\times}$. Then there exists a finite extension $K_r$ of $\bQ$ such that $K_r\bQ_p = L(\zeta_{p^r}, q^{1/{p^r}})$. In particular, $F_rK_r\bQ_p$ is unramified over $K_r\bQ_p$. Hence $F_rK_r$ is unramified over $K_r$. Note that, if we take $q_0 \in \bQ$ such that $q_0$ is close to $q$ enough, we may take a quadratic extension of $\bQ(\zeta_{p^r}, q_0^{1/{p^r}})$ as the field $K_r$ of the statement of this proposition. Next, to prove that $\Gal(F_rK_r/K_r)$ is isomorphic to $SL_2 (\bZ/p^r\bZ)$, we shall verify the conditions $(a)$,$(b)$,$(c)$ of Lemma \ref{gl2}. The condition $(a)$ is equivalent to the fact that $F_1$ contains $\zeta_p$. The condition $(b)$ is also satisfied by the assumption and Lemma\;1 of Serre \cite{jS68}, Ch.IV, 3.2. Next we shall show that the condition $(c)$ is satisfied for $p$. Assume $E[p]$ is reducible as $G_p$-module. Then there is a one-dimensional subspace $X$ of $E[p]$ which is a cyclic group of order $p$ and stable under the action of $G_p$. Now we put $E'$ = $E/X$, then we have an exact sequence \begin{equation*} \begin{CD} 0 @>>> X @>>> E @>\text{$\lambda$}>> E' @>>> 0. \end{CD} \end{equation*} Here $\lambda$ is a separable isogeny of degree $p$. $E'$ is not isomorphic to $E$, because $E$ has no complex multiplication. $E'$ has a non-trivial rational 2-division point, and we see that $E'$ has conductor $p$ by Serre-Tate \cite{ST68}, \S 1, Corollary\;1. However we know that there exist just two such curves up to isomorphism. Hence we see that they are $E$ and $E'$. Since they are connected by a 2-isogeny $\lambda'$, we have \begin{equation*} \begin{CD} E @>\text{$\lambda$}>> E' @>\text{$\lambda'$}>> E. \end{CD} \end{equation*} Then $\lambda' \circ \lambda$ is an endomorphism of $E$ of degree $2p$. But this is impossible since $E$ has no complex multiplication. Hence we obtain $$ \Gal(F_1/\bQ(\zeta_p)) \cong SL_2 (\bF_p). $$ Since $K_r \cap F_1 = \bQ(\zeta_p)$ for any positive integer $r$, we see $$ \Gal(F_1 K_r/K_r) \cong SL_2 (\bF_p). $$ By Lemma \ref{srr}, we obtain $$ \Gal(F_rK_r/K_r) \cong SL_2 (\bZ/p^r\bZ). $$ \end{proof} \bigskip %%%%%%%%%%%%%%%%%%%%%%%% Bibliography %%%%%%%%%%%%%%%%%%%%%%% \begin{thebibliography}{99} \bibitem{mA85} Asada,\;M., \textit{On unramified Galois extensions over maximum abelian extensions of algebraic number fields}, Math. Ann. \textbf{270} (1985), 477-487. \bibitem{mA85'} Asada,\;M., \textit{Construction of certain non-solvable unramified Galois extensions over the total cyclotomic field}, J. Fac. Sci. Univ. Tokyo. sect.IA, Math. \textbf{32} (1985), 397-415. \bibitem{aB81} Brumer,\;A., \textit{The class group of all cyclotomic integers}, J. Pure Appl. 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Grothendieck, ed., Lecture Notes in Math. \textbf{288}, Springer-Verlag, 1972, pp.313-523. \bibitem{kH90} Horie,\;K., \textit{CM-fields with all roots of unity}, Compositio Math. \textbf{74} (1990), 1-14. \bibitem{mK99} Kurihara,\;M., \textit{On the ideal class groups of the maximal real subfields of number fields with all roots of unity}, J. Eur. Math. Soc. \textbf{1} (1999), 35-49. \bibitem{sL70} Lang,\;S., \textit{Elliptic functions}, Addison Wesley, Reading. Mass, 1970. \bibitem{bM77} Mazur,\;B., \textit{Modular curves and the Eisenstein ideal}, Publ. Math. I.H.E.S. \textbf{47} (1977), 33-186. \bibitem{jS68} Serre,\;J.-P., \textit{Abelian $\mathit{l}$-adic representations and elliptic curves}, Benjamin, NewYork, 1968. \bibitem{jS72} Serre,\;J.-P., \textit{ Propri\'et\'es galoisiennes des points d'ordre fini des courbes elliptiques}, Invent. Math. \textbf{15} (1972), 259-331. \bibitem{jS87} Serre,\;J.-P., \textit{Sur les repr\'esentation modulaires de degr\'e $2$ de $\Gal(\overline{\bQ}/\bQ)$}, Duke Math. J. \textbf{54} (1987), 179-230. \bibitem{ST68} Serre,\;J.-P. and Tate,\;J., \textit{Good reduction of abelian varieties}, Ann. of Math. \textbf{88} (1968), 492-517. \bibitem{bS75} Setzer,\;B., \textit{Elliptic curves of prime conductor}, J. London Math. Soc.(2), \textbf{10} (1975), 367-378. \bibitem{AEC} Silverman,\;J.H., \textit{The arithmetic of elliptic curves}, Graduate Texts in Math. \textbf{106}, Springer-Verlag, 1986. \bibitem{kU82} Uchida,\;K, \textit{Galois groups of unramified solvable extensions}, Tohoku Math. J. (2) \textbf{34} (1982), 311-317. \end{thebibliography} \end{document} From emerton@math.uchicago.edu Fri Sep 22 18:25:55 2000 Received: from giambelli.uchicago.edu (giambelli.uchicago.edu [128.135.72.176]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id e8MMQ1B20278 for ; Fri, 22 Sep 2000 18:26:01 -0400 (EDT) Received: (from emerton@localhost) by giambelli.uchicago.edu (8.9.3+Sun/8.9.3) id RAA01607 for was@math.harvard.edu; Fri, 22 Sep 2000 17:25:55 -0500 (CDT) Date: Fri, 22 Sep 2000 17:25:55 -0500 (CDT) Message-Id: <200009222225.RAA01607@giambelli.uchicago.edu> From: emerton@math.uchicago.edu To: was@math.harvard.edu Subject: refined Eisenstein conjecture Content-Length: 316 Status: RO X-Status: A Hi William, I think that the methods in the note on optimal quotients that I sent you actually suffice to prove your entire refined Eisenstein conjecture. I have almost finished the write-up, and will send it to you when it's done, if you're interested. I hope things are going well in Cambridge, Regards, Matt From emerton@math.uchicago.edu Tue Sep 26 15:14:59 2000 Received: from giambelli.uchicago.edu (giambelli.uchicago.edu [128.135.72.176]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id e8QJF5B10745 for ; Tue, 26 Sep 2000 15:15:05 -0400 (EDT) Received: (from emerton@localhost) by giambelli.uchicago.edu (8.9.3+Sun/8.9.3) id OAA01080 for was@math.harvard.edu; Tue, 26 Sep 2000 14:14:59 -0500 (CDT) Date: Tue, 26 Sep 2000 14:14:59 -0500 (CDT) Message-Id: <200009261914.OAA01080@giambelli.uchicago.edu> From: emerton@math.uchicago.edu To: was@math.harvard.edu Subject: refined Eisenstein conjecture Content-Length: 921 Status: RO X-Status: A Hi William, I have finished writing up the proof of your conjecture. I've tried to eliminate all the typos and spurious arguments, but I'm sure there are plenty still in there, for which I'll apologize in advance. The argument is slightly intricate, but basically elementary (once you grant Mazur and Ribet's work), so it shouldn't be too hard to decide if it's correct or not. (I believe that it is, but I'm always willing to be corrected!) The following email is a uuencoded version of the dvi file of the write-up. The email following that one is a uuencoded version of the final draft of my weight two theta series paper (I'm sending it just because I refer to a part of it in the refined Eisenstein paper, and the labels have probably changed from what they were in the version I sent you a while ago). I'm looking forward to hearing any comments you have, if you get a chance to look at it. Regards, Matt From mwatkins@msri.org Tue Oct 17 06:54:09 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id e9HAsHu04731 for ; Tue, 17 Oct 2000 06:54:17 -0400 Received: from ep1.msri.org (ep1.msri.org [198.129.64.226]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id e9HAsAK15777 for ; Tue, 17 Oct 2000 06:54:11 -0400 (EDT) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep1.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 13lUNl-0003sQ-00; Tue, 17 Oct 2000 03:54:09 -0700 Received: from alice.msri.org [198.129.65.103] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 13lUNl-0001Sd-00; Tue, 17 Oct 2000 03:54:09 -0700 Received: from mwatkins by alice.msri.org with local (Exim 3.12 #1 (Debian)) id 13lUNl-00043W-00; Tue, 17 Oct 2000 03:54:09 -0700 Subject: Re: Stuff In-Reply-To: <00101523435416.19609@modular> from William A Stein at "Oct 15, 2000 11:43:55 pm" To: was@math.harvard.edu Date: Tue, 17 Oct 2000 03:54:09 -0700 (PDT) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: F Many of the rank 5 examples with reasonable ss-conductor have large non-semistable part like 81 or 49 dividing the conductor. Now upon twisting some of these examples by the non-semistable prime, the conductor remains the same while the 2-Selmer rank remains high (if I weren't so lazy, I'd compute their analytic ranks, or use Womack's PARI code to do so). Is this a known phenomenon? A specific example is [0,0,0,-4572,121185], which has 2^2 and 3^4 dividing the conductor. It has rank 5, whilst its twist by -3 has 2-Selmer rank 5. And the twist by -4 has 2-Selmer rank 3. Another example is [0,0,0,-5607,183690] whose twist by -3 has 2-Selmer rank of 4. And the -3 twist of [0,0,0,-8967,235690] has 2-Selmer rank 3, while the -7 twist has 2-Selmer rank 4. The Stephens curve [0,0,1,0,-3997] has rank 3 as does its twist by -3. The 5th twist of the rank five [0,0,0,125,47650] has 2-Selmer rank 4. The 5th twist of [0,-1,0,-5833,167437] has 2-Selmer rank 3. The -3rd twist of [0,0,0,-6012,188820] has 2-Selmer rank 4. The -7th twist of [0,-1,0,-5700,181476] has 2-Selmer rank 4. The 5th twist of [0,0,0,-9100,358900] has 2-Selmer rank 3. The -7th twist of [0,0,0,-3724,82516] has 2-Selmer rank 4. Yet the -3rd twist of [0,0,0,-9723,257722] has rank 1 (perhaps better to twist by 12, as the original is not minimal; gives 2-Selmer rank of 2) and the 5th twist of [0,-1,0,-8633,366237] has 2-Selmer rank only 2. Unfortunately, I can't compute the 2-Selmer rank of the -19th twist of [0,0,0,-8892,731025]. Could ask Nigel Smart (cf ANTS III paper) I suppose, or just compute the analytic rank, which shouldn't be too hard, since I already have the a_p computed. Plus the conductor isn't going up when the twist is made. Cremona's code is certainly inoptimal for the rank-finding here, since it is rarely able to find enough points. It usually finds one or zero rational points, depending on the parity of the 2-Selmer rank. BTW, I found out that the paper "Iwasawa theory for the symmetric square" by Coates and Schmidt (Crelle 375) has a bug in it; specifically, for elliptic curves with 2^8 dividing the conductor, there can be another type of exotic behaviour than those they describe. In yet another use of HECKE, I can give an explicit example; e.g. 256B does *not* have a quadratic twist at level 64 as their tables imply, but only one at level 128. The example that I'd like to understand is 768H, where the local factor at 2 in the symmetric square L-function has a different sign from what they claim; there the form has a quadratic twist at level 24, which is very different from their claim. I don't know enough Iwasawa theory to tell where they go wrong; the error might be in the Atkin-Li paper on newforms that they reference. === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Wed Oct 18 07:44:43 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id e9IBiou11548 for ; Wed, 18 Oct 2000 07:44:50 -0400 Received: from ep1.msri.org (ep1.msri.org [198.129.64.226]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id e9IBigK09745 for ; Wed, 18 Oct 2000 07:44:42 -0400 (EDT) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep1.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 13lreF-00076S-00; Wed, 18 Oct 2000 04:44:43 -0700 Received: from alice.msri.org [198.129.65.103] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 13lreF-0001HV-00; Wed, 18 Oct 2000 04:44:43 -0700 Received: from mwatkins by alice.msri.org with local (Exim 3.12 #1 (Debian)) id 13lreF-0004KX-00; Wed, 18 Oct 2000 04:44:43 -0700 Subject: More Stuff In-Reply-To: <00101523435416.19609@modular> from William A Stein at "Oct 15, 2000 11:43:55 pm" To: was@math.harvard.edu Date: Wed, 18 Oct 2000 04:44:43 -0700 (PDT) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: O I have stuck up a webpage with the beginnings of the modular degree data. http://www.msri.org/members/people/mwatkins/degphi.html I have figured out where Coates/Schmidt went wrong; they applied Atkin-Li incorrectly (though the latter has a quite confusing remark following Theorem 4.4 which makes an iff statement w/o recalling the conditions of Theorem 4.3). Cremona sent me a short note giving an idea of how to compute the Manin constant, but wasn't sure it would work well in practise. Fortunately, it does. The idea is to compute the analytic lattice periods w1 and w2 from the c_4 and c_6 invariants, and then compute periods of "random" elements of Gamma_0(N). These periods will be of the form a*w1+b*w2 for some (a,b) in Z^2. Once you have enough (a,b) pairs to generate Z^2, you know the Manin constant is one. (It took me a little while to convince myself that this was indeed the right direction --- Manin constant is [AnalyticLattice:AlgebraicLattice]). This worked rather well for [0,0,1,-79,342]. Here Period(b,c,d) computes the period of (a,b;cN,d) in d*sqrt(N) time. Period(1,5,7) -> (5,10)=A Period(1,5,8) -> (8,0) =B Period(1,6,11)-> (2,-1)=C So the Manin constant is one, since A-3B+10C and -6A+19B-61C are (1,0) and (0,1) respectively. Excerpt from email to Cremona regarding modular degrees etc.: --- These curves with the higher powers in the conductor have some interesting properties; take for instance [0,0,0,-3724,82516] of conductor 232044008 = 8.49.277.2137. This has rank 5, and the twists by -4, -8, and -56 have 2-Selmer rank 3, while the twists by -7, 8, 28, and 56 have 2-Selmer rank 4. The twists all have rank one or zero (computed analytically). Is this phenomenon known? I asked Elkies and Silverman, but neither knew of it. There might be some 2-visilibilty of Sha going on here, since the curves satisfy a 2-congruence. If that is the reason, then any twist by a supersingular prime should have a similar property. CM curves have a lot of supersingular primes, so this would be the place to look. I don't know of any CM curves with rank 5, but the Stephens rank 3 CM curve [0,0,1,0,-3997] of conductor 3^5 73^2 will do. The twists by -3 and -73 both have rank 3. Supersingular primes include 2,5,11,17,23,29,41,47,53,59,71,83,89. The twist by -8 has rank 2, whilst the twist by 8 has 2-Selmer rank 2, and the twist by -4 has 2-Selmer rank 3. The twist by 5 possesses 2-Selmer rank 2, as do the twist by -11 and 17 (though I killed the computation in these last two cases; it would probably be better to use the Djabri/Smart code for computing 2-Selmer ranks here). The -23rd and 29th twist both have rank two. Some mix-and-match techniques work also, though not always. The -15th, -20th, and 33rd twist have 2-Selmer rank 2, and the 12th has 2-Selmer rank 3, but the 40th and -40th twists (where no subtwist has 2-Selmer rank 3) each have a 2-Selmer rank of 1. Another issue is that of the power of two that divides the modular degree. If the conductor is highly divisible, the 2-divisiblity of the modular degree will go up. But Elkies and I noticed that all of the rank 5 *prime* conductor examples extant have 2^5 dividing the modular degree. And 2^4 divides it in the rank 4 examples, correspondingly 2^3 in the rank 3 cases. Is a result like this known? [Cremona didn't know much about either question.] === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Wed Oct 18 19:48:03 2000 Return-Path: Received: from ep1.msri.org (ep1.msri.org [198.129.64.226]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id e9INm8u12371 for ; Wed, 18 Oct 2000 19:48:08 -0400 Received: from rafiki.msri.org [198.129.65.127] (mail) by ep1.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 13m2wF-0000ND-00; Wed, 18 Oct 2000 16:48:03 -0700 Received: from alice.msri.org [198.129.65.103] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 13m2wF-0007wr-00; Wed, 18 Oct 2000 16:48:03 -0700 Received: from mwatkins by alice.msri.org with local (Exim 3.12 #1 (Debian)) id 13m2wF-0004RE-00; Wed, 18 Oct 2000 16:48:03 -0700 Subject: Re: More Stuff In-Reply-To: from William A Stein at "Oct 18, 2000 01:19:32 pm" To: William A Stein Date: Wed, 18 Oct 2000 16:48:03 -0700 (PDT) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: A William A Stein wrote: > Do you have anything similar for rank 4? I'm interested in rank 4 because > I could then immediately deduce the existence of large Sha in some > quotient of J_0(p), which the audience of my next talk might enjoy. Similar in what sense? I only have Brumer/McGuinness information here, so all the conductors will be (boringly) prime. Here are smallest 10 prime conductor rank 4 examples (I also have the N=234446 example done). [0, 1, 1, -72, 210] 501029 271872 2^9 3^2 59 [0, 0, 1, -7, 36] 545723 514560 2^9 3 5 67 [1, 0, 0, -202, 1089] 602461 233664 2^6 3 1217 [0, 1, 1, -2, 42] 797611 365712 2^4 3 19 401 [0, 1, 1, -32, 72] 812011 322992 2^4 3^2 2243 [1, 0, 0, -49, 120] 887657 227936 2^5 17 419 [0, 0, 1, -67, 216] 953243 925664 2^5 28927 [1, 0, 0, -25, 66] 1000033 333488 2^4 19 1097 [0, 0, 1, -19, 60] 1129211 687616 2^9 17 79 [0,-1, 1, -35, 72] 1241437 309312 2^6 3^3 179 On the other hand, prime conductor is better for some things, such as I don't have to figure out the functional equation by educated guessing. There are 796 more rank 4 curves in their table, but I might stop at conductor 10^7 instead of 10^8. The rank 5 list should be completed in another couple of hours. > > I have figured out where Coates/Schmidt went wrong; they > > applied Atkin-Li incorrectly (though the latter has a quite > Is this regarding special values of weight 4 forms or the inaccuracy when > 2^8 divides N that you mention on your "Degrees of Modular..." table? This is regard to when 2^8|N. > > Fortunately, it does. The idea is to compute the analytic lattice > > periods w1 and w2 from the c_4 and c_6 invariants, and then compute > > periods of "random" elements of Gamma_0(N). These periods will be of > > the form a*w1+b*w2 for some (a,b) in Z^2. Once you have enough (a,b) > > pairs to generate Z^2, you know the Manin constant is one. (It took > > me a little while to convince myself that this was indeed the right > > direction --- Manin constant is [AnalyticLattice:AlgebraicLattice]). > > I can see why. After 30 seconds thought I'm still not convinced. > Since the c_4 and c_6 invariants are minimal, the w1 and w2 above > are the periods of the NERON lattice. If you compute the image > of Gamma_0(N) under the period mapping defined by a newform attached > to the elliptic curve you get a different lattice that *must* > contain the NERON lattice by a theorem of Edixhoven. Thus I > definitely do not believe your asserion that "These periods will be > of the form a*w1+b*w2 for some (a,b) in Z^2." I would be very > happy if you were to convince me that I'm wrong. Hmm. Maybe I got the inclusion backwards. I shouldn't believe Cremona so readily. So Edixhoven says that the AlgebraicLattice is the bigger one? Not sure why I convinced myself of the opposite. > > twists by -4, -8, and -56 have 2-Selmer rank 3, while the twists > > by -7, 8, 28, and 56 have 2-Selmer rank 4. The twists all have > > rank one or zero (computed analytically). Is this phenomenon > > known? I asked Elkies and Silverman, but neither knew of it. > > There might be some 2-visilibilty of Sha going on here, since > > the curves satisfy a 2-congruence. If that is the reason, then > > Probably so; this might be an example of visibility in the restriction > of scalars. Nils Bruin could maybe verify this, since I saw him do > some similar examples with Victor Flynn in Durham. If it has only to do with a_p being congurent to 0 mod p, then it should work for, say, the -3rd twist of [0,0,1,-79,342]. And it does! Here the 2-Selmer rank is 5; as does the -4th twist, while the +/- 8th twists each have 2-Selmer rank 4, the -8th twist having rank 2. === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Thu Oct 19 00:31:43 2000 Return-Path: Received: from ep1.msri.org (ep1.msri.org [198.129.64.226]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id e9J4Vmu12501 for ; Thu, 19 Oct 2000 00:31:48 -0400 Received: from rafiki.msri.org [198.129.65.127] (mail) by ep1.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 13m7Ml-0000wa-00; Wed, 18 Oct 2000 21:31:43 -0700 Received: from alice.msri.org [198.129.65.103] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 13m7Ml-0000DQ-00; Wed, 18 Oct 2000 21:31:43 -0700 Received: from mwatkins by alice.msri.org with local (Exim 3.12 #1 (Debian)) id 13m7Ml-0004Xc-00; Wed, 18 Oct 2000 21:31:43 -0700 Subject: Re: More Stuff In-Reply-To: from William A Stein at "Oct 18, 2000 01:19:32 pm" To: William A Stein Date: Wed, 18 Oct 2000 21:31:43 -0700 (PDT) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: O William A Stein wrote: > Do you have anything similar for rank 4? I'm interested in rank 4 because > I could then immediately deduce the existence of large Sha in some > quotient of J_0(p), which the audience of my next talk might enjoy. (all conductors primes, from the Brumer/McGuinness data) This one has the biggest Sha (though I'll beat this soon): [0, 0, 1, -37, 126] 3643883 3223536 2^4 3 67157 and this one has no large primes dividing the modular degree: [0, 1, 1, -60, 180] 3280363 1658880 2^12 3^4 5 http://www.msri.org/people/members/mwatkins/currexp.html I now believe that practically *every* prime twist of a high rank curve should have large 2-Sha. It might have something to do with all the powers of 2 in the modular degree. For instance, twisting [0,0,1,-79,342] by either -3 or -4 gives a curve whose 2-Selmer rank is 5, while the +/- 8th twists each have 2-Selmer rank 4. Twisting by either 5 or 13 gives a curve with 2-Selmer rank 5, the -7th and -11th twists give 2-Selmer rank 4. (Some with 2-Selmer rank 4 have rank 2.) === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Thu Oct 19 19:36:53 2000 Return-Path: Received: from ep1.msri.org (ep1.msri.org [198.129.64.226]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id e9JNavu13484 for ; Thu, 19 Oct 2000 19:36:58 -0400 Received: from rafiki.msri.org [198.129.65.127] (mail) by ep1.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 13mPF0-0003WI-00; Thu, 19 Oct 2000 16:36:54 -0700 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 13mPEz-0007pR-00; Thu, 19 Oct 2000 16:36:53 -0700 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 13mPEz-0003Ee-00; Thu, 19 Oct 2000 16:36:53 -0700 Subject: Re: More Stuff In-Reply-To: from William A Stein at "Oct 18, 2000 01:19:32 pm" To: William A Stein Date: Thu, 19 Oct 2000 16:36:53 -0700 (PDT) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: O William A Stein wrote: > Do you have anything similar for rank 4? I'm interested in rank 4 because > I could then immediately deduce the existence of large Sha in some > quotient of J_0(p), which the audience of my next talk might enjoy. I have completed the rank 4 curves thru conductor 10^7. This one has the biggest modular degree factor: [0, 0, 1, -109, 450] 4695371 4356544 2^6 68071 Can you show that this one implies the existence of a really big 17-Sha? [0, -1, 1, -24, 120] 4513931 1100512 2^5 7 17^3 As soon as I automate the program more, I will commence computations on elliptic curves with conductor in a similar range but rank<=3. === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Sat Oct 21 02:48:34 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id e9L6mbu14511 for ; Sat, 21 Oct 2000 02:48:38 -0400 Received: from ep1.msri.org (ep1.msri.org [198.129.64.226]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id e9L6mVK05688 for ; Sat, 21 Oct 2000 02:48:31 -0400 (EDT) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep1.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 13msSJ-0007X8-00; Fri, 20 Oct 2000 23:48:35 -0700 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 13msSI-0007VI-00; Fri, 20 Oct 2000 23:48:34 -0700 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 13msSI-0003Uz-00; Fri, 20 Oct 2000 23:48:34 -0700 Subject: Re: More Stuff In-Reply-To: <00101911054502.03132@form> from "William A. Stein" at "Oct 19, 2000 11:05:45 am" To: was@math.harvard.edu Date: Fri, 20 Oct 2000 23:48:34 -0700 (PDT) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: A William A. Stein wrote: > On Wed, 18 Oct 2000, you wrote: > > Hmm. Maybe I got the inclusion backwards. I shouldn't believe > > Cremona so readily. So Edixhoven says that the AlgebraicLattice > > is the bigger one? Not sure why I convinced myself of the opposite. > > The Manin constant is the index of the Neron lattice N in the lattice > L got by computing the image of Gamma_0(N) under the period map > attached to the newform. Edixhoven proves that the Manin constant is > an integer; equivalently, N is contained in L. That there is an > inclusion in either direction is very far from obvious without knowing > about [Katz-Mazur]. > > Maybe you can forward Cremona's argument to me. > Email from Cremona On checking that the Manin constant is 1, you could try this. Precompute a whole lot of coefficients a_n (or just a_p, depending on your method for summing the series). Find the periods w1,w2 of the (minimal model of the) curve, so you know its period lattice L. Now take lots of random elements of Gamma_0(N), and for each compute the associated period of the newform. You only need to compute it to sufficient accuracy to determine the integer coefficients (a,b) expressing it in the form a*w1+b*w2. This gives you lots of integer pairs (a,b). When you have enough to generate all of Z^2 you know that the manin constant is 1. I have not tried this for real, and have not thought about how many such periods you would need to compute, so this might be useless. I hope it's not as bad as factoring an integer N by picking random a and computing gcd(a,N)! === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Mon Oct 23 10:35:48 2000 Return-Path: Received: from ep1.msri.org (ep1.msri.org [198.129.64.226]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id e9NEZnu23318 for ; Mon, 23 Oct 2000 10:35:50 -0400 Received: from rafiki.msri.org [198.129.65.127] (mail) by ep1.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 13nihZ-0006wE-00; Mon, 23 Oct 2000 07:35:49 -0700 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 13nihZ-0004i4-00; Mon, 23 Oct 2000 07:35:49 -0700 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 13nihZ-0006fo-00; Mon, 23 Oct 2000 07:35:49 -0700 Subject: Re: Manin Constant To: was@modular.fas.harvard.edu Date: Mon, 23 Oct 2000 07:35:48 -0700 (PDT) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: O > > On checking that the Manin constant is 1, you could try this. > > Precompute a whole lot of coefficients a_n (or just a_p, depending on > > your method for summing the series). Find the periods w1,w2 of the > > (minimal model of the) curve, so you know its period lattice L. Now > > take lots of random elements of Gamma_0(N), and for each compute the > > associated period of the newform. You only need to compute it to > > sufficient accuracy to determine the integer coefficients (a,b) > > expressing it in the form a*w1+b*w2. This gives you lots of integer > > pairs (a,b). When you have enough to generate all of Z^2 you know that > > the manin constant is 1. > > William points out this is the wrong direction --- it only shows the Manin > constant is an integer. The Neron Lattice from w1,w2 was shown by Edixhoven > to be contained in the lattice from elements of Gamma_0(N), which is exactly > what the algorithm above does. > Oops, sorry. In my haste to answer at least one of your questions, I did not stop to do the necessary mental exercise to get this the right way round. In which case there does not seem to be a short-cut, as without using modular symbols to cut down the number of perriods to check, one would have to check all the generators, of which there are approximately twice the index. Unless William has any better ideas... John ----- End of forwarded message from John Cremona ----- === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Mon Oct 23 16:19:23 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id e9NKJOu23529 for ; Mon, 23 Oct 2000 16:19:24 -0400 Received: from ep1.msri.org (ep1.msri.org [198.129.64.226]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id e9NKJIK07302 for ; Mon, 23 Oct 2000 16:19:18 -0400 (EDT) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep1.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 13no43-0007nx-00; Mon, 23 Oct 2000 13:19:23 -0700 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 13no43-0001vI-00; Mon, 23 Oct 2000 13:19:23 -0700 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 13no43-0006ji-00; Mon, 23 Oct 2000 13:19:23 -0700 Subject: Re: Old Stuff In-Reply-To: <00101911054502.03132@form> from "William A. Stein" at "Oct 19, 2000 11:05:45 am" To: was@math.harvard.edu Date: Mon, 23 Oct 2000 13:19:23 -0700 (PDT) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: O I'm trying to figure out that level 122, weight 2, nontrivial character example again. Can you compute L(1) for, say, one of the conjugate forms of level 29, weight 2, nontrivial character? There is some code for doing so in the old HECKE distribution (intrinsic called LAnalytic), but it doesn't seem to be in the MAGMA release, and reading in the file causes MAGMA to complain. I tried using PeriodMapping, but it seems to take forever. === Mark Watkins mwatkins@math.uga.edu From mwatkins@msri.org Mon Oct 23 18:09:42 2000 Return-Path: Received: from ep1.msri.org (ep1.msri.org [198.129.64.226]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id e9NM9gu23672 for ; Mon, 23 Oct 2000 18:09:42 -0400 Received: from rafiki.msri.org [198.129.65.127] (mail) by ep1.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 13npmo-00085T-00; Mon, 23 Oct 2000 15:09:42 -0700 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 13npmo-0002LA-00; Mon, 23 Oct 2000 15:09:42 -0700 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 13npmo-0006ks-00; Mon, 23 Oct 2000 15:09:42 -0700 Subject: Re: Old Stuff In-Reply-To: from William A Stein at "Oct 23, 2000 05:06:53 pm" To: William A Stein Date: Mon, 23 Oct 2000 15:09:42 -0700 (PDT) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: O William A Stein wrote: > > doing so in the old HECKE distribution (intrinsic called LAnalytic), > > The analogous command in Magma is "LSeries". Try "LSeries;" to get > documentation on it. > > > I tried using PeriodMapping, but it seems to take forever. > > Help me make it more efficient :-) It seems to have slowed down; I remember being able to compute with LAnalytic, but now PeriodMapping and LSeries can't even handle precision=1, for e.g. 389k2. > N:=NewformDecomposition(CuspidalSubspace(ModularSymbols(389,2,1))); > LSeries(N[1],1,10); [Interrupted] > LSeries(N[1],1,1); [Interrupted] > quit; Total time: 176.500 seconds > Just out of curiosity, would you have any interest in going to Sydney some > month (like May) as a MAGMA fellow? If so, maybe you should talk to John > Cannon. We could try to go at the same time and just spend a month working > on optimizing modular symbols... Any idea on how far ahead of time I'd have to ask him? Before I get to Penn State in January, I don't know much about my summer plans. I think I actually have summer money there, but if I don't use it, I believe I can blow it on fast Athlons or the like instead. (There's also a Dagstuhl on ANT in May, though I probably won't be invited) === Mark Watkins mwatkins@msri.org From maru@math.jussieu.fr Wed Dec 31 19:00:00 1969 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id eA5IKS730131 for ; Sun, 5 Nov 2000 13:20:28 -0500 Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id eA5IJRq26410 for ; Sun, 5 Nov 2000 13:19:28 -0500 (EST) Received: from moka.ccr.jussieu.fr (moka.ccr.jussieu.fr [134.157.1.23]) by shiva.jussieu.fr (8.10.0/jtpda-5.3.3) with ESMTP id eA5IJRc76717 for ; Sun, 5 Nov 2000 19:19:27 +0100 (CET) Received: from math.jussieu.fr (147.ext.jussieu.fr [134.157.81.147]) by moka.ccr.jussieu.fr (8.10.0/jtpda-5.3.3) with ESMTP id eA5IJPO136406 for ; Sun, 5 Nov 2000 19:19:25 +0100 Message-ID: <83E25BED.43F54B9F@math.jussieu.fr> Date: Thu, 07 Jan 1904 02:31:14 +0000 From: Marusia Rebolledo X-Mailer: Mozilla 4.5 [fr] (Macintosh; I; PPC) X-Accept-Language: fr,fr-FR MIME-Version: 1.0 To: was@math.harvard.edu Subject: calculus Content-Type: text/plain; charset=us-ascii; x-mac-type="54455854"; x-mac-creator="4D4F5353" Content-Transfer-Encoding: 7bit Status: RO X-Status: A Hello William, I'm doing calculus about 13-torsion. Coul'd you send me the detail of your proof of : L(f,chi,1)=/=0 for all chi and for the two conjugate form which are a basis of S2(g1(13)) , please? Thank you. Marusia. PS: How do you feel in your new university? From mwatkins@msri.org Sun Dec 3 19:43:11 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id eB40haJ25153 for ; Sun, 3 Dec 2000 19:43:36 -0500 Received: from ep2.msri.org (ep2.msri.org [198.129.64.227]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id eB40hC421701 for ; Sun, 3 Dec 2000 19:43:13 -0500 (EST) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep2.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 142jiq-0001oe-00; Sun, 03 Dec 2000 16:43:12 -0800 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 142jiq-00032V-00; Sun, 3 Dec 2000 16:43:12 -0800 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 142jip-0003DH-00; Sun, 03 Dec 2000 16:43:11 -0800 Subject: Re: more modular degrees. In-Reply-To: <00120221504918.00895@modular> from William A Stein at "Dec 2, 2000 09:51:01 pm" To: was@math.harvard.edu Date: Sun, 3 Dec 2000 16:43:11 -0800 (PST) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: A William A Stein wrote: [Charset iso-8859-1 unsupported, filtering to ASCII...] > Mark, > > At http://shimura.math.berkeley.edu/~oisin/Examples.html > you will find the "Setzer-Neumann" elliptic curve > > [1, 1, 0, -103324, -12826653] > > which has prime conductor 4959593. The BSD conjecture predicts that > #Sha=289. Is this Sha visible? If 17 does *not* divide the modular > degree, then we can conclude that this Sha is invisible. > > Can you compute the modular degree of that curve, or do none of the > tricks you know work? > > -- William > This is the strong isogeny I take it? Or is [1,1,0,-103319,-12827950]? I can never figure out the allisog output. In any case, factors of 2 do not change the 17-divisibility. My program returns 20935043 as the modular degree, so I guess the III is invisible. Here's the modular degree factoid of the week: look at the Brumer-McGuinness curves of conductor less than 10^6. For p>3, approximately 1/(p-1) of them have modular degree divisible by p. Why this ratio? Maybe a Cohen-Lenstra type heuristic for congruence primes? (For p=3 the ratio is 4/9, while for p=2 it depends on the rank). === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Mon Dec 4 17:05:32 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id eB4M5tJ24450 for ; Mon, 4 Dec 2000 17:05:55 -0500 Received: from ep2.msri.org (ep2.msri.org [198.129.64.227]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id eB4M5X418620 for ; Mon, 4 Dec 2000 17:05:33 -0500 (EST) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep2.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 1433jo-0003kD-00; Mon, 04 Dec 2000 14:05:32 -0800 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 1433jo-00028l-00; Mon, 4 Dec 2000 14:05:32 -0800 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 1433jo-0003MZ-00; Mon, 04 Dec 2000 14:05:32 -0800 Subject: Re: Modular degrees In-Reply-To: <0012041646110F.20459@modular> from William A Stein at "Dec 4, 2000 04:46:11 pm" To: was@math.harvard.edu Date: Mon, 4 Dec 2000 14:05:32 -0800 (PST) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: A William A Stein wrote: [Charset iso-8859-1 unsupported, filtering to ASCII...] > Mark, > > If you send me a table of modular degrees of prime-level elliptic > curves that you've computed, I'll enter them in the database > > http://modular.fas.harvard.edu/mfd/x0n/ > > This database now contains (almost) all of Cremona's tables and > Brumer-McGuinness's tables, amongst other things. I guess I haven't done the Setzer curves, since they need not be the strong Weil isogeny. Your table for e.g. N=21089 could list the other isogeny [1,0,0,-434,3589]. I'll come up with some reasonable way to package the modular degree data and send it later. Also, I sent the following to B/McG regarding their work: --- What are the rules for isogenies? [[0, -1, 1, -7820, -263580], 11, even, 0, 0.253842, [[0, -1, 1, 0, 0], 11, even, 0, 6.346047, 0.253842, [[1, -1, 1, -1, 0], 17, even, 0, 6.188319, 0.386770, [[1, -1, 1, -91, -310], 17, even, 0, 1.547080, 0.386770, [[0, 1, 1, -769, -8470], 19, even, 0, 0.453253, 0.453253, [[0, 1, 1, 1, 0], 19, even, 0, 4.079279, 0.453253, [[0, 1, 1, -1873, -31833], 37, even, 0, 0.725681, 0.725681, [[0, 1, 1, -3, 1], 37, even, 0, 6.531130, 0.725681, I am also worried that the accuracy of the L-function computation is not always sufficiently high. [[1, 0, 0, -1563492, -752604497], 75047633, setzer, 0, 0.135013, -0.029860, -0.884646, 0, 1.000000, [ ]], I compute the L-series as 0.03375; your predicted regulator is -0.884646 (this curve is not the strong Weil isogeny; [1,0,0,-1563487,-752609550] is) [[1, 1, 0, -42744, -3419297], 90585107, even, 0, 0.166016, 0.240223, 1.446991, 0, 1.000000, [ ]], Here I get 0.166016, in accordance with your real period, but not your computed L-value of 0.240223 --- McGuinness said that because they have such a large data set, they didn't care about these four examples of isogenies. I still don't have a good idea about the strong isogeny for Setzer curves. He didn't really address the fact that their L-value calcs were wrong; only some mumbo-jumbo about the superiority of the 68040 math chip. I also sent this to them: --- Don Zagier recently asked me about curves which are purportedly rank 2 (from L-series computations), but for which no generators are known. Of the 61515 curves identified as analytic rank 2 in your list, 70 of them have no generators listed (and 2669 more with only one generator). However, it seems that 11 of these 70 curves in fact have analytic (and algebraic) rank zero. [0, 1, 1, -24382, -1473544] [0, 1, 1, -622, -6166] [0, 1, 1, -3312, -74478] [0, 1, 1, 74, -382] [0, 1, 1, -354, -2730] [0, -1, 1, -358395, -82463721] [0, -1, 1, -36, -437] [1, 0, 0, -36, -473] [0, 1, 1, -1442430, -667272815] [0, 0, 1, -133, -738] [0, 0, 1, -2842, -58314] John Cremona's mwrank quickly finds two generators for each of the other 59 curves (sometimes of height greater than 30). I have not tried the 2669 one-generator curves to try to find another generator. === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Mon Dec 4 17:50:37 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id eB4Mp1J31309 for ; Mon, 4 Dec 2000 17:51:01 -0500 Received: from ep2.msri.org (ep2.msri.org [198.129.64.227]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id eB4Moc421029 for ; Mon, 4 Dec 2000 17:50:39 -0500 (EST) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep2.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 1434RS-0003s3-00; Mon, 04 Dec 2000 14:50:38 -0800 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 1434RR-0002JK-00; Mon, 4 Dec 2000 14:50:37 -0800 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 1434RR-0003N9-00; Mon, 04 Dec 2000 14:50:37 -0800 Subject: Re: Modular degrees In-Reply-To: <0012041714320G.20459@modular> from William A Stein at "Dec 4, 2000 05:14:32 pm" To: was@math.harvard.edu Date: Mon, 4 Dec 2000 14:50:37 -0800 (PST) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: A William A Stein wrote: > On Monday 04 December 2000 05:05 pm, you wrote: > > I guess I haven't done the Setzer curves, since they need not > > be the strong Weil isogeny. Your table for e.g. N=21089 could > > list the other isogeny [1,0,0,-434,3589]. I'll come up with some > > reasonable way to package the modular degree data and send it later. > > I assume you proved that [1,0,0,-434,3589] is optimal using some method > you once told me about. > Now when you point your browser at 21089A, you'll get the updated > Weierstrass equation. Now I'm confused. Maybe I should ask what your rules for isogenies are... Your numbering like 11A 11B 11C is different from Cremona's 11A1 11A2 11A3. My contention would be that 11A 11B and 11C are all the same newform factor (same L-series) --- if you view them as different, then both isogenies for 21089A should similarly be listed as A and B. Here it seems that the strong isogeny is [1,0,0,-434,3589] which has modular degree 11000. > I hope we discuss this project a lot at the MSRI workshop. This means I better finish the latest issue of my SCRABBLE(r) periodical before the conference... === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Mon Dec 4 19:31:41 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id eB50W5J12413 for ; Mon, 4 Dec 2000 19:32:05 -0500 Received: from ep2.msri.org (ep2.msri.org [198.129.64.227]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id eB50Vg425393 for ; Mon, 4 Dec 2000 19:31:42 -0500 (EST) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep2.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 14361G-00044R-00; Mon, 04 Dec 2000 16:31:42 -0800 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 14361F-0002ia-00; Mon, 4 Dec 2000 16:31:41 -0800 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 14361F-0003O8-00; Mon, 04 Dec 2000 16:31:41 -0800 Subject: Re: Modular degrees In-Reply-To: <0012041834520H.20459@modular> from William A Stein at "Dec 4, 2000 06:34:52 pm" To: was@math.harvard.edu Date: Mon, 4 Dec 2000 16:31:41 -0800 (PST) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: A William A Stein wrote: > No. However, there is a currently some redundancy with the entries of > small prime level in my table, which will be corrected in phase two of > my integration of the Brumer data into my database. Brumer and > McGuiness evidently list different representatives for the same > isogeny class more than once; in particular, they don't always list > the optimal curve. As per previous email, their isogeny-based errors are with N=11,17,19,37 (Setzer-like curves with N <> u^2+64), and the Setzer curves (N=u^2+64). I am unsure, but I think they always list the non-strong isogeny for Setzer curves. If this is true, it will be easy to clean up. In what form would like modular degree information? Is '[0,1,1,-23,-50] 2' sufficient, or should I include the conductor? I could also include the Sym^2 L-value and fundamental volume, though the second is easy to compute and hence gives you the first. > I added their data on top of my data, except when > the Weierstrass equations are the same, and in phase two, when I sort Can't you start your numbering with phase zero like everyone else?? :) Oh yeah, my prog assumes the Manin constant is one; you might want to mention this somewhere. === Mark Watkins mwatkins@msri.org From mwatkins@msri.org Mon Dec 4 23:38:12 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id eB54cbJ04989 for ; Mon, 4 Dec 2000 23:38:37 -0500 Received: from ep2.msri.org (ep2.msri.org [198.129.64.227]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id eB54cE405278 for ; Mon, 4 Dec 2000 23:38:14 -0500 (EST) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep2.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 1439rp-0004S2-00; Mon, 04 Dec 2000 20:38:13 -0800 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 1439rp-0003KS-00; Mon, 4 Dec 2000 20:38:13 -0800 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 1439ro-0003R7-00; Mon, 04 Dec 2000 20:38:12 -0800 Subject: Setzer Curves In-Reply-To: <0012042003280K.20459@modular> from William A Stein at "Dec 4, 2000 08:03:28 pm" To: was@math.harvard.edu Date: Mon, 4 Dec 2000 20:38:12 -0800 (PST) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: O Here are the 119 Setzer curves up to conductor 10^6. To get from my [a1,a2,a3,a4,a6] to the [A1,A2,A3,A4,A6] of Brumer/McGuinness, take A1=a1, A2=a2, A3=a3, A4=a4-5, A6=a6-sign(a6)*(u-5k) where N=u^2+64 and k=floor((u+5)/12). For instance, with N=37313, u=193, k=16, my curve is [1,0,0,-772,8385] while B/McG have [1,0,0,-777,8272] My curve is always the strong Weil curve (it has disc -N^2, while theirs has disc N). Curve Level ModDeg [1,-1,0,4,-3] 73 3 [1,1,0,4,5] 89 5 [1,1,1,3,-4] 113 6 [1,0,1,0,11] 233 27 [1,1,1,-2,16] 353 24 [1,0,0,-7,-30] 593 58 [1,-1,1,-19,68] 1153 96 [1,0,1,-22,-75] 1289 219 [1,0,1,-25,81] 1433 165 [1,1,0,-34,-135] 1913 309 [1,-1,0,-38,145] 2089 219 [1,0,0,-42,-155] 2273 392 [1,1,1,-59,-254] 3089 710 [1,-1,1,-64,268] 3313 650 [1,1,1,-84,326] 4289 872 [1,0,0,-107,490] 5393 1142 [1,-1,0,-113,-510] 5689 1601 [1,-1,0,-176,1037] 8713 1547 [1,0,0,-192,1105] 9473 3648 [1,-1,0,-311,-2158] 15193 5005 [1,1,1,-332,-2594] 16193 7740 [1,-1,1,-376,-2844] 18289 3702 [1,0,0,-434,3589] 21089 11000 [1,-1,0,-446,-3663] 21673 4843 [1,-1,1,-484,4368] 23473 9250 [1,1,0,-549,-5350] 26633 12899 [1,0,0,-577,-5550] 27953 18602 [1,1,1,-634,-6584] 30689 12420 [1,-1,1,-649,6698] 31393 11000 [1,0,0,-772,8385] 37313 19384 [1,1,0,-804,8645] 38873 25185 [1,0,1,-872,10035] 42089 26679 [1,-1,0,-941,11562] 45433 14469 [1,0,1,-1070,-13779] 51593 29571 [1,-1,0,-1226,-16463] 59113 23879 [1,1,0,-1246,16665] 60089 27905 [1,-1,1,-1351,-19024] 65089 27200 [1,0,0,-1437,-21350] 69233 57078 [1,0,0,-1459,21594] 70289 59970 [1,-1,0,-1481,-21838] 71353 46853 [1,0,1,-1572,-24385] 75689 41671 [1,1,0,-1664,-27115] 80153 54057 [1,-1,1,-1909,-31922] 91873 50940 [1,1,1,-1934,32236] 93089 69092 [1,-1,0,-2063,-35870] 99289 45689 [1,0,1,-2170,-39399] 104393 79823 [1,-1,1,-2224,-40132] 106993 53810 [1,0,0,-2334,-43931] 112289 128484 [1,-1,1,-2476,48376] 119089 72850 [1,0,1,-2505,-48799] 120473 98637 [1,1,1,-2592,50066] 124673 97956 [1,1,1,-2802,-58624] 134753 148128 [1,-1,1,-2931,-60334] 140689 74066 [1,1,1,-2957,61036] 142193 167634 [1,0,1,-3247,-71865] 156089 123173 [1,1,0,-3379,-77430] 162473 142359 [1,-1,0,-3413,78010] 164089 178045 [1,0,0,-3447,-78590] 165713 141414 [1,1,0,-3549,80330] 170633 190759 [1,1,0,-3974,95225] 191033 176269 [1,-1,1,-4159,-102632] 199873 162372 [1,1,1,-4467,-116776] 214433 234212 [1,-1,1,-4501,117806] 216289 148048 [1,1,0,-4696,-126315] 225689 346193 [1,-1,0,-4736,127117] 227593 187599 [1,-1,0,-4856,-129523] 233353 142599 [1,1,0,-4896,130325] 235289 205793 [1,1,1,-4937,-136064] 237233 204954 [1,1,1,-5142,140276] 247073 386112 [1,0,1,-5565,159821] 267353 464981 [1,0,0,-5782,-170235] 277793 289296 [1,1,0,-5914,173145] 284153 458037 [1,1,1,-5959,-180074] 286289 289798 [1,-1,1,-6004,181088] 288433 278238 [1,0,0,-6367,195570] 305873 271602 [1,1,0,-6459,197690] 310313 347091 [1,0,1,-6647,208575] 319289 322245 [1,-1,1,-6694,-209682] 321553 234330 [1,0,0,-6884,-220991] 330689 463868 [1,0,0,-6932,222145] 332993 443752 [1,1,1,-7672,-262454] 368513 625336 [1,0,1,-7825,266361] 375833 351129 [1,-1,1,-7876,-267664] 378289 390626 [1,1,1,-7927,268966] 380753 525938 [1,-1,1,-8344,296068] 400753 442494 [1,0,1,-8397,-297475] 403289 397781 [1,-1,0,-8663,313170] 416089 445885 [1,1,0,-9264,-347755] 444953 614145 [1,0,1,-9715,-370029] 466553 585441 [1,1,1,-10587,415586] 508433 1038522 [1,1,0,-10646,-427975] 511289 923841 [1,1,1,-11312,459106] 543233 670972 [1,0,0,-11497,-476190] 552113 1076330 [1,-1,0,-12188,521725] 585289 605487 [1,1,0,-12444,529925] 597593 815613 [1,1,1,-12509,-544484] 600689 1036998 [1,-1,1,-12574,546598] 603793 570354 [1,-1,1,-12769,-552942] 613153 1277652 [1,1,1,-12834,555056] 616289 1166524 [1,0,0,-13097,576610] 628913 1094650 [1,-1,0,-13163,-578790] 632089 817881 [1,0,0,-13834,-628251] 664289 766236 [1,0,0,-13902,630565] 667553 1005444 [1,-1,0,-14591,682882] 700633 919749 [1,-1,0,-14801,-690238] 710713 1058009 [1,0,1,-15155,717651] 727673 1259477 [1,0,0,-15512,-745775] 744833 1353500 [1,-1,0,-15656,-750963] 751753 1460727 [1,0,0,-16387,-809670] 786833 1132926 [1,1,0,-17134,-871335] 822713 1500937 [1,0,0,-17362,880005] 833633 1622320 [1,-1,1,-17899,-918132] 859393 1027816 [1,1,1,-18522,-978904] 889313 1463128 [1,1,0,-18996,-1016635] 912089 1942941 [1,1,0,-19396,1032645] 931289 2691909 [1,-1,1,-19801,-1068454] 950689 1145244 [1,0,0,-20127,-1101710] 966353 1269078 [1,-1,1,-20539,1139068] 986113 1822924 [1,0,1,-20705,1145961] 994073 1407177 From mwatkins@msri.org Mon Dec 4 23:51:49 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id eB54qFJ07224 for ; Mon, 4 Dec 2000 23:52:15 -0500 Received: from ep2.msri.org (ep2.msri.org [198.129.64.227]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id eB54pp405736 for ; Mon, 4 Dec 2000 23:51:51 -0500 (EST) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep2.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 143A50-0004TE-00; Mon, 04 Dec 2000 20:51:50 -0800 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 143A4z-0003Ls-00; Mon, 4 Dec 2000 20:51:49 -0800 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 143A4z-0003Rn-00; Mon, 04 Dec 2000 20:51:49 -0800 Subject: 9040 Curves with Modular Degree To: was@math.harvard.edu Date: Mon, 4 Dec 2000 20:51:49 -0800 (PST) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: O [0,-1,1,-10,-20] 1 [1,-1,1,-1,-14] 1 [0,1,1,-9,-15] 1 [0,0,1,-1,0] 2 [0,1,1,-23,-50] 2 [0,1,1,0,0] 2 [1,-1,1,0,0] 2 [1,0,0,-2,1] 2 [0,1,1,-12,-21] 5 [1,1,1,-2,0] 2 [1,1,1,1,0] 2 [1,1,1,-1,0] 2 [0,1,1,-1,-1] 2 [1,-1,0,-8,-7] 4 [0,-1,1,1,0] 2 [1,1,0,-3,-4] 6 [0,0,1,-2,1] 6 [0,0,1,-1,-1] 9 [0,0,1,-5,4] 10 [1,0,0,-2,-1] 8 [0,0,1,-2,-1] 6 [1,0,1,0,-1] 10 [0,-1,1,2,-1] 15 [0,0,1,-8,-9] 13 [0,0,1,1,-1] 11 [1,1,0,0,-1] 10 [1,0,0,-5,4] 12 [0,1,1,2,0] 14 [1,-1,1,-7,8] 8 [1,0,1,-23,39] 16 [0,1,1,-2,-2] 22 [0,1,1,-2,0] 40 [1,-1,1,-9,-8] 40 [1,0,0,0,-1] 10 [1,0,0,0,1] 28 [0,1,1,1,1] 12 [1,0,0,-3,-2] 14 [1,0,1,-84,-301] 62 [0,0,1,-4,3] 34 [1,-1,0,2,-1] 38 [1,0,0,-210,-1189] 100 [1,0,1,-32,-71] 22 [0,-1,1,-268,1781] 182 [1,1,0,0,1] 14 [1,1,1,-15,16] 52 [0,-1,1,-929,-10595] 120 [0,1,1,-4,2] 48 [1,0,1,-2,1] 22 [1,0,0,-4,3] 32 [0,1,1,-372,2641] 141 [1,1,0,-79,-306] 50 [1,1,1,2,0] 22 [0,-1,1,-2,1] 42 [0,-1,1,-2,0] 44 [1,1,0,-75,-284] 48 [0,0,1,1,1] 33 [1,0,0,2,1] 24 [0,0,1,-2,-2] 24 [0,0,1,-10,12] 44 [0,0,1,-4,-3] 66 [0,-1,1,-18,36] 80 [0,-1,1,-24,54] 96 [0,-1,1,-5,-3] 48 [0,1,1,-1961,32779] 192 [0,1,1,-6,4] 72 [0,1,1,-5,3] 40 [1,0,0,-4,-3] 38 [0,-1,1,-4,5] 81 [1,-1,1,-3,0] 88 [0,0,1,2,1] 117 [0,1,1,-22,33] 91 [1,-1,1,0,-2] 80 [0,0,1,-1,-2] 159 [0,1,1,3,1] 30 [1,-1,1,-3,-2] 68 [1,1,0,1,2] 40 [1,1,0,3,2] 88 [1,0,0,-18,-31] 52 [1,1,0,2,-1] 38 [0,1,1,2,2] 120 [0,0,1,-14,20] 96 [1,-1,1,-1,-2] 54 [1,0,0,-7,-8] 50 [0,-1,1,-92,-311] 304 [0,1,1,-3,0] 80 [1,-1,1,-4,4] 68 [1,1,1,-3,-2] 72 [0,0,1,-5,-5] 90 [0,-1,1,3,0] 62 [0,-1,1,-2,-2] 177 [1,-1,1,-1,2] 64 [1,1,0,-29,-74] 116 [0,-1,1,-2,3] 165 [1,-1,1,-46,130] 112 [1,1,0,-202,1025] 300 [1,0,0,1,-2] 84 [0,1,1,-7,-10] 114 [0,1,1,-8,-12] 126 [0,0,1,-4,-4] 55 [0,-1,1,-3,4] 62 [1,1,0,-10,9] 102 [0,-1,1,-8,12] 116 [1,-1,0,-214,-1153] 480 [1,-1,1,-39,102] 352 [1,0,1,-20,-35] 228 [1,1,1,-5,-6] 84 [1,-1,0,-1,-2] 206 [1,-1,0,-16,29] 216 [1,-1,0,1,2] 54 [1,-1,0,-26,-45] 152 [0,-1,1,-11,18] 54 [1,1,0,1,-2] 116 [0,0,1,1,2] 148 [0,0,1,-7,-7] 98 [0,1,1,-6,-8] 138 [1,0,1,-5,-5] 138 [0,-1,1,-3,1] 70 [1,-1,0,-7,-6] 80 [0,0,1,-11,14] 160 [1,0,0,-5,-4] 156 [1,1,0,-1,2] 110 [1,1,1,-4,0] 96 [0,-1,1,1,-3] 171 [0,0,1,-88,-318] 413 [0,0,1,2,2] 70 [1,0,1,-2,-3] 122 [0,1,1,-9,-14] 176 [0,1,1,-4,-4] 184 [1,-1,1,3,0] 448 [1,-1,0,-4,-1] 444 [0,1,1,-4,-6] 205 [1,0,0,-9,10] 116 [1,-1,1,-4,0] 98 [0,-1,1,-7,-5] 138 [1,0,1,3,-1] 144 [0,1,1,-49,-150] 218 [0,0,1,-1,-3] 401 [1,1,1,-9,-14] 104 [0,-1,1,-37,100] 322 [1,0,1,-9,9] 280 [0,-1,1,-54,172] 510 [0,-1,1,3,1] 149 [0,-1,1,4,-2] 290 [1,0,1,-6,5] 162 [0,0,1,1,-3] 263 [0,1,1,2,3] 130 [0,-1,1,-25,-41] 247 [1,0,1,0,-3] 186 [1,0,0,-4,1] 160 [0,1,1,-3,-5] 117 [1,0,0,-26,49] 270 [0,0,1,-35,-80] 409 [0,1,1,3,3] 164 [1,-1,0,4,-1] 102 [1,1,0,-5,-8] 162 [0,0,1,2,-3] 292 [0,0,1,-71,230] 568 [1,0,1,-581,5335] 492 [0,-1,1,-92,372] 578 [0,0,1,-4,-1] 162 [1,0,0,3,-2] 186 [0,0,1,-5,5] 171 [0,1,1,1,-3] 184 [0,-1,1,-3,-3] 90 [1,0,0,2,3] 156 [0,1,1,-6,3] 430 [0,1,1,-4,0] 422 [0,0,1,-2,3] 90 [0,-1,1,4,0] 268 [1,-1,0,-7303,242050] 3504 [1,1,0,-11,10] 200 [1,-1,1,-2091,-36272] 2184 [1,-1,0,-5,-2] 274 [1,-1,1,-3,4] 704 [0,0,1,4,-1] 223 [0,0,1,-10,-12] 364 [0,-1,1,-2,-3] 327 [1,0,1,0,3] 156 [1,-1,1,-13,20] 180 [1,0,1,-197,-1077] 784 [1,1,1,-6,2] 196 [1,0,0,-32,67] 260 [0,-1,1,-2,4] 374 [0,-1,1,-43,124] 292 [0,-1,1,2,-4] 340 [0,0,1,-5,-3] 318 [1,0,0,-31,64] 336 [1,0,0,4,-1] 304 [1,1,1,-19,24] 312 [0,1,1,-4,-1] 518 [0,0,1,-7,6] 1984 [0,1,1,-17,22] 292 [0,-1,1,-4,1] 490 [0,0,1,-20,34] 234 [1,1,0,-4,-3] 434 [0,0,1,-17,-27] 302 [1,-1,0,-7,10] 430 [1,0,0,-13012,570217] 2496 [1,1,1,3,4] 224 [0,0,1,4,-2] 245 [0,-1,1,-28,-49] 546 [1,1,0,4,-1] 186 [1,0,0,-2893,-60134] 1264 [0,0,1,-2,-4] 308 [0,0,1,-4,-5] 701 [1,-1,1,-10,-10] 336 [1,1,0,-4,-7] 392 [1,-1,0,-5,4] 288 [0,0,1,-5,2] 296 [1,-1,0,-1,4] 420 [0,-1,1,-6,9] 897 [0,1,1,3,4] 205 [1,1,0,-5,4] 228 [0,0,1,-1,-4] 308 [0,0,1,-172,868] 917 [1,1,0,1,4] 198 [1,0,0,-3,4] 216 [0,1,1,-2,3] 515 [0,1,1,-2,-5] 492 [1,0,0,-146,667] 660 [1,1,1,1,4] 308 [1,-1,0,1,-4] 194 [1,0,1,-58,-173] 746 [0,-1,1,-17,33] 392 [1,1,0,-7,-10] 506 [0,1,1,-242,-1533] 2340 [1,-1,1,-1,-4] 192 [1,0,1,3,-3] 346 [0,1,1,5,2] 330 [1,0,0,1,4] 312 [1,-1,1,-385,3000] 774 [0,1,1,1,4] 451 [1,1,1,-250,-1626] 776 [1,-1,0,-17,32] 292 [0,1,1,4,4] 288 [1,0,0,-5,-2] 360 [1,-1,0,-7,8] 528 [1,-1,1,-75,-230] 1136 [1,0,0,-438,-3565] 976 [1,1,0,-9195,-343232] 2736 [0,0,1,-14,-20] 450 [1,1,0,-13,14] 394 [0,1,1,-192,-1091] 984 [0,-1,1,-117,528] 518 [1,-1,1,-1,4] 220 [0,1,1,5,0] 676 [0,0,1,-7,8] 732 [0,0,1,-23,42] 620 [0,0,1,-7,-6] 1356 [0,1,1,5,3] 345 [1,1,0,-16,-33] 580 [0,0,1,-65,-202] 891 [1,-1,0,1,4] 398 [0,0,1,-5,-1] 474 [1,1,1,-160,-846] 636 [0,0,1,1,4] 560 [1,1,1,-5,-2] 432 [0,0,1,-16,24] 818 [1,0,1,-14,-21] 796 [0,-1,1,-21,45] 336 [1,1,0,2,5] 524 [0,0,1,5,-1] 501 [0,-1,1,-9,-9] 417 [1,0,1,-86,297] 912 [0,-1,1,4,-5] 540 [0,1,1,2,-4] 672 [0,-1,1,-7,11] 429 [1,1,0,-8,5] 950 [1,1,1,-8,4] 396 [1,0,1,-11,-15] 490 [1,1,0,-32,-85] 1118 [0,1,1,-31,57] 456 [0,-1,1,1,-5] 503 [0,-1,1,-111,489] 1494 [0,-1,1,3,3] 416 [0,0,1,-1609,-24842] 7459 [0,0,1,-205,-1130] 3960 [0,-1,1,1,4] 540 [0,0,1,-11,-15] 765 [0,-1,1,-7,-4] 294 [1,-1,1,-16,28] 666 [1,-1,1,-97,-342] 1208 [1,0,1,-2,-5] 382 [0,1,1,-20,28] 810 [1,-1,1,-4,6] 436 [1,-1,1,-40,106] 636 [1,0,1,-20,31] 570 [1,0,0,-17,26] 548 [0,-1,1,-3,6] 388 [0,-1,1,-13,-14] 316 [0,0,1,5,-2] 1195 [0,0,1,-11,13] 1106 [1,1,0,-5,-4] 400 [1,0,0,-19,30] 748 [1,0,0,-7,-6] 912 [1,0,0,3,-4] 472 [0,-1,1,-4,7] 676 [1,1,0,-6,-7] 1178 [0,1,1,-12,13] 958 [0,1,1,-31,-78] 510 [0,0,1,-26,51] 588 [0,0,1,1,-5] 987 [1,1,1,5,4] 578 [0,0,1,-187,984] 3760 [0,0,1,-52,144] 1344 [0,1,1,-22,-48] 1328 [0,0,1,-373,-2773] 2026 [0,1,1,-13,-24] 731 [0,-1,1,-6,-2] 678 [1,0,1,4,-3] 578 [0,0,1,-8,7] 450 [1,-1,1,-33,80] 888 [1,0,0,-74,239] 736 [0,1,1,4,-3] 779 [1,0,1,-7,-5] 392 [1,0,1,-381,-2889] 1508 [0,0,1,2,-5] 663 [0,0,1,-19,32] 1152 [0,-1,1,-28,67] 798 [0,0,1,-139,-631] 1750 [0,0,1,-40,97] 1338 [0,0,1,-94,-351] 1578 [1,0,0,-17,-28] 652 [1,1,1,-2,4] 716 [0,-1,1,2,4] 816 [0,-1,1,-11,-12] 561 [1,-1,0,-5,8] 684 [1,-1,0,4,-5] 512 [0,1,1,-19,-39] 456 [1,1,1,-4,4] 768 [1,-1,0,-4,7] 1480 [1,0,1,-6,-1] 532 [0,1,1,-37,75] 902 [1,0,1,5,1] 456 [0,1,1,-356,-2708] 2376 [0,1,1,1,-5] 592 [1,0,1,-13,15] 632 [1,1,1,-12,-22] 572 [0,0,1,-28,57] 2636 [0,-1,1,-7,-7] 450 [1,1,1,-3,4] 384 [0,-1,1,-14,-17] 1769 [1,1,0,-9,-14] 590 [0,-1,1,-4,-5] 893 [1,-1,1,-45,126] 1252 [1,-1,1,-6,0] 2080 [0,1,1,-25,-58] 843 [1,0,0,-26,-53] 692 [1,1,0,-13,-24] 528 [1,1,0,6,1] 698 [1,0,1,-7,-9] 698 [1,-1,1,-6,2] 516 [0,0,1,-112,456] 4760 [1,0,1,0,5] 708 [1,1,1,-7,-8] 1280 [0,-1,1,4,-6] 1688 [0,1,1,-10,8] 1080 [0,1,1,-59,156] 1056 [0,-1,1,-57,186] 660 [0,-1,1,4,3] 1023 [0,-1,1,6,-3] 854 [0,0,1,1,5] 1177 [1,1,0,-6,1] 514 [0,-1,1,-9,15] 512 [1,-1,1,5,-4] 422 [1,1,0,-12,-23] 600 [0,-1,1,-10,17] 2744 [0,1,1,-53,-168] 1045 [1,1,1,-4,-8] 740 [1,0,0,-24,43] 880 [0,-1,1,-230,1422] 3128 [1,-1,0,-521,4710] 2286 [0,-1,1,-2,-5] 2451 [1,1,0,-365,-2842] 1712 [1,-1,1,2,-6] 990 [1,0,0,-81,274] 1382 [1,1,0,-6,-11] 810 [1,-1,0,-2,-5] 612 [1,1,0,-1,-6] 704 [0,-1,1,-6,10] 1152 [0,0,1,-8,10] 436 [1,-1,1,-811,-8682] 3078 [1,-1,1,-7,10] 736 [0,-1,1,1,-6] 540 [1,0,1,-35,-81] 1258 [1,0,0,-8,-11] 928 [1,0,1,-70,217] 1136 [1,0,0,5,4] 664 [0,-1,1,-68,-195] 1946 [1,0,0,-33,-76] 624 [1,-1,1,-19,36] 1088 [1,1,1,-47,-144] 2028 [1,1,1,-7,6] 764 [0,-1,1,6,0] 1000 [1,1,1,-10481,408636] 16176 [1,1,1,6,2] 1320 [1,0,0,6,1] 944 [0,-1,1,-3,7] 991 [0,0,1,-7,4] 3410 [0,-1,1,-13,22] 514 [0,0,1,1,-6] 1155 [1,1,1,-22,30] 1628 [0,-1,1,-62,210] 1696 [1,1,1,5,-2] 816 [0,-1,1,-6,0] 920 [1,0,0,-122,-529] 1916 [0,0,1,-11,-13] 1326 [0,0,1,-25,-48] 2850 [0,1,1,-34,66] 1560 [0,-1,1,6,-5] 1928 [0,0,1,-401,-3091] 4258 [0,1,1,3,-5] 936 [1,1,1,5,6] 1180 [1,0,0,-13,16] 824 [0,1,1,-13,15] 686 [0,1,1,-27,46] 1061 [1,1,0,-6,-5] 652 [1,1,0,6,-1] 1526 [1,-1,0,-76,-237] 1486 [0,1,1,1,-6] 989 [1,-1,1,-106,-392] 2520 [1,0,0,-5116,-141273] 6232 [1,0,1,-12,15] 1458 [1,0,1,-7,-3] 1512 [1,-1,0,-25,-42] 1902 [0,-1,1,-32,81] 1646 [0,-1,1,-6,3] 1446 [0,-1,1,-669,6888] 2946 [0,-1,1,-9,12] 986 [1,0,0,-23,-44] 1280 [0,0,1,5,4] 3471 [1,-1,1,-741,-7574] 9052 [0,0,1,-7,-4] 1448 [1,1,1,-9,-16] 1262 [0,-1,1,-10,-11] 1390 [0,-1,1,-33,-63] 1276 [1,0,0,-13,18] 880 [0,1,1,-6,-3] 1806 [1,0,0,-37,-90] 1178 [0,-1,1,4,4] 2205 [0,0,1,-2,6] 1068 [1,1,1,-12,-20] 1376 [1,1,0,-4,-9] 1600 [1,-1,1,-3,-6] 3162 [1,-1,0,-194,-993] 2064 [1,-1,0,-806,-8609] 3808 [1,-1,0,1,6] 1036 [0,0,1,-1,6] 4160 [1,1,1,1,-6] 780 [0,1,1,-203,-1184] 1862 [0,1,1,-3,-8] 531 [1,-1,1,-195,1094] 1552 [1,0,1,-89,313] 1670 [1,-1,0,-1,-6] 894 [0,1,1,-178,857] 4354 [1,0,1,-62,-191] 1284 [0,1,1,-7,-7] 778 [1,0,1,4,-5] 724 [0,1,1,-11,12] 760 [1,-1,1,-87,332] 1240 [1,-1,1,-145,-634] 2232 [0,-1,1,-196,1124] 7398 [0,-1,1,-2,-6] 1136 [0,0,1,2,6] 2710 [1,-1,1,0,6] 1592 [0,0,1,-7,3] 3582 [1,-1,0,-8,-9] 734 [0,1,1,6,-2] 1084 [0,-1,1,-2,7] 3033 [0,-1,1,-42,-92] 2320 [1,0,1,-619,5869] 4108 [1,0,1,-17,25] 1152 [0,-1,1,-3,-6] 516 [0,0,1,5,-5] 2358 [1,-1,1,-15,26] 1020 [0,-1,1,-17,34] 1186 [0,-1,1,-48,145] 1830 [1,1,1,-7,0] 816 [0,-1,1,-47,141] 1182 [1,1,0,7,4] 852 [1,1,1,-10,6] 2160 [1,1,1,-54,130] 1776 [1,0,1,6,-1] 1340 [1,1,0,-44,-133] 2260 [0,1,1,-17,-33] 1752 [0,1,1,-10,-18] 1342 [1,0,0,-40,-101] 1112 [1,-1,1,-19,-26] 1344 [1,-1,1,-7,0] 1908 [1,-1,0,-28,65] 2376 [0,0,1,4,-6] 783 [0,-1,1,-9,-6] 564 [0,0,1,-181,937] 7548 [1,1,1,-7,-6] 940 [0,1,1,-124,492] 3760 [0,-1,1,-22,-34] 2052 [1,-1,0,5,4] 1714 [1,0,0,-4,7] 1894 [1,-1,1,-7,-8] 672 [1,0,1,-5,7] 1448 [0,1,1,-363,-2787] 3566 [0,1,1,7,3] 754 [0,1,1,7,1] 827 [1,0,0,-82,279] 2916 [1,1,1,2,-6] 974 [1,0,0,-10,-11] 1012 [1,0,1,0,-7] 1292 [0,0,1,-406,-3149] 6807 [1,-1,1,-4,8] 1134 [0,1,1,-4,6] 1968 [0,1,1,-11,9] 1288 [1,-1,0,7,-2] 1248 [0,-1,1,-1390,20417] 18564 [1,-1,0,-10,13] 3312 [0,0,1,-11,12] 1260 [0,1,1,-2,-8] 1484 [1,1,1,-2,-8] 2398 [0,0,1,-43,-109] 2859 [0,1,1,7,0] 740 [1,1,1,-39,-110] 1530 [1,0,0,-22,-41] 1460 [1,1,1,-2,6] 1252 [0,-1,1,-47159,3957565] 21770 [0,0,1,-101,-391] 2551 [1,-1,1,-9,14] 936 [0,1,1,-40,-112] 2622 [0,0,1,-7,1] 7206 [1,0,0,-7,-2] 1172 [0,-1,1,-40,112] 3104 [1,-1,0,-7,12] 992 [1,0,0,-16,-27] 1908 [0,-1,1,-66,-186] 4786 [1,1,0,-19,-42] 1720 [1,0,1,-3652,84625] 8260 [1,0,1,-36,-85] 1672 [0,-1,1,7,0] 654 [1,0,0,-23,40] 1488 [1,0,1,-664,6523] 4308 [0,1,1,-13,-22] 1372 [0,0,1,-4,-8] 3105 [1,1,1,-77,228] 1584 [1,-1,1,-6,10] 3436 [0,1,1,-7,0] 968 [1,-1,1,-96,384] 4974 [1,-1,1,-76,272] 1332 [0,0,1,5,-6] 3656 [0,1,1,-39,-108] 1916 [1,0,0,6,5] 1722 [1,1,1,7,4] 1608 [1,-1,1,6,2] 2152 [1,1,1,-4,6] 844 [1,-1,0,7,-4] 1294 [0,-1,1,-26,60] 2268 [1,0,0,-5,8] 1210 [1,0,0,-10,9] 1956 [1,0,1,-25,45] 1852 [0,0,1,-1,7] 3930 [0,-1,1,-12,22] 3324 [0,0,1,-716,7374] 4137 [1,-1,1,-43,118] 1250 [1,-1,0,-10,17] 2356 [0,-1,1,-9,16] 2446 [0,1,1,-23,36] 2816 [0,1,1,-54,136] 3080 [0,-1,1,-9202,342845] 30615 [1,1,1,-29,-72] 2208 [0,0,1,-8,-5] 870 [1,0,1,-2,7] 2268 [1,-1,0,-38,-81] 1488 [0,-1,1,3,6] 1176 [0,-1,1,-4,-7] 2778 [1,-1,1,6,-6] 3080 [0,0,1,-1684,-26599] 5358 [1,-1,1,-3,8] 1428 [0,0,1,-14,-19] 1650 [1,-1,0,-7,2] 2098 [0,-1,1,-7,4] 1570 [1,-1,1,-360,-2536] 2484 [0,-1,1,-13,-16] 1421 [0,1,1,-7,8] 1422 [1,0,1,-9,-7] 1984 [1,-1,0,-2,-7] 1122 [1,0,1,-42,99] 2020 [1,1,0,-17,20] 2332 [0,1,1,-41,-116] 1324 [0,1,1,-462,3672] 9320 [1,0,1,3,-7] 1348 [1,1,1,-10,-14] 3520 [1,0,1,-9,5] 1468 [1,0,0,-9,-8] 1920 [0,0,1,-19,-33] 6510 [1,-1,0,-173,-834] 3360 [0,-1,1,1,7] 1455 [0,1,1,4,-6] 2064 [1,0,1,-5,-9] 2096 [1,-1,1,-6,-8] 3712 [0,-1,1,-13,-13] 1742 [0,-1,1,-46,-106] 7258 [0,0,1,-179,-922] 4426 [1,-1,0,-1571,24364] 5548 [0,0,1,-47,124] 2208 [0,1,1,-9,10] 1803 [0,-1,1,-12744,-549515] 33021 [1,1,0,-17,22] 1826 [0,1,1,-4,-10] 3979 [1,0,0,-8,-5] 1448 [1,1,1,7,0] 1364 [1,1,0,-996,11693] 7542 [1,1,0,-19,26] 1600 [0,1,1,-7,-3] 1196 [0,0,1,7,-3] 2292 [0,-1,1,-7,2] 1530 [0,0,1,-4,8] 1262 [0,0,1,-32,69] 1306 [0,1,1,-110,-483] 4606 [1,-1,1,-574,5432] 6364 [0,0,1,-11,-12] 2042 [0,-1,1,-9,-5] 2102 [0,1,1,-41,88] 4502 [1,-1,0,-13,20] 3772 [1,-1,0,-22,45] 1252 [1,-1,1,-57,178] 3924 [0,0,1,-49,132] 8752 [1,0,1,4,7] 1472 [1,-1,0,7,2] 1678 [1,0,0,-7,10] 2254 [0,0,1,2,-8] 1804 [1,1,0,-26,41] 2516 [0,0,1,-8,-4] 1974 [0,-1,1,5,5] 1508 [1,1,1,-939,10684] 8000 [0,1,1,-19,27] 1404 [1,-1,1,-9,-4] 1184 [0,-1,1,-73,266] 1934 [0,1,1,-13,-25] 1869 [1,0,0,-2016,34673] 12838 [1,0,0,1,8] 1478 [1,0,0,-3,8] 1328 [1,0,0,3,8] 1792 [1,1,1,-8,0] 1324 [0,1,1,-15,-27] 2064 [1,-1,0,-34567,-2465042] 31340 [1,-1,1,-12,20] 2276 [1,-1,1,-21,42] 5536 [0,0,1,7,-4] 3036 [0,-1,1,6,4] 2874 [1,-1,0,-5,-8] 1314 [1,-1,1,-1,8] 1440 [1,-1,1,-28,-50] 1464 [1,0,0,-4,-9] 1968 [1,0,0,1,-8] 2432 [1,0,1,-167,813] 3816 [1,1,0,-79,240] 2828 [1,1,0,-9,4] 2724 [1,-1,1,-10,-12] 1456 [1,-1,1,-25,-42] 1710 [0,-1,1,-4,10] 4240 [1,1,0,-6,-13] 1700 [0,0,1,-16,23] 1780 [1,0,0,-12,17] 1374 [1,0,1,-10,-15] 2340 [1,-1,1,-561,5250] 19280 [0,0,1,-166,823] 7886 [0,0,1,-4,-9] 2656 [1,-1,0,-3107,67442] 11028 [0,1,1,8,3] 1770 [0,1,1,-51,124] 2216 [0,-1,1,-498,-4116] 7254 [0,-1,1,8,-2] 2404 [0,0,1,-1,8] 2734 [0,0,1,-8,-3] 1384 [1,1,0,1,-8] 1148 [1,1,1,-74,-276] 2692 [0,1,1,7,7] 1136 [0,-1,1,-9,10] 2502 [0,1,1,7,-3] 2332 [1,1,1,-3,-10] 1284 [1,-1,0,-46,-109] 4052 [0,-1,1,-14,-18] 2720 [0,-1,1,3,7] 2064 [0,1,1,-552,4812] 9056 [1,0,1,-63,185] 3040 [0,1,1,-196815,33541918] 47580 [0,-1,1,-17,-21] 1534 [1,-1,0,-23,-38] 2406 [1,-1,1,-46,-108] 2432 [1,1,1,-20,-44] 3988 [0,-1,1,8,-1] 4236 [1,-1,0,-2,9] 2134 [1,1,1,-8,-6] 1228 [0,1,1,-88,290] 4040 [0,-1,1,-2,9] 5376 [0,0,1,-19,-31] 4162 [0,0,1,-11,11] 2926 [0,0,1,5,7] 4820 [1,-1,0,-35,-72] 3620 [1,1,0,-149,-766] 5328 [0,-1,1,8,-5] 2944 [0,0,1,-64,197] 2596 [0,1,1,-39,82] 1924 [1,-1,1,6,4] 3592 [0,0,1,-49,-132] 13510 [1,1,0,-8,1] 1864 [0,1,1,-25,-57] 1740 [1,-1,0,8,-3] 1296 [0,-1,1,-4,-8] 2686 [1,0,0,-42,101] 3168 [1,-1,0,-19,36] 1692 [0,1,1,3,9] 2816 [0,-1,1,-3,-8] 2511 [1,-1,0,-8,5] 2064 [0,1,1,-11,13] 2355 [1,-1,1,-10,10] 1572 [0,0,1,7,-5] 4729 [1,0,0,-75,244] 3908 [0,1,1,8,5] 4105 [1,1,0,-19,24] 3380 [1,1,1,-8,-2] 2796 [0,-1,1,-51,-125] 3109 [0,0,1,-26,-52] 2728 [0,0,1,-8,12] 1352 [0,1,1,-27,45] 2280 [1,1,1,-60,154] 3114 [0,0,1,-161,786] 8514 [0,1,1,-15,-30] 3466 [1,0,1,-11,-11] 1908 [1,-1,0,-274,-1679] 7920 [1,0,0,8,1] 3168 [1,1,1,-1360,18738] 6496 [0,0,1,-2,-9] 2120 [1,1,1,-630,5824] 4688 [1,1,0,-66,181] 2822 [0,0,1,-22,-39] 5510 [1,1,1,2,-8] 2224 [1,1,1,-5,-12] 1216 [1,1,0,-51,-164] 1814 [0,1,1,-10,-13] 3078 [0,1,1,-57,148] 3136 [0,0,1,-29,-61] 3724 [0,1,1,-115,438] 2564 [0,0,1,2,-9] 5845 [0,-1,1,-29,70] 2576 [1,0,0,-14,17] 2358 [1,-1,1,-15,-20] 3656 [1,0,0,-7,-12] 3090 [1,0,0,2,9] 2916 [1,-1,1,-55,-142] 2820 [1,1,1,-10,4] 2822 [0,-1,1,-36,-72] 3774 [1,-1,0,8,-5] 3380 [0,1,1,-37,-100] 3846 [0,0,1,-15023,-708734] 58892 [0,1,1,-1081,-14047] 15124 [1,-1,0,-8,1] 1678 [1,-1,0,-8,3] 2592 [1,-1,1,-13,-12] 1628 [0,1,1,-9,3] 2796 [0,1,1,6,9] 3669 [1,1,1,5,-6] 1272 [0,1,1,1,9] 2702 [0,-1,1,-7738,264590] 25984 [0,1,1,-4,8] 6096 [0,1,1,1,-9] 1732 [0,-1,1,-74,271] 7254 [1,1,0,3,10] 2884 [0,-1,1,5,-10] 2093 [0,1,1,-10,12] 4410 [1,1,0,-19,-40] 2080 [0,-1,1,-15,-20] 1686 [0,0,1,-16,26] 1554 [0,0,1,-25,47] 2812 [1,0,0,-10,-9] 3072 [1,0,0,-242,1429] 8076 [1,0,1,-1140,-14901] 8088 [0,-1,1,-13,25] 1960 [0,1,1,-348,2386] 7956 [1,0,1,1,9] 4990 [1,1,1,3,10] 2040 [0,-1,1,8,-7] 2984 [0,0,1,-2,9] 1712 [0,0,1,-10,15] 4104 [0,1,1,-5528,-160055] 26560 [1,1,1,-15,14] 2716 [0,-1,1,2,-10] 3752 [1,-1,0,-14,-19] 1820 [0,1,1,5,-7] 3160 [0,-1,1,-4,11] 3673 [0,1,1,-44,99] 14928 [1,1,1,-14,-24] 3168 [0,-1,1,-826,9418] 10952 [0,1,1,-431,3304] 4656 [0,1,1,-6914,-223601] 34944 [0,1,1,-766,-8421] 10608 [1,0,0,2,-9] 2184 [1,1,0,3,-8] 1620 [1,1,1,8,0] 2920 [0,0,1,-1,9] 5636 [1,1,1,-9,10] 2480 [1,0,0,-22,39] 3984 [1,1,1,-59,150] 3634 [0,0,1,4,-9] 2144 [1,1,1,3,-8] 2560 [1,1,0,-788,-8851] 10866 [0,0,1,-7,-12] 13443 [0,-1,1,-6,13] 6655 [0,0,1,-113,462] 5816 [0,-1,1,-10,19] 4647 [1,0,1,-12,-13] 4680 [1,0,1,-11,15] 2438 [0,0,1,-10,-8] 4834 [0,0,1,-28,56] 2938 [0,1,1,-47,109] 2970 [1,1,0,-78,235] 4464 [0,1,1,4,-8] 2740 [1,-1,0,-4,11] 5918 [0,1,1,-10,5] 9054 [1,1,0,-1,-10] 2114 [1,1,1,2,10] 3054 [1,0,0,8,-3] 2970 [0,-1,1,-597,5818] 8766 [0,0,1,-16,-23] 5566 [1,1,0,-8,-5] 3040 [0,-1,1,-7,-10] 3590 [0,-1,1,-25,-40] 2670 [1,-1,1,-69,-202] 5340 [0,-1,1,8,2] 4554 [1,-1,0,-25,54] 2076 [1,-1,1,-6,12] 5880 [1,-1,0,2,9] 1862 [1,0,0,-103,394] 4272 [1,1,0,-52,125] 6588 [0,1,1,-18,-35] 3568 [0,1,1,-273,1648] 3552 [1,-1,1,8,0] 5540 [0,1,1,-6,9] 4089 [1,1,1,-10,-12] 2296 [1,0,1,7,-5] 2388 [1,0,0,-3,-10] 2516 [0,-1,1,-34,88] 8264 [0,-1,1,-3,-9] 1856 [0,0,1,-14,-18] 2582 [1,1,1,-17,18] 2792 [1,1,1,-9,-8] 2792 [0,1,1,-14,18] 6200 [1,-1,1,-15,-16] 1578 [0,0,1,-20,-36] 2245 [1,-1,1,5,-10] 2568 [1,0,0,-58,165] 6740 [1,1,0,9,2] 3358 [0,1,1,-78,-293] 6300 [0,-1,1,-112,-421] 6638 [0,-1,1,5,7] 4608 [1,-1,1,0,-10] 1702 [1,0,1,-5,-11] 3996 [1,-1,0,-1,10] 5454 [1,-1,0,-5,12] 2262 [1,1,0,2,-9] 2750 [0,1,1,-9,-18] 3456 [0,0,1,1,-10] 5232 [1,-1,0,8,3] 1764 [1,0,1,-1368761,-616480979] 151376 [1,-1,0,8,-7] 3480 [0,-1,1,-19,-28] 2634 [0,0,1,2,-10] 2378 [0,1,1,6,10] 5285 [1,-1,1,-9,-12] 5194 [1,-1,0,-12803,560804] 27188 [0,1,1,3,-9] 2028 [1,0,1,8,3] 4464 [1,-1,1,-10,-4] 1976 [0,0,1,-119,-500] 9540 [1,-1,1,-9,2] 2412 [0,-1,1,-9,-2] 3738 [1,1,0,-30,53] 3590 [0,1,1,-314,-2250] 14710 [0,1,1,-2,9] 5493 [1,-1,1,-1,-10] 2176 [1,0,1,-26,-51] 3108 [0,1,1,-2,-11] 5126 [1,1,1,-281,1696] 16296 [0,1,1,-9,-8] 2518 [0,0,1,-17,-29] 6460 [0,1,1,8,8] 4648 [1,-1,1,6,6] 5482 [0,-1,1,-23,52] 4340 [1,-1,0,-19,-26] 9960 [0,1,1,-43,-124] 3312 [1,-1,0,-248,1567] 6528 [1,1,1,4,-8] 3192 [0,-1,1,-9,-12] 4276 [1,-1,0,-43,-98] 9084 [1,1,0,8,-3] 3080 [1,1,1,-18,20] 2970 [0,-1,1,-22,49] 5531 [1,0,1,-15,-25] 3944 [1,1,1,-24,34] 3126 [0,0,1,-23,41] 7278 [1,0,1,-687,-6981] 9192 [1,0,0,-3,10] 4432 [1,1,0,-16,17] 2502 [0,0,1,-19,30] 11176 [1,0,0,-9,2] 3100 [1,1,0,-15,-32] 4034 [1,0,0,1,-10] 2552 [1,0,1,8,-3] 3408 [1,-1,0,-16,-23] 7964 [1,1,1,-1274,16972] 13108 [1,1,1,-185,-1046] 7080 [0,-1,1,-9,7] 3634 [1,1,1,7,10] 3072 [1,-1,1,-48,138] 11472 [1,1,0,-12,-25] 5380 [1,1,1,-32,-84] 2924 [1,0,0,-55,-162] 4344 [0,1,1,-29,52] 2892 [0,-1,1,2,-11] 4233 [0,0,1,5,9] 9451 [1,0,1,-1934,32563] 9944 [1,1,1,-9,-2] 3256 [1,1,0,-16,21] 4112 [1,-1,0,-1,-10] 5792 [1,0,1,-9,-15] 2416 [1,1,0,1,-10] 2674 [0,-1,1,-32,82] 7242 [0,1,1,-29,-72] 3432 [0,0,1,-4,-11] 4312 [1,1,1,-1025,12204] 11488 [0,1,1,8,-4] 4584 [1,0,0,-9,-2] 3792 [1,-1,0,4,9] 2716 [0,1,1,2,-10] 5570 [0,1,1,-3,-12] 3476 [1,-1,0,4,-11] 3188 [1,1,1,-152,658] 6756 [0,0,1,8,-6] 4474 [1,-1,1,-31,72] 4848 [0,0,1,-31,67] 13737 [1,-1,0,-31,74] 7360 [1,0,1,5,-9] 3724 [0,1,1,-52,-163] 14372 [0,1,1,-1333837,-593373237] 150392 [0,1,1,-138,580] 8010 [0,1,1,-83,265] 4952 [1,1,1,9,2] 2616 [0,-1,1,-9,18] 1986 [0,-1,1,1,-11] 3597 [1,0,0,-15,-26] 6314 [1,0,1,-34,-77] 4388 [0,1,1,-18,22] 2972 [0,0,1,-25,49] 9386 [1,1,1,-12,14] 2800 [1,1,1,-34,-90] 3280 [0,0,1,-23,-44] 8494 [0,0,1,-11,9] 4546 [1,-1,1,-181,-890] 5140 [1,1,0,6,-7] 3726 [1,0,1,-6,11] 3588 [1,0,0,-10,-17] 2544 [1,0,1,-2,-11] 2630 [1,-1,1,-30,-56] 8362 [0,-1,1,-4,-10] 6490 [0,-1,1,-9,0] 5000 [0,1,1,-165,763] 5808 [1,-1,1,-949,-11010] 8640 [1,-1,0,-13,-18] 3440 [0,0,1,-8,-14] 3342 [1,1,1,-19,22] 3662 [1,0,0,-12,-13] 2940 [1,-1,0,-26,57] 3250 [1,1,0,-96,325] 3934 [0,1,1,2,11] 4063 [0,0,1,-2,-11] 1955 [0,0,1,-74,245] 4668 [0,1,1,-9,12] 4760 [0,0,1,-22,38] 2964 [1,-1,0,-10,19] 7980 [0,-1,1,-7,-11] 2550 [1,0,0,9,-2] 5344 [0,1,1,9,7] 2300 [1,0,0,-79,264] 6840 [1,-1,1,-24,-40] 2738 [1,1,0,-49,114] 5700 [0,0,1,-275,1755] 9654 [1,-1,0,-3412,-75865] 32056 [1,0,1,6,9] 4250 [1,0,1,-29,55] 3956 [1,1,1,-207,1060] 8136 [0,1,1,-118,-535] 12962 [0,0,1,-211,-1180] 11181 [1,1,1,-142,592] 10438 [1,0,0,-91,-342] 10452 [1,-1,1,8,-8] 2208 [0,0,1,-83,291] 15573 [1,-1,1,-10,-2] 2592 [1,0,0,-1367,-19568] 21468 [1,1,0,-34,63] 5144 [0,1,1,-9,-5] 3150 [0,-1,1,4,9] 5894 [0,0,1,-11,-18] 3460 [0,0,1,-1901,31902] 15852 [0,1,1,-13,17] 2920 [0,-1,1,-1464925,682939109] 231336 [0,-1,1,8,-10] 8973 [0,1,1,-87,285] 7824 [0,1,1,-82,-315] 15702 [1,0,1,1,-11] 5328 [1,1,1,-5,-14] 5680 [1,1,1,-17,-32] 4928 [0,-1,1,-18,34] 6922 [0,-1,1,-103,-370] 5790 [0,-1,1,-9,4] 2860 [0,1,1,-160,-835] 19158 [1,1,0,-10,3] 2796 [0,1,1,-34,65] 9238 [1,-1,1,-10,6] 3636 [0,-1,1,3,-12] 2004 [1,-1,1,9,-2] 10008 [1,1,1,4,12] 2872 [0,-1,1,-9,3] 7826 [1,1,1,-7,10] 4938 [0,0,1,-18800,992167] 23764 [0,1,1,-38,-105] 8264 [0,-1,1,-17,31] 5538 [0,0,1,-40,-97] 2584 [0,1,1,1,-11] 2852 [0,-1,1,-13,-11] 2156 [0,-1,1,-661361,207237598] 148678 [0,1,1,-364,2555] 20210 [1,-1,1,9,-4] 4034 [1,0,0,-73,-246] 5580 [1,0,1,-12,9] 3672 [1,-1,0,-11,12] 3112 [1,1,1,-217,1140] 7150 [1,1,0,-11,14] 4102 [1,1,0,-18,25] 7726 [0,0,1,-7,13] 12738 [0,-1,1,-11,13] 4320 [0,0,1,-2,11] 7176 [1,1,1,-19,-38] 4236 [0,-1,1,-13,26] 3475 [0,0,1,-14,-17] 2930 [1,-1,1,-102,420] 23712 [0,-1,1,-11,-15] 2308 [1,0,0,-17,-26] 5316 [1,-1,1,2,-12] 3200 [1,-1,1,-10,-14] 1932 [1,0,1,3,11] 4446 [1,0,1,-10,-17] 4462 [0,-1,1,-6,-11] 10746 [0,-1,1,-22,-35] 7456 [0,-1,1,-15,25] 3558 [1,-1,0,-58,185] 22764 [0,-1,1,3,10] 5484 [0,0,1,-17,29] 5267 [1,-1,0,-14,27] 2548 [1,1,0,-9,-4] 5388 [0,1,1,-7,11] 3622 [0,1,1,4,12] 5372 [0,-1,1,8,5] 6000 [0,1,1,-117,450] 7140 [0,0,1,-4,-12] 8021 [0,1,1,9,-3] 6257 [0,-1,1,-12,24] 6360 [1,0,1,-79499,8620909] 133204 [0,0,1,-25,-47] 9444 [1,1,1,-19,-42] 3650 [0,-1,1,-28,-50] 5355 [0,0,1,-10,4] 6530 [0,-1,1,-149,-653] 9923 [1,1,0,-1,-12] 4574 [1,-1,1,5,-12] 2368 [1,0,0,-1100,13951] 29272 [1,0,0,-78,259] 8836 [0,1,1,-88,-349] 11218 [0,-1,1,-13,19] 2860 [1,1,1,-4,10] 3048 [0,1,1,3,12] 3840 [0,0,1,-19,-30] 12280 [1,1,0,9,10] 3786 [0,-1,1,-172,928] 8536 [1,-1,0,-2,-11] 3824 [0,0,1,-5,12] 3528 [1,0,1,-16,19] 5114 [0,-1,1,1,-12] 3620 [1,-1,0,-55,172] 10788 [0,0,1,-10,-17] 6968 [0,1,1,-30,55] 9696 [1,-1,0,2,11] 6594 [1,-1,0,-880,-9831] 12458 [1,-1,0,5,-12] 7166 [1,0,1,-25,43] 3676 [1,0,1,-57,159] 5358 [1,-1,0,-2858,59529] 13790 [1,1,1,-99,338] 10480 [1,1,1,-8,-18] 4932 [0,-1,1,8,-11] 4769 [0,1,1,-35,-92] 4420 [0,0,1,-16,-22] 6198 [0,-1,1,10,-3] 7989 [0,0,1,5,-11] 13109 [0,-1,1,-3,-11] 4312 [0,-1,1,7,7] 3141 [1,-1,1,-94,372] 8240 [1,0,0,-10,3] 6320 [1,0,1,-1504,-22565] 10984 [0,1,1,-12,16] 6254 [1,-1,0,-778,-8161] 26520 [1,0,1,-2192,-39671] 14440 [0,0,1,-11,18] 6986 [0,0,1,-551,4978] 19790 [0,-1,1,10,-5] 5224 [0,0,1,4,11] 3420 [1,-1,1,-34,-68] 4352 [0,1,1,8,-6] 6237 [1,0,1,7,9] 4364 [1,0,1,4,-11] 3554 [0,-1,1,-141,-600] 4821 [1,-1,0,-1,12] 3182 [1,0,1,-411,3167] 12828 [1,0,0,-59,170] 4782 [1,-1,1,-12,22] 7768 [1,0,0,-42,-109] 6460 [0,1,1,-44,-128] 8690 [1,0,0,4,-11] 3670 [1,1,1,-4,-14] 5170 [0,-1,1,-12,-8] 13298 [0,0,1,1,-12] 4184 [0,0,1,-61666,5894090] 52541 [0,0,1,-7,-14] 15719 [0,0,1,-29,61] 6573 [1,1,1,-510,-4646] 9600 [0,0,1,-1331,18690] 20520 [0,1,1,3,-11] 2982 [1,0,0,-83,284] 6028 [1,-1,0,10,-3] 4816 [1,-1,1,6,-12] 11890 [1,1,0,-11,-14] 4076 [0,0,1,-10,-3] 5984 [1,1,0,-1928,-33401] 17028 [1,1,0,-898,-10741] 13980 [0,-1,1,-22,46] 6532 [1,1,0,-229,-1434] 14248 [0,1,1,-23,37] 2974 [1,-1,1,-13,24] 6078 [1,1,0,-31,54] 5892 [1,0,0,-12,19] 6576 [0,1,1,10,0] 17060 [0,0,1,-10,2] 2562 [1,0,0,-31,-68] 3624 [1,1,0,-12,7] 3192 [0,1,1,1,12] 3128 [1,-1,1,-28,64] 6066 [0,0,1,8,8] 2824 [1,-1,1,-7086,-227802] 86644 [1,1,1,-3749,-89916] 30132 [1,1,0,-822,8737] 16506 [0,1,1,1,-12] 3540 [1,0,0,-82,-293] 5230 [0,0,1,-11,7] 5246 [1,0,1,-15,23] 5304 [1,1,0,-6,11] 3882 [0,1,1,-11,-23] 3406 [1,0,1,9,5] 4420 [0,-1,1,10,0] 5312 [0,-1,1,-10,-1] 8568 [1,-1,1,-39,-84] 10214 [1,0,0,-455,3698] 8040 [0,1,1,-17,19] 5374 [1,1,0,-39,80] 7656 [1,1,1,10,4] 5208 [0,-1,1,9,4] 8215 [0,-1,1,-59,-156] 4800 [0,-1,1,-10,7] 8838 [1,0,1,-7,13] 5184 [0,1,1,-3114,65857] 31785 [1,1,0,-8,-19] 4216 [1,1,1,1,-12] 4740 [1,-1,0,-14,-13] 5004 [0,0,1,-1,12] 13296 [0,0,1,1,12] 6084 [1,1,0,-10,-9] 4108 [1,0,1,-16,25] 3774 [0,0,1,-184,-961] 6344 [0,1,1,4,13] 10908 [1,-1,0,8,7] 9040 [0,0,1,-37,-86] 14158 [0,0,1,10,-2] 2992 [1,1,0,10,-1] 6286 [1,0,1,-17,27] 6978 [1,-1,0,-3917,95344] 30120 [1,1,1,-14,10] 6060 [0,1,1,-85,275] 13248 [0,1,1,-11,4] 3014 [0,0,1,-73,-240] 18372 [0,1,1,-10,-7] 9180 [0,0,1,8,-9] 9464 [1,-1,0,7,-12] 3308 [0,1,1,2,-12] 6746 [0,-1,1,-12,15] 7968 [0,-1,1,-925,-10526] 14864 [0,-1,1,10,-8] 12804 [1,-1,0,-1174,-15193] 35712 [1,-1,1,-27,58] 10456 [0,-1,1,-108,-398] 15962 [0,-1,1,-2,-12] 7024 [1,1,0,-12335,-532472] 28698 [1,1,0,-36,-101] 3994 [0,1,1,-18,-39] 8885 [0,1,1,-3,-14] 3492 [0,1,1,-50,-155] 12450 [0,1,1,6,13] 6095 [1,-1,0,-11,10] 3186 [1,-1,1,-240,1488] 17372 [1,0,0,-5342,-150727] 22416 [1,0,0,-156,737] 8460 [0,0,1,7,10] 5726 [0,1,1,-62,-210] 12496 [0,1,1,-12672,-553301] 95452 [1,0,1,-17,21] 8728 [0,1,1,-86,-338] 10725 [0,1,1,-74,222] 12740 [0,-1,1,1,12] 5518 [0,0,1,-8,15] 3712 [0,0,1,-5,13] 7534 [1,1,1,-69,-250] 7844 [1,1,0,-1,12] 9916 [1,0,0,-12,-11] 5736 [0,0,1,-14,-24] 5283 [0,1,1,-9,-20] 9735 [1,-1,0,2,-13] 9312 [0,-1,1,-84,-270] 7898 [1,-1,1,-289,-1816] 12308 [1,1,0,-37,72] 7896 [1,1,1,-35,64] 6068 [1,1,0,5,14] 3710 [0,-1,1,-17,36] 4256 [1,1,0,-692,-7303] 20832 [1,0,1,-57,-169] 9408 [1,0,0,-14,-25] 4140 [1,-1,0,-31,76] 12428 [0,-1,1,-10,5] 10598 [0,-1,1,-109,476] 8280 [0,0,1,-41,100] 7136 [1,-1,0,10,-7] 7740 [1,1,1,2,-12] 3516 [0,1,1,-768,7941] 20568 [1,0,0,-2068523,1144915064] 330396 [1,0,1,-23,37] 8384 [1,0,1,-19,-35] 6808 [0,0,1,-128,557] 7910 [0,0,1,-29,-59] 8862 [0,1,1,-104,375] 18338 [1,-1,1,-19,38] 3964 [1,-1,1,-9,18] 8748 [0,-1,1,-7,17] 5338 [0,1,1,-13,18] 4353 [0,0,1,-1,-13] 5819 [0,-1,1,-10,4] 5328 [0,1,1,-62,168] 10580 [0,0,1,-13,-13] 5834 [0,1,1,-10,-3] 7578 [1,0,1,-9,15] 4308 [0,-1,1,-52,-128] 15070 [1,0,0,-2,-13] 4356 [0,1,1,3,-12] 7500 [0,-1,1,-14,21] 7554 [1,1,0,-14,-23] 3888 [0,0,1,-11,-6] 5782 [0,-1,1,-14,-12] 12726 [0,-1,1,-313,2239] 9660 [0,1,1,7,13] 3802 [1,0,0,-18,25] 4428 [1,1,0,-5,-16] 5004 [1,0,0,-6197,187252] 52292 [1,0,1,-12,-21] 6198 [1,1,0,-14,11] 7284 [0,0,1,-20,-37] 3885 [0,-1,1,-19,-29] 3970 [0,1,1,-74,-272] 7572 [1,0,0,-5,-14] 11064 [1,0,1,-12,-9] 5020 [0,1,1,-4,-15] 8270 [0,1,1,-18,21] 12266 [0,1,1,-11,-11] 2812 [0,-1,1,6,-14] 7124 [0,1,1,-11,3] 8196 [0,1,1,-2,12] 11712 [1,-1,0,-31,-58] 13208 [0,-1,1,-39,-83] 6928 [0,-1,1,-882,-9794] 42900 [0,-1,1,-4887,133138] 29370 [0,1,1,-6,12] 9656 [1,0,1,-11,1] 3120 [1,1,0,-182,-1025] 17256 [1,1,1,6,14] 9360 [1,1,1,-2,12] 5536 [1,-1,0,-125,-508] 6898 [1,0,0,8,-9] 5544 [0,0,1,-35,-81] 7326 [0,0,1,-43,-108] 22188 [0,1,1,1,13] 5752 [1,0,1,-11,-3] 4576 [0,-1,1,-136,658] 10128 [1,0,0,-520,4521] 31950 [1,1,0,-15,-26] 4826 [0,0,1,-13,22] 19605 [0,0,1,-11,5] 6094 [1,0,0,-470,3883] 19162 [1,-1,0,10,3] 5750 [1,1,0,-14,19] 4836 [0,-1,1,-12,25] 11625 [0,1,1,-70,203] 13824 [1,0,1,10,1] 6696 [1,1,1,-7,-18] 4992 [1,1,0,-110,-493] 9144 [0,1,1,-12,6] 9760 [0,1,1,-867,-10121] 14968 [1,1,1,-117,-536] 12326 [1,0,1,-5,13] 4560 [0,0,1,-86,307] 7350 [0,0,1,4,-13] 3204 [0,1,1,-3,12] 5188 [1,0,0,-2,13] 6110 [1,1,0,-24,-55] 6260 [1,1,0,-14,-31] 8028 [1,0,0,-507,4352] 13368 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[0,-1,1,10,4] 7712 [1,0,0,-8,-17] 6762 [0,1,1,-8591,303642] 42500 [0,0,1,-47,123] 10606 [0,1,1,-13,8] 7176 [1,-1,0,-17362,-876211] 115090 [1,-1,0,8,9] 4754 [0,1,1,11,0] 7176 [0,-1,1,8,-14] 8550 [0,0,1,-16,28] 16420 [1,-1,0,5,12] 5986 [0,-1,1,-10,22] 19800 [0,1,1,-9,-21] 8382 [1,-1,1,8,8] 6910 [0,0,1,-11,2] 8552 [1,1,0,-9,14] 8836 [1,0,0,-26,47] 6800 [1,-1,1,-16,-24] 6490 [1,0,1,-5,-15] 8014 [1,-1,1,9,6] 9600 [0,0,1,5,13] 22981 [0,-1,1,-22,-31] 12894 [1,-1,0,-266,-1605] 11888 [1,1,0,-138,-685] 26064 [1,-1,1,-10,-16] 4942 [1,1,1,-8,-20] 6580 [0,-1,1,-3,15] 5470 [0,1,1,-118,456] 15172 [0,0,1,-11,1] 6954 [1,1,0,-21,32] 10976 [0,-1,1,-34,-65] 12570 [0,1,1,-17,18] 6892 [1,1,0,-44,95] 10504 [0,0,1,-32,68] 4624 [1,0,1,-13,-11] 4044 [1,0,1,-46,115] 5560 [0,1,1,-18,27] 11498 [0,0,1,-268,-1689] 29602 [0,0,1,11,-1] 7428 [1,0,1,9,9] 6058 [0,1,1,10,-5] 16475 [0,-1,1,-30,-56] 17445 [0,1,1,-15,13] 4674 [0,-1,1,-12,-18] 14388 [0,1,1,-45,103] 6300 [0,1,1,11,-1] 7056 [0,1,1,-434,-3629] 20956 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[0,1,1,-111,-492] 290844 [1,-1,1,5,46] 87196 [0,1,1,25,0] 296264 [0,0,1,-54014,4831785] 1224208 [1,0,0,-56,-173] 146046 [1,0,1,23,21] 131902 [1,-1,0,-13,-48] 175948 [1,-1,0,-121,546] 181358 [1,-1,1,18,-42] 247530 [1,0,0,-81,278] 353156 [0,-1,1,-25,19] 79016 [0,-1,1,-79,-250] 160500 [1,1,1,-112,412] 209368 [0,-1,1,20,-41] 135013 [0,1,1,-29,-48] 91540 [0,-1,1,10,-50] 157730 [0,-1,1,25,0] 83464 [1,0,0,-79,268] 158928 [1,1,0,-36,55] 87136 [1,0,0,-34,-93] 164048 [0,-1,1,-259,-1522] 184104 [0,0,1,-92,336] 182904 [1,-1,1,-42,102] 145952 [0,-1,1,-9,52] 194280 [0,-1,1,17,-46] 150454 [0,1,1,-102,367] 245746 [0,-1,1,7,-50] 105638 [0,1,1,-18,-63] 173036 [1,0,1,-106,-429] 129814 [1,-1,1,-78,-240] 180360 [0,0,1,-8,-49] 113143 [0,-1,1,-30,-33] 633406 [0,0,1,-13,51] 805754 [0,0,1,-82,-282] 196728 [0,-1,1,15,-48] 123172 [1,-1,0,-202,-1055] 346208 [1,0,1,-55,-167] 105576 [1,1,1,21,-22] 156048 [1,0,0,-61,172] 114972 [0,1,1,1,48] 186300 [1,-1,1,15,38] 102120 [0,-1,1,-32,96] 153096 [0,-1,1,-51,150] 133692 [0,1,1,-532,4550] 468592 [0,-1,1,-27,-18] 81992 [1,-1,0,20,29] 198662 [1,1,0,-524,4405] 210214 [1,1,1,-1334,-19310] 303212 [0,1,1,-104,378] 337088 [0,0,1,17,-40] 141720 [1,-1,1,-13,54] 176886 [0,1,1,8,-45] 222444 [0,0,1,16,41] 66295 [1,1,1,-4,-50] 108298 [0,-1,1,23,-32] 94110 [0,1,1,-15,48] 98132 [1,1,1,-829,8842] 356660 [0,-1,1,-1571,24498] 259488 [0,-1,1,-76,-227] 190320 [1,-1,1,-27,78] 272088 [0,0,1,-25,2] 144760 [1,0,1,13,45] 115408 [0,-1,1,-76,-236] 260025 [1,1,0,25,22] 121056 [1,0,1,-4267,-107621] 415836 [0,1,1,-24,-75] 268410 [1,1,1,-5,46] 192762 [1,1,0,5,-46] 75956 [0,-1,1,-67,-185] 133602 [0,1,1,18,-33] 160609 [1,0,0,25,4] 128112 [1,-1,1,-1,48] 103952 [0,0,1,-74,240] 158572 [0,-1,1,-34,-80] 186056 [1,-1,0,-101,420] 159612 [0,0,1,-112,-459] 329976 From mwatkins@msri.org Wed Dec 13 00:07:28 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id eBD57lJ10997 for ; Wed, 13 Dec 2000 00:07:48 -0500 Received: from ep2.msri.org (ep2.msri.org [198.129.64.227]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id eBD57X400697 for ; Wed, 13 Dec 2000 00:07:34 -0500 (EST) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep2.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 14648W-0003gW-00; Tue, 12 Dec 2000 21:07:28 -0800 Received: from rebirth.msri.org [198.129.65.36] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 14648W-0006d7-00; Tue, 12 Dec 2000 21:07:28 -0800 Received: from mwatkins by rebirth.msri.org with local (Exim 3.12 #1 (Debian)) id 14648W-0007CS-00; Tue, 12 Dec 2000 21:07:28 -0800 Subject: Fifty Sha=25 Curves In-Reply-To: <0012042003280K.20459@modular> from William A Stein at "Dec 4, 2000 08:03:28 pm" To: was@math.harvard.edu Date: Tue, 12 Dec 2000 21:07:28 -0800 (PST) X-Mailer: ELM [version 2.4ME+ PL66 (25)] MIME-Version: 1.0 Content-Type: text/plain; charset=US-ASCII Content-Transfer-Encoding: 7bit Message-Id: From: Mark Watkins Status: RO X-Status: O Here are fifty rank=0 sha=25 curves with their modular degree, (assuming c=1) from the Brumer/McGuinness list. [0,-1,1,-12744,-549515] 33021 [0,-1,1,-104,-377] 100049 [0,0,1,-17866,-919156] 402087 [1,1,0,-223,-1380] 252882 [0,-1,1,-6,-41] 318333 [1,-1,0,-307,-1996] 308486 [0,0,1,-7,-47] 413711 [0,0,1,-139,-629] 510606 [0,-1,1,24,-41] 721341 [1,1,0,-1322,-19063] 449580 * [0,1,1,-15890,-776285] 3804750 * [0,0,1,-13,-59] 1160563 [0,-1,1,-106,-391] 553819 [1,-1,0,-1078,-13357] 1400504 [1,1,0,-13759,-626964] 3212752 [0,0,1,-8194,-285491] 2187358 [1,1,0,-1110,-14707] 722808 [1,0,1,-7412,-246207] 1306204 [0,-1,1,-216,-1151] 1270434 [0,0,1,-733,-7639] 1569133 [1,1,0,-8529,-306758] 1435660 * [0,-1,1,-221,-1196] 1049189 [0,0,1,29,-41] 1127307 [0,1,1,-8598,-309747] 2873727 [0,0,1,-166,-827] 1002351 [1,-1,0,-51766,-4520389] 7738158 [0,1,1,-4756,-127843] 4730187 [0,0,1,-7,-89] 5960437 [0,0,1,-315422,-68184567] 11204926 [0,1,1,-647551,-200782822] 38717959 [0,-1,1,-97315,-11652328] 11274961 [0,0,1,-491,-4189] 1660781 [0,0,1,-1688,-26694] 1953625 * [1,1,0,-818,-9355] 1499872 [1,1,1,-1010247,-391252366] 25024278 [0,-1,1,-314,-2043] 2547045 * [1,1,0,-1947,-33892] 1954730 * [0,1,1,-16512,-822201] 6800715 * [0,0,1,-436,-3506] 2977965 * [0,1,1,4,-107] 1841393 [0,0,1,-43,1] 3575318 [0,-1,1,-13860,-623451] 8293902 [0,-1,1,-58084,-5368751] 10663517 [1,-1,0,-109688,-13955177] 10383408 [1,1,0,-7101,-233302] 7050968 [0,0,1,41,-59] 3204619 [1,1,0,-43,-180] 1909334 [0,-1,1,-342,-2327] 3206755 * [0,0,1,-5585,-160651] 4794910 * [0,1,1,-177053,-28734074] 12024909 From coleman@math.berkeley.edu Wed Dec 13 14:55:46 2000 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id eBDJu5J11828 for ; Wed, 13 Dec 2000 14:56:05 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id eBDJtq427693 for ; Wed, 13 Dec 2000 14:55:52 -0500 (EST) Received: from [136.152.194.62] (as3-1-140.HIP.Berkeley.EDU [136.152.194.62]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id LAA03320; Wed, 13 Dec 2000 11:55:48 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: References: Date: Wed, 13 Dec 2000 11:55:46 -0800 To: =?iso-8859-1?Q?Lo=EFc?= Merel From: "Robert F. Coleman" Subject: Re: Infinite slope Cc: was@math.harvard.edu Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O > > I have a question about the corollary of the main theorem. Did you >consider proving this using more elementary techniques >(Eichler-Shimura isomorphism etc)? I'm not quite sure what you mean. I didn't try proving that one could approximate twist by finite slope forms by other techniques. One might try to generalize Katz's "A result on modular forms mod p" to modular forms mod p^n and then lift which I don't know how to do. Is this what you had in mind? -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Wed Jan 3 00:17:52 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f035Ise04002 for ; Wed, 3 Jan 2001 00:18:54 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f035Ibm25739 for ; Wed, 3 Jan 2001 00:18:37 -0500 (EST) Received: from [136.152.194.6] (as3-1-4.HIP.Berkeley.EDU [136.152.193.182]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id VAA22054 for ; Tue, 2 Jan 2001 21:18:36 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010218412305.08428@form> References: <01010218412305.08428@form> Date: Tue, 2 Jan 2001 21:17:52 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Re: Hatada Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: A > > >2. His space S^- is the image by integration of cusp forms in an >appropriate Euclidean space R^n. He gives the relations that >cut out S^- in R^n. The modular symbols theory proves that the >equations he gives are enough to cut out the image of cusp forms, >and not something bigger. > Is this "modular symbols theory" explained in your notes? -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Thu Jan 4 14:32:45 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f04JX6t06078 for ; Thu, 4 Jan 2001 14:33:06 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f04JWpm25833 for ; Thu, 4 Jan 2001 14:32:51 -0500 (EST) Received: from [136.152.193.228] (as3-1-50.HIP.Berkeley.EDU [136.152.193.228]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id LAA15929 for ; Thu, 4 Jan 2001 11:32:49 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <00122616201611.01632@form> References: <00122616201611.01632@form> Date: Thu, 4 Jan 2001 11:32:45 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Re: Systems of mod-9 eigenvalues of level 1 Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O >Robert, > >I did a computation that suggests the followin conjecture: > >-------------------------------------------------------------- >CONJECTURE: Consider the set of all normalized eigenforms >F(q) = sum a_n q^n on SL_2(Z) with all a_n in Z_3. Then >the set of mod-9 reduction { F(q) mod 3^2 } has cardinality five. >The five elements are: > > q + 6*q^2 + 4*q^4 + 3*q^5 + 5*q^7 + 3*q^8 + 6*q^11 + 8*q^13 + 3*q^14 + > 7*q^16 + 2*q^19 + 3*q^20 + 3*q^23 + 7*q^25 + 3*q^26 + 2*q^28 >+ 6*q^29 + > 8*q^31 + 6*q^35 + 2*q^37 + 3*q^38 + O(q^40), > q + 3*q^2 + 7*q^4 + 6*q^5 + 8*q^7 + 6*q^8 + 3*q^11 + 5*q^13 + 6*q^14 + > 4*q^16 + 2*q^19 + 6*q^20 + 6*q^23 + 4*q^25 + 6*q^26 + 2*q^28 >+ 3*q^29 + > 5*q^31 + 3*q^35 + 2*q^37 + 6*q^38 + O(q^40), > q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + >2*q^31 + 2*q^37 + > O(q^40), > q + 3*q^2 + 4*q^4 + 6*q^5 + 5*q^7 + 6*q^8 + 3*q^11 + 8*q^13 + 6*q^14 + > 7*q^16 + 2*q^19 + 6*q^20 + 6*q^23 + 7*q^25 + 6*q^26 + 2*q^28 >+ 3*q^29 + > 8*q^31 + 3*q^35 + 2*q^37 + 6*q^38 + O(q^40), > q + 6*q^2 + 7*q^4 + 3*q^5 + 8*q^7 + 3*q^8 + 6*q^11 + 5*q^13 + 3*q^14 + > 4*q^16 + 2*q^19 + 3*q^20 + 3*q^23 + 4*q^25 + 3*q^26 + 2*q^28 >+ 6*q^29 + > 5*q^31 + 6*q^35 + 2*q^37 + 3*q^38 + O(q^40) > >-------------------------------------------------------------- > >For comparison, the mod-9 q-expansion of the weight-2 form on X_0(27) is: > > q + 7*q^4 + 8*q^7 + 5*q^13 + 4*q^16 + 2*q^19 + > 4*q^25 + 2*q^28 + 5*q^31 + 2*q^37 + O(q^40) > >The only one that looks at all similar is > > q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + >2*q^31 + 2*q^37 + ... > >which comes S_{16}(SL_2(Z)). > > Lookinmg at the two forms >F(q) = q + q^4 + 2*q^7 + 2*q^13 + q^16 + 2*q^19 + q^25 + 2*q^28 + >2*q^31 + 2*q^37 + O(q^40), and >G(q) = q + 7*q^4 + 8*q^7 + 5*q^13 + 4*q^16 + 2*q^19 + > 4*q^25 + 2*q^28 + 5*q^31 + 2*q^37 + O(q^40) It looks like one is a twist of the other. Namely if a_n are the coefficients of F(q) and b_n are the coefficients of G(q), b_n \con \chi(n)a_n (9) where \chi(n) = n^{2} \mod 9. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Fri Jan 5 12:43:03 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f05HhT401207 for ; Fri, 5 Jan 2001 12:43:29 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f05Hh6m04331 for ; Fri, 5 Jan 2001 12:43:07 -0500 (EST) Received: from [136.152.194.41] (as3-1-119.HIP.Berkeley.EDU [136.152.194.41]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id JAA12532 for ; Fri, 5 Jan 2001 09:43:05 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: References: Date: Fri, 5 Jan 2001 09:43:03 -0800 To: "William A. Stein" From: "Robert F. Coleman" Subject: Re: if so... Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: A > > If we change our definition of approximate. > >But, you proved that twists of finite slope forms can be approximated by >finite slope forms. If we can show that, modulo 9, the form on X_0(27) is >equal to a twist of a form of level 1, then we can use your theorem to >approximate the twist of the form of level 1 modulo 9 by a finite-slope >form. I don't see why a change in definition is needed. > >William But this is a twist by an infinite order character, the square of the cyclotomic character. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Fri Jan 5 13:16:13 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f05IGh401261 for ; Fri, 5 Jan 2001 13:16:43 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f05IGJm06143 for ; Fri, 5 Jan 2001 13:16:20 -0500 (EST) Received: from [136.152.194.41] (as3-1-119.HIP.Berkeley.EDU [136.152.194.41]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id KAA14600 for ; Fri, 5 Jan 2001 10:16:17 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010515082807.15849@form> References: <01010515082807.15849@form> Date: Fri, 5 Jan 2001 10:16:13 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Re: if so... Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: A Let F_27 be the weight 2 form on X_0(27) and G the weight 16 form of level 1. It seems like \theta^2G\con F_27\mod 27. The next question is, is there a finite slope form H such that \theta^2H\con F_27\mod 81. One should first look at forms of weight 52=\phi(81)-2 (16=\phi(27)-2). z -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Fri Jan 5 18:18:00 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f05NIU401967 for ; Fri, 5 Jan 2001 18:18:30 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f05NI7m17596 for ; Fri, 5 Jan 2001 18:18:07 -0500 (EST) Received: from [169.229.58.143] (robert-1.Math.Berkeley.EDU [169.229.58.143]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id PAA05070 for ; Fri, 5 Jan 2001 15:18:06 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010519224301.18287@form> References: <01010519224301.18287@form> Date: Fri, 5 Jan 2001 15:18:00 -0800 To: was@math.harvard.edu From: Robert Coleman Subject: Re: if so... Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O >On Fri, 05 Jan 2001, you wrote: >> >But, you proved that twists of finite slope forms can be approximated by >> >finite slope forms. If we can show that, modulo 9, the form on X_0(27) is >> >equal to a twist of a form of level 1, then we can use your theorem to >> >approximate the twist of the form of level 1 modulo 9 by a finite-slope >> >form. I don't see why a change in definition is needed. >> >> But this is a twist by an infinite order character, the square of the >> cyclotomic character. > >I sort of disagree, though probably only in the way I think about chi >and not with your conclusion. I.e., I am going to demonstrate how >naive I am about p-adic characters today. For me, the twist is by a >character chi of order 3. You defined chi as the map > > Z ---> Z/9Z > >sending > > n to (n mod 9)^2. > >Equivalently, chi is defined by a map > > (Z/9Z)^* ---> {1,omega,omega^2}, where omega^3=1, > >so chi is a mod-9 Dirichlet character of order 3. The reason that the >order is 3 is because (Z/9Z)^* is cyclic of order phi(9)=6 and >squaring takes a cyclic group order 6 onto a cyclic group of >order 3. > >Of course I agree with you that the approximation Theorem 1.1 from our >paper doesn't apply. However, I think it is because Theorem 1.1 only >applies to twists by powers of the mod p cyclotomic character > (Z/pZ)^* -----> Z_p^*. >The order 3 character chi above is not a mod 3 character; instead, >it's a mod 9 character. > >In any case, can we obtain lots of examples of infinite-slope forms >from finite-slope forms in this way? Simply take a finite-slope >p-adic form F and twist it by a character mod p^r, for some r>= 1. >The resulting form will have infinite slope. > > QUESTION (as you suggested when you said "modify the definition"): > Can every infinite slope form be approximated by arbitrary twists > of finite-slope forms? > >The Theorem 1.1 of our paper says nothing about this question because >the conclusion is automatic for forms constructed in this way. Do we >have any interesting examples? What do you think? > > -- William I am thinking the character \chi from Z_9^* into Z_9^* which takes a to a^2. If \sum anq^n is the form \sum_{n,p)=1}\chi(n)n^2a^n is the new form. I am not making mod 9 forms. See Mazur's paper on the infinite fern. I am writing a setion on this for our paper. The questions are, what is the topological closure of the "modular locus" and what is the closure of the "finite slope modular locus" What part of the former is contained in the latter AND DITTO if you allow twisting? From coleman@math.berkeley.edu Fri Jan 5 18:57:42 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f05NwC402002 for ; Fri, 5 Jan 2001 18:58:12 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f05Nvnm18365 for ; Fri, 5 Jan 2001 18:57:49 -0500 (EST) Received: from [169.229.58.143] (robert-1.Math.Berkeley.EDU [169.229.58.143]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id PAA07306 for ; Fri, 5 Jan 2001 15:57:48 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010520163307.18287@form> References: <01010515082807.15849@form> <01010520163307.18287@form> Date: Fri, 5 Jan 2001 15:57:42 -0800 To: was@math.harvard.edu From: Robert Coleman Subject: Re: I'm puzzled by 52. Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: A >If F_{27} = sum a_n q^n and G = sum b_n q^n, then I think > > a_7 / b_7 = 76 (mod 81) > a_{13} / b_{13} = 34, > a_{19} / b_{19} = 64, > a_{31} / b_{31} = 16, > a_{37} / b_{37} = 19. > > Here's a bizarre observation. In all the above cases, a_p/b_p*p^2 = 28 mod 81. From coleman@math.berkeley.edu Sat Jan 6 12:42:20 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f06Hgk403095 for ; Sat, 6 Jan 2001 12:42:46 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f06HgPm09952 for ; Sat, 6 Jan 2001 12:42:25 -0500 (EST) Received: from [136.152.194.13] (as3-1-91.HIP.Berkeley.EDU [136.152.194.13]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id JAA02803 for ; Sat, 6 Jan 2001 09:42:23 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010614302107.00882@form> References: <01010515082807.15849@form> <01010520163307.18287@form> <01010614302107.00882@form> Date: Sat, 6 Jan 2001 09:42:20 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Re: I'm puzzled by 52. Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: A >On Fri, 05 Jan 2001, you wrote: >> >If F_{27} = sum a_n q^n and G = sum b_n q^n, then I think >> > >> > a_7 / b_7 = 76 (mod 81) >> > a_{13} / b_{13} = 34, >> > a_{19} / b_{19} = 64, >> > a_{31} / b_{31} = 16, >> > a_{37} / b_{37} = 19. >> >> Here's a bizarre observation. In all the above cases, >> > > a_p/b_p*p^2 = 28 mod 81. > >This doesn't continue forever. Here's what happens for p<500. >When p = 2 mod 3, we have a_p = 0. >When p = 61, 67, 73, 103, 151, 193, 271, 307, 367, 439, 499, we have > > a_p/b_p*p^2 = 1 mod 81. > >For the remaining p<500, we have > > a_p/b_p*p^2 = 28 mod 81. > >I don't know what is special about the set of primes > > { 61, 67, 73, 103, 151, 193, 271, 307, 367, 439, 499, ... } > >-- William There is a one out of three chance that a_p/b_p*p^2 = 28 mod 81.since we (at least I) believe a_p/b_p*p^2 = 1 mod 27 (It would be nice to prove this.) so I don't think what we have found means anything. The next easy (I hope) thing to look for is a finite slope at 7 form congruent to the weight 2 form F_(49) on X_0(49) mod 49.. I looked at the forms of weight 40 but didn't see any. I would also like to see how you do the search since my Pari skills are limited and my magma skills are non-existent. -- Robert http://math.berkeley.edu/~coleman From jdmitrikessler@yahoo.com Sat Jan 6 18:36:49 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f06NbC403564 for ; Sat, 6 Jan 2001 18:37:12 -0500 Received: from web9008.mail.yahoo.com (web9008.mail.yahoo.com [216.136.128.170]) by abel.math.harvard.edu (8.11.0/8.11.0) with SMTP id f06Naom15986 for ; Sat, 6 Jan 2001 18:36:50 -0500 (EST) Message-ID: <20010106233649.99236.qmail@web9008.mail.yahoo.com> Received: from [209.79.182.230] by web9008.mail.yahoo.com; Sat, 06 Jan 2001 15:36:49 PST Date: Sat, 6 Jan 2001 15:36:49 -0800 (PST) From: John Kessler Reply-To: jdmitrikessler@yahoo.com Subject: Story-Parts... To: Krissy Brown , Jasmine , Laura , Clarita Lefthand , William A Stein MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: RO X-Status: O Helmsman sat there in front of the screen of the computer monitor, playing an electronic version of the game "solitaire". He had beat the computer's random card sequence only once in the last 3 hours. Helmsman was stuck, unable to break himself away from the game, unable to perform all the other tasks of daily living which so-despaerately needed his attention. In a sudden fit of frustration, he began to pray: Oh, please! Please let me win, just this one game! Ah, but then he realized it was not just any god he was praying to; no, he was praying to the God of Games, who had suddenly begun to wxist, having been summoned out of the dark recesses of Helmsman's unconcious. Helmsman suddenly felt stupid and self-ashamed as he realized that he had become so pathetic as to pray to a non-existant god. (or, as up until this moment, had not previously existed.) ....................Your turn. Write about what happened next... ===== Blessed be. __________________________________________________ Do You Yahoo!? Yahoo! Photos - Share your holiday photos online! http://photos.yahoo.com/ From coleman@math.berkeley.edu Mon Jan 8 20:14:00 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f091EQ413354 for ; Mon, 8 Jan 2001 20:14:26 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f091E6m27456 for ; Mon, 8 Jan 2001 20:14:07 -0500 (EST) Received: from [169.229.58.143] (robert-1.Math.Berkeley.EDU [169.229.58.143]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id RAA26069 for ; Mon, 8 Jan 2001 17:14:06 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010901211415.02217@form> References: <01010901211415.02217@form> Date: Mon, 8 Jan 2001 17:14:00 -0800 To: was@math.harvard.edu From: Robert Coleman Subject: latex Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: A I tried making the introduction in latex. I used your outer file and macros. unfortunately it took so much longer than plain tex to compile, I didn't want to check my progress. More precisely, I cut out the body of your text from a copy of your file of our paper and stuck my introduction in, tried to make it look like latex, told textures the file was in latex and told textures to typeset the file. It did but took many time longer than it did when I typeset the introduction in plaintex. Is there any way to speed this up. From coleman@math.berkeley.edu Mon Jan 8 20:32:11 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f091Wb413498 for ; Mon, 8 Jan 2001 20:32:37 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f091WHm28015 for ; Mon, 8 Jan 2001 20:32:17 -0500 (EST) Received: from [169.229.58.143] (robert-1.Math.Berkeley.EDU [169.229.58.143]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id RAA26867 for ; Mon, 8 Jan 2001 17:32:17 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010902172117.02217@form> References: <01010901211415.02217@form> <01010902172117.02217@form> Date: Mon, 8 Jan 2001 17:32:11 -0800 To: was@math.harvard.edu From: Robert Coleman Subject: Re: latex Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: A We should also than Merel. From coleman@math.berkeley.edu Mon Jan 8 20:36:00 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f091aQ413504 for ; Mon, 8 Jan 2001 20:36:26 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f091a7m28093 for ; Mon, 8 Jan 2001 20:36:07 -0500 (EST) Received: from [169.229.58.143] (robert-1.Math.Berkeley.EDU [169.229.58.143]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id RAA27039 for ; Mon, 8 Jan 2001 17:36:06 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <0101090235091B.02217@form> References: <01010901211415.02217@form> <01010902172117.02217@form> <0101090235091B.02217@form> Date: Mon, 8 Jan 2001 17:36:00 -0800 To: was@math.harvard.edu From: Robert Coleman Subject: Re: latex Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: A >On Mon, 08 Jan 2001, you wrote: > > We should also than Merel. > >This sentence doesn't make sense. than = thank From coleman@math.berkeley.edu Tue Jan 9 01:01:15 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0961b413927 for ; Tue, 9 Jan 2001 01:01:37 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0961Im03052 for ; Tue, 9 Jan 2001 01:01:18 -0500 (EST) Received: from [136.152.194.23] (as3-1-101.HIP.Berkeley.EDU [136.152.194.23]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id WAA07393 for ; Mon, 8 Jan 2001 22:01:17 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010520163307.18287@form> References: <01010515082807.15849@form> <01010520163307.18287@form> Date: Mon, 8 Jan 2001 22:01:15 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Re: I'm puzzled by 52. Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O It would be really nice to find an eigenform G such that \theta^2G = F_27 mod 81. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Tue Jan 9 01:12:45 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f096D6413933 for ; Tue, 9 Jan 2001 01:13:06 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f096Clm03185 for ; Tue, 9 Jan 2001 01:12:47 -0500 (EST) Received: from [136.152.194.23] (as3-1-101.HIP.Berkeley.EDU [136.152.194.23]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id WAA07708 for ; Mon, 8 Jan 2001 22:12:46 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010901211415.02217@form> References: <01010901211415.02217@form> Date: Mon, 8 Jan 2001 22:12:45 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Re: our paper Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O Immediate reactions:The Katz an Hida section should go at the end, we should have a section of questions and we should include the proof of the Mazur Jochnowitz observation. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Tue Jan 9 01:19:08 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f096JT413937 for ; Tue, 9 Jan 2001 01:19:29 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f096JAm03223 for ; Tue, 9 Jan 2001 01:19:10 -0500 (EST) Received: from [136.152.194.23] (as3-1-101.HIP.Berkeley.EDU [136.152.194.23]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id WAA07900 for ; Mon, 8 Jan 2001 22:19:09 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <01010901211415.02217@form> References: <01010901211415.02217@form> Date: Mon, 8 Jan 2001 22:19:08 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Re: our paper Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O Other things I've thought about including are the assertion that the twist of an overconvergent form is overconvergent and the statement that we don't know whether the twist of a non-classical overconvergent form can be classical (one thinks about Buzzard-Taylor). -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Tue Jan 9 12:19:15 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f09Hlq415046 for ; Tue, 9 Jan 2001 12:47:52 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f09HlXm24044 for ; Tue, 9 Jan 2001 12:47:33 -0500 (EST) Received: from [136.152.193.241] (as3-1-174.HIP.Berkeley.EDU [136.152.194.96]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id JAA00579 for ; Tue, 9 Jan 2001 09:47:31 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <0101091505571N.02217@form> References: <01010515082807.15849@form> <01010520163307.18287@form> <0101091505571N.02217@form> Date: Tue, 9 Jan 2001 09:19:15 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Re: I'm puzzled by 52. Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O >Quick question: > >On Tue, 09 Jan 2001, you wrote: >> It would be really nice to find an eigenform G such that \theta^2G = >> F_27 mod 81. > >Can one rewrite this as > > theta^{-2}F_27 = G (mod 81), > >at least away from the a_{3n} coefficients? > >What sort of object is theta^{-2}F_27 (mod 81)? Its a twist. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Tue Jan 9 15:08:59 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f09K9M415205 for ; Tue, 9 Jan 2001 15:09:22 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f09K93m00878 for ; Tue, 9 Jan 2001 15:09:03 -0500 (EST) Received: from [136.152.193.241] (as3-1-174.HIP.Berkeley.EDU [136.152.194.96]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id MAA09201 for ; Tue, 9 Jan 2001 12:09:01 -0800 (PST) Mime-Version: 1.0 Message-Id: Date: Tue, 9 Jan 2001 12:08:59 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: theta Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O One thing that is true is: If \chi is a Grossencharacter associated to a quadratic imaginary field K of infinity type \sigma^{k-1} and j\ge2 is an integer congruent to 2-k mod \phi(p^n) and F_\chi is the associated modular form, then if p is prime to the conductor of \chi and splits in K there exists a classical modular form G of weight j and slope 0 such that \theta^{k-1}G(q)\con F_\chi(q) \mod p^n. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Wed Jan 10 23:03:35 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0B440416454 for ; Wed, 10 Jan 2001 23:04:00 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0B43hm03725 for ; Wed, 10 Jan 2001 23:03:43 -0500 (EST) Received: from [136.152.194.110] (as3-1-188.HIP.Berkeley.EDU [136.152.194.110]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id UAA24313 for ; Wed, 10 Jan 2001 20:03:38 -0800 (PST) Mime-Version: 1.0 Message-Id: Date: Wed, 10 Jan 2001 20:03:35 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: twisting Content-Type: multipart/mixed; boundary="============_-1232936677==_============" Status: RO X-Status: O --============_-1232936677==_============ Content-Type: multipart/alternative; boundary="============_-1232936677==_ma============" --============_-1232936677==_ma============ Content-Type: text/plain; charset="us-ascii" ; format="flowed" Here's a write up of twisting which allows us to twist overconvergent forms and thus replace classical with overconvergent in Theorem 1.1 and state a corollary for classical forms which is, on appearance anyway, stronger than this theorem. Namely, \begin{corollary} Let~$F$ be a classical eigenform on $X_1(Np^{t})$, $t\ge 1$, over $\Qbar_p$ of infinite slope at~$p$, which is a tame twist o an overconvergent form of finite slope. Suppose $A,s\in\Z$, $A>0$ and $0\le s twisting

Here's a write up of twisting which allows us to twist overconvergent forms and thus replace classical with overconvergent in Theorem 1.1 and state a corollary for classical forms which is, on appearance anyway, stronger than this theorem.
Namely,



\begin{corollary}
Let~$F$ be a classical eigenform on
$X_1(Np^{t})$, $t\ge 1$, over $\Qbar_p$ of
infinite slope at~$p$, which is a tame twist o an overconvergent form of finite slope.  Suppose $A,s\in\Z$, $A>0$ and $0\le s<p-1$.
Then there exists a classical eigenform~$G$ on $X_1(Np^t)$
of finite slope such that~$G$ has character $\omega^s$ for the
action of $\zms p$ and $G(q)\con F(q)\mod p^A$.
\end{corollary}


Of course we know of no examples where the overconvergent form is not classical.

By the way, Shimura's Proposistion 3.64 does not apply to \omega^{p-1}.
--============_-1232936677==_ma============-- --============_-1232936677==_============ Content-Id: Content-Type: text/plain; name="overtwist"; charset="us-ascii" Content-Disposition: attachment; filename="overtwist" ; modification-date="Wed, 10 Jan 2001 20:00:37 -0800" \beginsection Twisting of Overconvergent Modular Forms In this section, we rewrite Shimura's disscussion of twisting ffom a geomeric point of view so we can apply it to overconvegent forms. Suppose $f$ is a modular form of level $N$ and integral weight $k$ (so a function on pairs $(E,\iota)$ where $E$ is an elliptic curve and $\iota$ is an embedding of $\mu_N$ into $E$ to differentials on $E$). Suppose $\chi$ is a Dirichlet character of conductor M. I will define a modular form $f^\chi$ of level $L(M,N)$, where $L(a,b)$ is the LCM of $a^2$, $b$ and $ab$. Let $\alp \colon \mu_{L(M,N)}\ra E$, be such that $\alp|_{\mu_N}=\iota$. Then $\b(E/\alp(\mu_M)\b)[M]$ is isomorphic to $\Z/M\Z\times \mu_M$ via the Weil pairing $(\ ,\ )_M$: Indeed. suppose $\zeta\in\mu_M$ and $b\in \Z/M\Z$. Let $\zeta'\in\mu_{M^2}$ such that $(\zeta')^M=\zeta$. Let $P_b\in E[M]$ such that $$(\alp(\eps),P_b)_M=\eps^b$$ for all $\eps\in \mu_M$. Let $\rho_\alp$ be the isogeny from E to $E/\alp(\mu_M)$. Then the isomorphism from $\Z/M\Z\times \mu_M$ to $\rho_\alp(E)$ takes $(b,\zeta)$ to $\rho_\alp(P_b+\alp(\zeta'))$. Let $\gamma_{\eps}$ be the natural isogeny from $E$ to $E_{\alp,\eps}=:\rho_\alp(E)/((1,\eps)) $. Let $\iota_x\colon \mu_N\ra E_{\alp, \zeta^x}$ be the embedding such that $$\iota_\eps(\bet)=\gamma_{\eps}(\bet')$$ where $\bet'\in\mu_{L(M,N)}$ and $\bet'^M=\bet$. Now if $\chi$ is primitive of conductor $M$, $G(\bar\chi,\zeta)f^\chi(E,\alp)$ is $${1\ov M^k}\sum \bar\chi(x)\gamma_{\zeta^x}^*f(E_{\alp,\zeta^x},\iota_{\zeta^x})$$ where the sum runs over $x\in \Z/M\Z$. Indeed, if $E=\Gm/q^\Z$ and $\alp\colon \mu_{L(M,N)}\ra \Gm$ is the inclusion, $\rho_\alp(E)=\Gm/q^{M\Z}$, $\rho_\alp(E)/((1,\zeta^x)) = \Gm/(\zeta^xq)^{\Z}$, $\iota_x(\eps)=\eps$ and if $f(q)=\sum_na_nq^n$, $$f(\Gm/(\zeta^xq)^{\Z},\iota_x)=\sum_na_n(\zeta^xq)^n.$$ It also follows that if $(M,N)=1$, $$f^{\chi}|\dia{d}_{{M}}=\chi(d)^2f^{\chi}.$$ If $\alp$ is changed to $\alp^d$, $((1,\zeta))$ is changed to $((d\iv,\zeta^{d}))=((1,\zeta^{d^2}))$. What is going on is that we have commutative diagrams $$\matrix{E_1\b(L(M,N)\b)&\lmr{S_\eps}&E_1(N)\cr \downarrow&&\downarrow\cr X_1\b(L(M,N)\b)&\lmr{s_\eps}&X_1(N)}$$ for $\eps\in \mu_M$. Indeed $s_\eps$ takes th point corresponding to the pair $(E,\alp)$ to the point corresponding to the pair $(E_{\alp,\eps},\iota_\eps)$ and $S_\eps$ on the fibers above thesed points is the isogeny $\gamma_{\eps}$ discussed above. This geometric point of view allows us, since all these morphisms take the cusp at infinity to the cusp at infinity (see {[CO]}), to conclude that if $F$ is an overconvergent form weight $k$ and level $N$, then $F^\chi$ is an overconvergent form of weight $k$ and level $L(M,N)$. --============_-1232936677==_============ Content-Type: text/plain; charset="us-ascii" -- Robert http://math.berkeley.edu/~coleman --============_-1232936677==_============-- From watkins@math.psu.edu Sat Jan 13 16:31:48 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0DLW4418852 for ; Sat, 13 Jan 2001 16:32:04 -0500 Received: from math.psu.edu (leibniz.math.psu.edu [146.186.130.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0DLVnm10846 for ; Sat, 13 Jan 2001 16:31:50 -0500 (EST) Received: from jensen.math.psu.edu (jensen.math.psu.edu [146.186.131.84]) by math.psu.edu (8.9.3/8.9.3) with ESMTP id QAA17452; Sat, 13 Jan 2001 16:31:48 -0500 (EST) From: Mark J Watkins Received: (from watkins@localhost) by jensen.math.psu.edu (8.9.3/8.9.3) id QAA28362; Sat, 13 Jan 2001 16:31:48 -0500 (EST) Message-Id: <200101132131.QAA28362@jensen.math.psu.edu> Subject: Re: "Special values of newforms of weight > 2" To: verrill@math.ku.dk, was@math.harvard.edu Date: Sat, 13 Jan 2001 16:31:48 -0500 (EST) In-Reply-To: from "Helena A. Verrill" at Jan 10, 2001 09:38:45 AM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: RO X-Status: A > I propose that we three write a joint paper called maybe "Special > values of newforms of weight > 2". The contents would include: > > (1) Definition of the period map, period lattice, and complex > torus attached to a newform on Gamma_0(N) of integer > weight >= 2. > > (2) Formula for the the rational number (or module) L(M_f,i)/Omega_i. > > (3) How computing L(1) and L(2) (usually) gives the period lattice. > > (4) Examples, conjectural arithmetic interpretation in terms of > Bloch-Kato, visibility of Sha for motives. Mark and I have > lots of data in this direction. > > What do you two think? As per Helena's comment, (1), (2), and (3) seem to be the background, and (4) is the heart of the endeavour. My main contribution seems to be a program which quickly computes a special L-value, which in particular allows a faster test to tell whether it is zero. Perhaps we should be more specific about the data and examples in (4) (I am not sure I remember everything we have). I can think of: (a) Examples where the critical value is zero without being forced by functional equation (like 127k4A1) -- since this is like rank 2, it indicates that visible Sha might be floating around. (b) Examples with (large) squares in the numerator of the special rational number, in particular ones which are not just derived from (a). [These might include some invisible examples.] (c) Central values of quadratic twists of 8k4 (computed via the coefficients of the weight 5/2 form which is its Shintani lift), in particular how many are zero without being forced. Do either of you know how to compute Chow groups of motives? When I mentioned to K. Ono that the L-function of 127k4A1 has a double zero at its critical point, he mumbled something about it being useful to know whether the Chow group was trivial. I recently refereed a paper of Dummigan's on constructing visible Sha elements in symmetric square motives associated with level 1 rational newforms (k=12,16,18,20,22,26), but I don't know how relevant it is here. === Mark Watkins watkins@math.psu.edu From rpollack@math.harvard.edu Sun Jan 14 03:51:19 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0E8pWH20005 for ; Sun, 14 Jan 2001 03:51:32 -0500 Received: from localhost (rpollack@localhost) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0E8pJW22689 for ; Sun, 14 Jan 2001 03:51:19 -0500 (EST) Date: Sun, 14 Jan 2001 03:51:19 -0500 (EST) From: Robert Pollack To: "William A. Stein" Subject: Re: p-adic L-series person In-Reply-To: <01011222063204.08830@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A Hi William, I received four copies of your e-mail. Is this a suggestion that I should send you my write-up on those computations? (I will soon...I am currently in New Orleans for the joint conferences and then I have a talk to prepare for next Friday. But hopefully I will find time this next week.) Can your program compute modular symbols for weight 2 modular forms with non-rational coefficients? That would be very cool for me because I imagine the rest of the computation of the p-adic L-series of such things would be identical. For me in particular, one can find lots of examples of such forms that are supersingular at p yet a_p is non-zero. Anways, hope Germany is treating you well. Robert On Sun, 14 Jan 2001, William A. Stein wrote: > Georg, > > Robert Pollack is also running lots of p-adic L-series computations. > > rpollack@math.harvard.edu > > > -- William > From coleman@math.berkeley.edu Sun Jan 14 21:22:12 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0F2MVJ27661 for ; Sun, 14 Jan 2001 21:22:31 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0F2MIm15251 for ; Sun, 14 Jan 2001 21:22:18 -0500 (EST) Received: from [136.152.194.85] (as3-1-163.HIP.Berkeley.EDU [136.152.194.85]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id SAA19753 for ; Sun, 14 Jan 2001 18:22:16 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <0101150109090K.12578@form> References: <0101150109090K.12578@form> Date: Sun, 14 Jan 2001 18:22:12 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Re: Eichler-Selberg Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O >Robert, > >I've just implemented some classical Eichler-Selberg trace formula >type algorithms for computing modular forms, because they are >potentially useful when the weight is large. Here's a funny >observation, which is true for k<=52, at least: > >Fix a weight k. Let f = sum_{n>=1} Tr(T_n)*q^n, where T_n is the n-th >Hecke operator on S_k(SL_2(Z)). Let f_n = T_n(f). Then > > 1) f_1, ..., f_d form a basis for S_k > > 2) det(a_i(f_j)) = discriminant of the Hecke algebra. > >Is this observation trivial? > >Note that if you, e.g., replace f by 2*f, then part 2 of the above >is false, so there's something special about the choice of f. > >Yesterday I implemented an algorithm to compute the intersection >pairing on weight-2 modular symbols, i.e., on H_1(X_0(N);Q). > > -- William I assume d is the dimension of S_k. Let L be the map from S_k to \C^d g |-> (a_1(g),...,a_d(g)). Then 1 is the statement that L is an isomorphism. I don't know why this is true but it seems believable (its sort of saying \infty is not a Weie1rtrass point)...The statement 2 is more subtle. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Sun Jan 14 22:17:50 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0F3ICJ27742 for ; Sun, 14 Jan 2001 22:18:12 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0F3Hsm19500 for ; Sun, 14 Jan 2001 22:17:55 -0500 (EST) Received: from [136.152.194.85] (as3-1-163.HIP.Berkeley.EDU [136.152.194.85]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id TAA21285 for ; Sun, 14 Jan 2001 19:17:52 -0800 (PST) Mime-Version: 1.0 Message-Id: In-Reply-To: <0101150109090K.12578@form> References: <0101150109090K.12578@form> Date: Sun, 14 Jan 2001 19:17:50 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: Some thoughts Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O >Robert, > >I've just implemented some classical Eichler-Selberg trace formula >type algorithms for computing modular forms, because they are >potentially useful when the weight is large. Here's a funny >observation, which is true for k<=52, at least: > >Fix a weight k. Let f = sum_{n>=1} Tr(T_n)*q^n, where T_n is the n-th >Hecke operator on S_k(SL_2(Z)). Let f_n = T_n(f). Then > > 1) f_1, ..., f_d form a basis for S_k > > 2) det(a_i(f_j)) = discriminant of the Hecke algebra. > >Is this observation trivial? > >Note that if you, e.g., replace f by 2*f, then part 2 of the above >is false, so there's something special about the choice of f. > >Yesterday I implemented an algorithm to compute the intersection >pairing on weight-2 modular symbols, i.e., on H_1(X_0(N);Q). > > -- William OK, let g_1,...,g_d be a basis of normalized eigenform say over K. Let O be the ring of integers in K and M the O-linear span of the g_i. Let I be the module of eigenforms with integral coefficients. What is the relationship between |I/M| and the discriminant of Hecke? Now \ f=g_1+....+g_d and f_n=\sum a_n(g_i)g_i\ so the index of the span of f_n, n\le d in M, is |\det(a_i(g_j))| Also a_i(f_j)=\sum a_j(g_r)a_i(g_r) Thus |\det(a_i(f_j))|= |\det(a_i(g_j))|^2. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Tue Jan 16 13:40:34 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0GIepJ12586 for ; Tue, 16 Jan 2001 13:40:51 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0GIedm22324 for ; Tue, 16 Jan 2001 13:40:40 -0500 (EST) Received: from [136.152.194.47] (as3-1-125.HIP.Berkeley.EDU [136.152.194.47]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id KAA16665 for ; Tue, 16 Jan 2001 10:40:37 -0800 (PST) Mime-Version: 1.0 Message-Id: Date: Tue, 16 Jan 2001 10:40:34 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: warning Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O The statement I wrote about Gouvea-Mazur's density in the new introduction is not uite right. I'll send a new improved version in a few days. Also, twisting by powers of the cyclotomic character figures strongly in their proof. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Tue Jan 16 20:29:22 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0H1TbJ12906 for ; Tue, 16 Jan 2001 20:29:37 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0H1TQm08927 for ; Tue, 16 Jan 2001 20:29:27 -0500 (EST) Received: from [169.229.58.143] (robert-1.Math.Berkeley.EDU [169.229.58.143]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id RAA27251 for ; Tue, 16 Jan 2001 17:29:25 -0800 (PST) Mime-Version: 1.0 Message-Id: Date: Tue, 16 Jan 2001 17:29:22 -0800 To: was@math.harvard.edu From: Robert Coleman Subject: conjecture Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: A Let M be an integer. Then there exists a level 1 eigenform F and an integerr such that \theta^rF(q)\con F_0(27)(q) \mod 3^M. I don't know what r should but whatever it is, if a_p(F) is the p-th Fourier coeficient of F, this implies a_p(F)\con 0 \mod 3^M if p\con 2 \mod 3. This seems like something one could compute. From coleman@math.berkeley.edu Fri Jan 19 01:09:03 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0J69GJ18143 for ; Fri, 19 Jan 2001 01:09:16 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0J698m12754 for ; Fri, 19 Jan 2001 01:09:08 -0500 (EST) Received: from [136.152.193.241] (as3-1-63.HIP.Berkeley.EDU [136.152.193.241]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id WAA18066 for ; Thu, 18 Jan 2001 22:09:06 -0800 (PST) Mime-Version: 1.0 Message-Id: Date: Thu, 18 Jan 2001 22:09:03 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: the introduction Content-Type: multipart/mixed; boundary="============_-1232237949==_============" Status: RO X-Status: O --============_-1232237949==_============ Content-Type: text/plain; charset="us-ascii" ; format="flowed" Here is a corrected version. --============_-1232237949==_============ Content-Id: Content-Type: text/plain; name="Introduction-tex"; charset="us-ascii" ; format="flowed" Content-Disposition: attachment; filename="Introduction-tex" ; modification-date="Thu, 18 Jan 2001 22:06:47 -0800" \def\Qpbar{{\overline\Q_p}} \beginsection{Introduction} Fix a prime~$p$. Consider a classical newform~$f$ in $S_k(\Gamma_1(Np^t),\Qpbar)$ where $k$, $t\ge0$ and $N$ are integers, $(N,p)=1$. The {\bf slope} of~$f$ is $\ord_p(a_p(f))$, where $\ord_p(p)=1$. By [{\sevenrm This in not Prop.\ 3.6 of Shimura}], the twist of~$f$ by any Dirichlet character $\chi$ of conductor dividing $p$ is an eigenform $f^\chi$ of weight $k$ on $\Gamma_1({Np^{\max{\{t,2\}}} })$. This twist has infinite slope. In Section ?, we prove that if~$f$ has finite slope then it is possible to approximate $f^{\chi}$ arbitrarily closely by classical finite slope forms. Even assuming refinements of standard conjectures, the best estimate we can get for the smallest weight of an approximating forms is exponential in the approximating modulus $p^A$. Section ? contains computations that suggest that the best estimates should have weight that is linear in $p^A$. One motivation for the question of approximation of infinite slope forms by finite slope forms is the desire to understand the versal deformation space of a residual modular representations [Mazur, Deforming Galois representations] (the deformation space of an irreducible representation is universal [ibid.]\ as is the deformation space of a residual pseudorepresentation [C-Mazur, The Eigencurve]). The results in [Gouvea-Mazur, On the denstiy of modula representtions] (see also [Mazur, The ``infinite fern'' in the universal deformation space]), imply that the Zariski closure of the locus of totally wildly ramified twists of finite slope modular deformations of an absolutely irreducible ``totally unobstructed'' tame level one residual modular representation is Zariski dense in the associated representation scheme but very little is known about the topological closure of this locus or about the Zariski closure in the associated rigid space. For example, it is not known if either of these contain any open sets. Our result implies that they contain tamely ramified twists of modular deformations. We also show that a result of Hatada implies that in one (albeit non-irredeucible) case it doesn't contain all modular deformations. {\sevenrm There is definitely a problem here in the non-absolutely irreducible case, for in this case, only the semi-simplication of the representation attached to a modular form is well defined. } The investigation began with with our answer in\S? to a question of Jochnowitz. The idea of studying the \pad\ variation of modular forms began with Serre {[Formes modulaires et functions zeta $p$-adique]} and was since developed by Katz [Higher congruencesa betwween modular forms] and Hida [Iwaswa modules attached to congruences of modula forms] (see also [Gouv\^ea, Arithmetic of \pad\ modular forms]). It follows, in particular from their work, that one can approximate all eigenforms on $X_0(p^n)$ with forms on the $j$-line (but not necessarily with eigenforms). We prove the above result about twists in section ? We explain how to reinterpret Hatada's result in section ? We recall some rwesult of Katz and Hida in section ? and we present the results of our computations in section ? {\bf Acknowledgments.} The authors would like to thank Naomi Jochnowitz for initiating this line of thought and for interesting conversation Barry Mazur for helpful comments and interesting questions. , [and for suggesting that the form on $X_0(27)$ can not be approximated] {\sevenrm I don't think he did this. He gave an invalid reason why.} \ --============_-1232237949==_============ Content-Type: text/plain; charset="us-ascii" ; format="flowed" -- Robert http://math.berkeley.edu/~coleman --============_-1232237949==_============-- From coleman@math.berkeley.edu Tue Jan 23 19:04:55 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0O05ql02969 for ; Tue, 23 Jan 2001 19:05:52 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0O04wm13479 for ; Tue, 23 Jan 2001 19:04:58 -0500 (EST) Received: from [136.152.194.78] (as3-1-156.HIP.Berkeley.EDU [136.152.194.78]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id QAA07504 for ; Tue, 23 Jan 2001 16:04:57 -0800 (PST) Mime-Version: 1.0 Message-Id: Date: Tue, 23 Jan 2001 16:04:55 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: twisting Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O We've proved that the twist F' of form of level N, F, by Teichmuller can be approximated by finite sloe forms But is it the p-deprivation of a finite slope form. This would MEAN it can be approximated arbitrarily closely by the p-deprivations of,classical forms of bounded slope -- Robert http://math.berkeley.edu/~coleman From jpb@msri.org Wed Jan 24 17:28:07 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0OMT3l03907 for ; Wed, 24 Jan 2001 17:29:03 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0OMSAm21015 for ; Wed, 24 Jan 2001 17:28:10 -0500 (EST) Received: from ep2.msri.org (ep2.msri.org [198.129.64.227]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id OAA20722 for ; Wed, 24 Jan 2001 14:28:09 -0800 (PST) Received: from rafiki.msri.org [198.129.65.127] (mail) by ep2.msri.org with esmtp (Exim 3.12 #1 (Debian)) id 14LYOe-0008B7-00; Wed, 24 Jan 2001 14:28:08 -0800 Received: from moby.msri.org [198.129.65.72] (mail) by rafiki.msri.org with esmtp (Exim 2.05 #1 (Debian)) id 14LYOd-00080o-00; Wed, 24 Jan 2001 14:28:07 -0800 Received: from jpb by moby.msri.org with local (Exim 3.12 #1 (Debian)) id 14LYOd-0000xU-00; Wed, 24 Jan 2001 14:28:07 -0800 Date: Wed, 24 Jan 2001 14:28:07 -0800 To: kaltofen@math.ncsu.edu, was@math.berkeley.edu Subject: pseudo powerpoint starting from a TeX file Message-ID: <20010124142807.I2531@moby> Mime-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Disposition: inline User-Agent: Mutt/1.2.5i From: Joe Buhler Status: RO X-Status: O Dear William, Erich, Both of you have talked to me about giving laptop math talks, and I wanted to mention ppower4, which I find quite neat. (Though you may already know about it...) It starts from a TeX file and produces a pdf document that acroread reads, giving most of the functionality of powerpoint; it's pretty cool from my limited tests. You have to lightly annotate the TeX file (e.g., to tell acroread where to pause before displaying the next bullet), but pdftex then produces a pdf file that, when viewed with acroread in full page mode, is hard to distinguish from a powerpoint file (except that it has nice math formulas). Some, but not all, of the snazzy powerpoint functionality is available, e.g., it has only very limited animation. The home page is at www.tex.ac.uk/tex-archive/support/ppower4/readme.html Joe From schoof@wins.uva.nl Sat Jan 27 09:40:57 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0REfni07669 for ; Sat, 27 Jan 2001 09:41:49 -0500 Received: from mbox.wins.uva.nl (root@mbox.wins.uva.nl [146.50.16.20]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0REexm04147 for ; Sat, 27 Jan 2001 09:40:59 -0500 (EST) Received: from turing.wins.uva.nl [146.50.15.20] by mbox.wins.uva.nl with ESMTP (sendmail 8.10.2/config 10.1). id f0REewg17115; Sat, 27 Jan 2001 15:40:58 +0100 (MET) Received: from localhost by turing.wins.uva.nl (sendmail 8.10.2/config 9.4). id f0REevu26076; Sat, 27 Jan 2001 15:40:57 +0100 (MET) Message-Id: <200101271440.f0REevu26076@turing.wins.uva.nl> Date: Sat, 27 Jan 2001 15:40:57 +0100 (MET) From: schoof@wins.uva.nl (Rene Schoof) X-Organisation: Faculty of Science University of Amsterdam The Netherlands X-Address: See http://www.science.uva.nl/location To: was@math.harvard.edu Status: RO X-Status: A Hi, we would be interested in you adding an appendix to our paper containg Thm.3.3 of your note "generating the Hecke algebra as a Z-module" I'll send you our paper, which contains nothing more than an explicit example of one of Wiles's deformation rings, soon. Then you can decide whether you want to write the appendix. I have a question concerning "generating the Hecke algebra". In our note we always work with the Hecke operators T_p for prime p. The only time when we seem to need also Hecke operators T_n with n not prime is when we apply your Thm.3.3. Is it possible that Thm.3.3 is also true with T_n where n only runs over the primes less than "bound"? Would it be much harder to prove that? Best Rene' PS.. Of course the T_n with n prime should still generate the Hecke-algebra as a Z-module (rather than Z-algebra). Actually we always work with Z_3-modules and Z_3-algebras, but that won't help much, I think. From marlene@infomagic.com Sun Jan 28 13:57:54 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0SIg0i16275 for ; Sun, 28 Jan 2001 13:42:00 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0SIfCm17297 for ; Sun, 28 Jan 2001 13:41:12 -0500 (EST) Received: from infomagic.com (MAX1-Port22.Downtown.InfoMagic.NET [63.229.94.42]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f0SIfAd27946 for ; Sun, 28 Jan 2001 11:41:11 -0700 Message-ID: <3A746BB1.B5A35C1D@infomagic.com> Date: Sun, 28 Jan 2001 11:57:54 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: was@math.harvard.edu Subject: Re: AHCCCS ADDRESS FOR COMPUTER References: <01012800570002.02072@form> <3A738AAE.EB5C3EE5@infomagic.com> <0101281745030B.10286@form> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: RO X-Status: A Hi William: I told you that we want to fly backeast but I guess you have a lot on your mind as we all do. We are planning to go to an accordion/concertina weekend September 14-16th in Washington Manor, Massachusettes. It's called THE NORTHEAST SQUEEZE-IN put on by the Bottonbox in Massachusettes. Check this website and click on Northeast Squeeze-In which is near the top of the website page http://www.buttonbox.com/ It's an annual weekend of music and dance. We want to visit the Boston and vicinity area, see the northeast. We would LOVE to come to a class you are teaching. Mom "William A. Stein" wrote: > On Saturday 27 January 2001 21:57, you wrote: > > I remember meeting Matt Baker. We'll be in the northeast the last two > > weeks in Sept. 2001 so maybe we'll see you then. We need to see > > Harvard...you'll have a "minute" I am sure and dinner with us. > > Really? What are you doing in the North East? Maybe you can come to > the class I'll be teaching! > > -- William From marlene@infomagic.com Sun Jan 28 14:00:03 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0SIiAi16305 for ; Sun, 28 Jan 2001 13:44:10 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0SIhLm17310 for ; Sun, 28 Jan 2001 13:43:22 -0500 (EST) Received: from infomagic.com (MAX1-Port22.Downtown.InfoMagic.NET [63.229.94.42]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f0SIhKd28204 for ; Sun, 28 Jan 2001 11:43:20 -0700 Message-ID: <3A746C33.8090D3B2@infomagic.com> Date: Sun, 28 Jan 2001 12:00:03 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: was@math.harvard.edu Subject: snowing again, menu, thanks References: <01012802124101.02934@form> <3A738EF9.C0271D68@infomagic.com> <0101281753050E.10286@form> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: RO X-Status: O oh boy, a menu, thanks. I still collect them. There's a new Asian Restaurant on Milton, I have a menu and I think it's a NICE restaurant. IT'S ALMOST NOON AND SNOWING LOTS RIGHT NOW. MOM "William A. Stein" wrote: > On Saturday 27 January 2001 22:16, you wrote: > > I just looked at the photos for January 26th....Picture dscf1351 looks > > like Mathematical Petroglyphs. > > It's just some math on a blackboard that I was too lazy to copy down > by hand. > > > The staircase photos are fantastic, amazing > > It was fantastically hard work hiking up the stairs! > > > ...where is the handicapped ramp???? ha ha.... > > There's none, and no elevator either! France has to be one of the > most handicap-unfriendly countries I've ever visited. > > > the cheese dinner, mmmmm > > The resturaunt had an unusual and huge menu, with lots of pictures > and stories. Though they charged $6 for the menu, the waiter gave > me one for free, which I inten to give to you. > > > don't know if I have been to Bordeaux. When I went to Europe in 1997 on > > the trip to Spain, we drove through France coming home.. > > Now to look at Jan. 27th. SNOW TO CONTINUE TONIGHT. > > More snow! You are so lucky. > > I saw a picture in the Harvard paper, which showed a student walking > in shorts in the snow, so clearly Cambridge has received some nice > snow. Bonn had a tiny bit of snow once, but nothing interesting. > Argh. > > -- William From marlene@infomagic.com Sun Jan 28 14:08:31 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0SIqbi16332 for ; Sun, 28 Jan 2001 13:52:37 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0SIpnm17407 for ; Sun, 28 Jan 2001 13:51:49 -0500 (EST) Received: from infomagic.com (MAX1-Port22.Downtown.InfoMagic.NET [63.229.94.42]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f0SIpmd29387 for ; Sun, 28 Jan 2001 11:51:48 -0700 Message-ID: <3A746E2F.A88C92C5@infomagic.com> Date: Sun, 28 Jan 2001 12:08:31 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: was@math.harvard.edu Subject: Re: Instrument Convention References: <3A738FB0.4B698EC@infomagic.com> <0101281754310F.10286@form> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: RO X-Status: O My 3 concertinas are on top of the piano but hard to see. There are two to the right of my "blue" free bass Giulietti accordion (I am trying to learn to play this instrument) The two concertinas to the right are both "Made in England", the one on the far right end is an antique Lachenal (1805-1905) and to the right of the accordion on the piano is my favorite--made by Charles Wheatstone (1937). To the left of the accordion on top of the paino is my Italian made "Stagi" concertina (newly bought and my first concertina purchased from The Lark in the Morning Catalog in California which I rarely play anymore). The little accordion on the right is one Dad bought from E-bay and added more notes to it. "William A. Stein" wrote: > On Saturday 27 January 2001 22:19, you wrote: > > > > Just for fun, Dad just photographed all our instruments. The instruments > > are going to have a conference tonight. > > Very nice. I don't remember the mini-according in the lower right. > Where are your concertinas? > > William From marlene@infomagic.com Sun Jan 28 14:09:06 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0SIrCi16336 for ; Sun, 28 Jan 2001 13:53:12 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0SIqOm17417 for ; Sun, 28 Jan 2001 13:52:24 -0500 (EST) Received: from infomagic.com (MAX1-Port22.Downtown.InfoMagic.NET [63.229.94.42]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f0SIqNd29450 for ; Sun, 28 Jan 2001 11:52:23 -0700 Message-ID: <3A746E52.748AA189@infomagic.com> Date: Sun, 28 Jan 2001 12:09:06 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: was@math.harvard.edu Subject: Re: [Fwd: pictures] References: <3A739061.AC0EA39B@infomagic.com> <0101281755010G.10286@form> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: RO X-Status: O Dennis finally wrote me and told me this so I did pull them up and looked at the photos today. Mom "William A. Stein" wrote: > On Saturday 27 January 2001 22:22, you wrote: > > Have you been able to pull Dennis' photos up on your computer. I can't > > figure out where to go on this webpage. I wrote Dennis an e-mail but he > > hasn't written back to tell me what to do. Mom > > He made a mistake. Go to fotki instead of www.mbe.com. > > William From mbaker@math.harvard.edu Sun Jan 28 22:46:57 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T3lii16934 for ; Sun, 28 Jan 2001 22:47:45 -0500 Received: from localhost (mbaker@localhost) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T3kvn26393 for ; Sun, 28 Jan 2001 22:46:57 -0500 (EST) Date: Sun, 28 Jan 2001 22:46:57 -0500 (EST) From: Matt Baker To: "William A. Stein" Subject: Re: NSF? In-Reply-To: <0101290053330S.10286@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O Hi William, I was originally on a 3-year plan, hoping that one way or another Harvard would give me support for teaching. Then when I got the BP I decided to postpone the NSF for a year, which they let me do (so I've been a full-time BP this past year). They didn't seem to care too much, it was pretty simple. The main thing you need to decide is when to use your NSF money. For me, it seemed best to get some teaching experience in early, since I had been on fellowships many years in a row. But you might want to do things differently... In summary, I think you should decide based on your own personal wishes what to do in terms of combining your NSF and BP (make sure with Ruby and/or Dick that whatever you want to do is kosher with them), and then inform the NSF of your wishes via email. It would be very strange if they didn't go along. I believe their only rule is that you can't postpone the NSF for two years in a row... (but I'm not 100% sure if that is written in stone) -- Matt On Mon, 29 Jan 2001, William A. Stein wrote: > HI Matt, > > I assumed when you initially received your NSF postdoc that a two or > three year payment schedule was set. When you received the BP, this > payment schedule must have been changed. How did you go about > changing it, and what did the NSF say? Perhaps your experience will > help me to better formulate my strategy. > > Thanks, > William > From whitney@math.berkeley.edu Sun Jan 28 23:02:39 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T3tVi16944 for ; Sun, 28 Jan 2001 22:55:31 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T3shm26892 for ; Sun, 28 Jan 2001 22:54:43 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0T42d212406 for ; Sun, 28 Jan 2001 20:02:40 -0800 Date: Sun, 28 Jan 2001 20:02:39 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: Re: nice In-Reply-To: <0101290441130W.10286@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A On Mon, 29 Jan 2001, William A. Stein wrote: > Something even easier would be to not run the program until total > memory usage is reasonably low. I could make my own wrapper around > nohup that polls mf? every 30 seconds using the free command until > free is sufficiently large. Do you have any suggestions for what > the rules should be? It depends on the mean runtime and memory usage of a job. If the mean time run time is 10 hours and memory is 256M, then I would think you could check every 5-15 minutes, and if the "free" RAM (including buffers, etc) is at least 256M, then launch a new job. It's OK to have some swapping, as will happen when the two jobs together happen to go above the average. > Good point. In fact, I consider it a server in addition to a client, > though the boundaries might be slightly fuzzy in this case. It is OK > with me if he continues to run it if there is a truly compelling > reason to do so. Is there? I don't know, I'll ask when I see him next. Cheers, Wayne From whitney@math.berkeley.edu Mon Jan 29 00:58:26 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T5pGi17016 for ; Mon, 29 Jan 2001 00:51:16 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T5oSm01668 for ; Mon, 29 Jan 2001 00:50:28 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0T5wQu12467 for ; Sun, 28 Jan 2001 21:58:26 -0800 Date: Sun, 28 Jan 2001 21:58:26 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: Re: nice In-Reply-To: <0101290518040X.10286@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O On Mon, 29 Jan 2001, William A. Stein wrote: > I can probably write something like the above, but I'll wait for your > input, if you have any, before doing so. You might consider extending rwhod, rather than just hacking something together using scripts. You might also want to look at http://slashdot.org/article.pl?sid=01/01/27/0058252&mode=thread although I couldn't find any actual code. Cheers, Wayne From whitney@math.berkeley.edu Mon Jan 29 01:12:09 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T64xi17092 for ; Mon, 29 Jan 2001 01:04:59 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T64Bm01869 for ; Mon, 29 Jan 2001 01:04:11 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0T6C9N12483 for ; Sun, 28 Jan 2001 22:12:09 -0800 Date: Sun, 28 Jan 2001 22:12:09 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: Re: nice In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A On Sun, 28 Jan 2001, Wayne Whitney wrote: > http://slashdot.org/article.pl?sid=01/01/27/0058252&mode=thread Hmm, I guess Beowulf (what is described in this thread) is really about parallel computing using MPI. I guess that is a level of abstraction beyond what you consider that you need? Anyway, I wonder if any of these setups include an 'rtop' command. I read some of the slashdot comments and looked at a few links, these were the only ones I did not eliminate as not so interesting: http://www.beowulf-underground.org/ http://www.scyld.com/ They looked interesting but I did not really look at them. Cheers, Wayne From rpollack@math.harvard.edu Mon Jan 29 01:21:02 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T6Loi17140 for ; Mon, 29 Jan 2001 01:21:50 -0500 Received: from localhost (rpollack@localhost) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T6L2E01972 for ; Mon, 29 Jan 2001 01:21:02 -0500 (EST) Date: Mon, 29 Jan 2001 01:21:02 -0500 (EST) From: Robert Pollack To: "William A. Stein" Subject: Re: send me your code In-Reply-To: <01012220481105.00756@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A > Hey, send me some magma code that involves p-adic L-functions! Okay...but the code is really awful. I believe it works though. I will send you two more e-mails. One containing a write-up in TeX of computing the p-adic L-series and the other will be my code. The first will probably be more helpful than the later. As the code is not documented, let me at least lead you thru what it is supposed to do. The main procedure is called Lseries. It takes in five parameters - let's call them V,p,a_p,n,D. The V should be the space of modular symbols associated to E The p is the prime The a_p is the a_p value of E The n represents the level of accuracy. The D represents twisting by a quadratic character of conductor D. (you can always just take D=1) Then Lseries(V,p,a_p,n,D) returns in the ordinary case a polynomial of degree p^(n-1)-1 approximating the L-series. In the supersingular case, it returns two polynomials say G and H so that G + H * alpha approximates the L-series where alpha is a root of x^2 - a_p x + p. The other main procedure is iwasawa_invariants_w_curve. This takes 6 arguments, say N,V,p,a_p,n,D. Here N is the conductor of the curve V is the space of modular symbols of the curve p is some prime a_p is the a_p value of the curve n is the level of accuracy D represents twisting by a quadratic character of conductor D This procedure is supposed to return the iwasawa invariants of the L-series. Precisely, in the ordinary case it returns a 6-tuple of data, namely the rank of the curve, the p-adic valuation of the leading non-zero term of the L-series, the mu invariant of the L-series the lambda invariant of the L-series the newton polygon of the L-series. -this is returned as [ [m_1,s_1] , [m_2,s_2] , ... ] where m_1 is the number of roots of slope s_1...; a slope of 10000 represents the root 0. a boolean saying whether the above data is correct. (note if mu is positive then a "true" only means that the data is as accurate as it can be using a polynomial of degree p^(n-1)-1; that is it could have a another root of smaller slope) In the supersingular case, the p-adic L-series is not an iwasawa function so the above data couldn't make any sense. Instead it returns an 11-tuple of data consisting of the rank and then the same data for the g and h functions i construct in my thesis. i should have a rough draft of my thesis ready soon if you are interested in how they are defined. basically, they are the "real" and "imaginary" parts of the L-series with their trivial zeroes removed. They are in fact iwasawa functions (actually integral power series). I've been back and forth between NY and Boston for the last few weeks. Now I should be back in Boston full time for the next month. Okay...let me try and send out the code. Robert From rpollack@math.harvard.edu Mon Jan 29 01:23:33 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T6OLi17144 for ; Mon, 29 Jan 2001 01:24:21 -0500 Received: from localhost (rpollack@localhost) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T6NY002056 for ; Mon, 29 Jan 2001 01:23:34 -0500 (EST) Date: Mon, 29 Jan 2001 01:23:33 -0500 (EST) From: Robert Pollack To: "William A. Stein" Subject: TeX Lseries Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O \documentclass{article} \usepackage{amsmath,amsthm,amssymb} \theoremstyle{plain} \newfont{\cyrr}{wncyr10} \def\Sha{\mbox{\cyrr Sh}} \newtheorem{thm}{Theorem}[section] \newtheorem{prop}{Proposition}[section] \newtheorem{lemma}[thm]{Lemma} \newtheorem{cor}[thm]{Corollary} \theoremstyle{definition} \newtheorem{ex}[thm]{Example} \newtheorem{defn}[thm]{Definition} \newtheorem{remark}[thm]{Remark} %Letters \newcommand{\A}{{\mathcal A}} \renewcommand{\O}{{\mathcal O}} \newcommand{\ve}{\varepsilon} \newcommand{\oo}{\text{o}} \newcommand{\OO}{\text{O}} %Number systems \newcommand{\Z}{{\bf Z}} \newcommand{\Q}{{\bf Q}} \newcommand{\R}{{\bf R}} \newcommand{\C}{{\bf C}} \newcommand{\Zp}{\Z_p} \newcommand{\Zpx}{\Z_p^\times} \newcommand{\ZpM}{\Z_{p,M}} \newcommand{\ZpMx}{\Z_{p,M}^\times} \newcommand{\Zppsi}{\Zp[\psi]} \newcommand{\ZM}{\left( \Z / M \right)} \newcommand{\ZMx}{\left( \Z / M \right)^{\times}} \newcommand{\ZMpx}{\left( \Z / Mp \right)^{\times}} \newcommand{\Cn}{\Phi_{n}(T)} \newcommand{\Zmodp}{\left( \Z / p \right)} \newcommand{\Zmodpx}{\left( \Z / p \right)^{\times}} \newcommand{\oneZp}{1+p\Zp} \newcommand{\Qp}{\Q_p} \newcommand{\Qpx}{\Q_p^\times} \newcommand{\Cp}{\C_p} \newcommand{\Cpx}{\C_p^\times} \newcommand{\Fp}{{\bf F}_p} \newcommand{\Fpx}{{\bf F}_p^\times} \newcommand{\Qbar}{\overline{\Q}} \newcommand{\Qpbar}{\overline{\Q}_p} \newcommand{\Qppsi}{\Qp(\psi)} \newcommand{\Qpa}{\Qp(\alpha)} \newcommand{\Qpapsi}{\Qp(\alpha,\psi)} %Maps \newcommand{\inj}{\hookrightarrow} \newcommand{\surj}{\twoheadrightarrow} \newcommand{\maps}{\rightarrow} %Characters \newcommand{\w}{\omega} %L-series %Measures \newcommand{\mua}{\mu_{E,\alpha}^+} \newcommand{\modsym}[2]{\left[ \frac{#1}{#2} \right]^+} %Math Operators \DeclareMathOperator{\ord}{ord} \newcommand{\ordp}{\ord_p} \DeclareMathOperator{\Hom}{Hom} \DeclareMathOperator{\sign}{sgn} \DeclareMathOperator{\Tam}{Tam} \begin{document} \title{A brief note on computing $p$-adic $L$-series of elliptic curves} \author{Robert Pollack} \maketitle \section{Definition of $p$-adic $L$-series} Let $E/\Q$ be an elliptic curve. Fix $p$ a prime number. Denote by $\modsym{r}{s}$ the positive modular symbol of $E$ associated to $\frac{r}{s}$ (defined up to choice of sign). Let $\alpha$ be a root of $x^2 - a_p x + p$ with $\ord_p(\alpha) < 1$. Here $a_p := p+1-E(\Fp)$. Such an $\alpha$ is unique in the ordinary case and is actually in $\Zpx$. In the supersingular case, there are two choices for alpha both conjugate in a quadratic extension of $\Qp$. Define a distribution on $\Zpx$ by $$ \mua(a+p^n \Zp) = \frac{1}{\alpha^n} \modsym{a}{p^n} - \frac{1}{\alpha^{n+1}} \modsym{a}{p^{n-1}}. $$ Then the $p$-adic $L$-series of $E$ is defined as a function on the $\Cp$-valued characters of $\Zpx$ by integration with respect to $\mua$. Here, $$ L_p(E,\alpha,\chi) = \int_{\Zpx} \chi ~ d\mua $$ where $\chi:\Zpx \maps \Cp$ is a character. \section{Powers series expression for the $p$-adic $L$-series} Pick $\gamma$ a topological generator of $1+pZ_p$. Then characters of $1+pZ_p$ are defined uniquely by their value on $\gamma$. For $u \in \Cp$ with $|u-1|_p < 1$, define a character $\chi_u$ on $\Zpx$ by first taking the natural projection of $\Zpx$ onto $1+pZ_p$ and then mapping $\gamma$ onto $u$. The $p$-adic $L$-series $L_p(E,\alpha,\chi_u)$ is then analytic in the variable $u$ and we will denote its expansion about $u=1$ by $L_{E,p,\alpha}(T) \in \Qp(\alpha)[[T]]$. This power series is convergent on the open unit disc of $\Cp$. We have that $$ L_{E,p,\alpha}(u-1) = L_p(E,\alpha,\chi_u). $$ \section{Explicit polynomial approximations of $L_{E,p,\alpha}(T)$} We can approximate $\int_{\Zpx} \chi ~ d\mua$ via Riemann sums. The details of this calculation will not be done here. However, the end result is a sequence $L_n(T)$ of polynomials in $\Qp(\alpha)[T]$ that converge to the $p$-adic $L$-series. Here, $$ L_n(T) = \sum_{j=0}^{p^{n-1}-1} \left( \sum_{a=1}^{p-1} \mua \left( \{a\} \gamma^j + p^n Z_p \right) \right) \cdot (1+T)^j $$ where $\{a\}$ is the Teichmuller lifting of $a$. \section{Computer calculations} The above formula for $L_n(T)$ allows one to readily approximate them on a computer. One can only approximate them as they have $Q_p(\alpha)$ coefficients. In the ordinary case, $\alpha$ is in $\Zpx$ and $L_n(T)$ should have $\Zp$ coefficients. This is known when $E[p]$ is irreducible, but it is conjectured to happen generally. Analyzing the rate of convergence of the $L_n(T)$, one can see that it suffices to compute everything mod $p^n$ (taking care with possible $p$'s in the denominators of the modular symbols). Computing $\{a\},\alpha$ and $(1+T)^j$ mod $p^n$ is very easy. All that is left to compute is $\modsym{a}{p^n}$ and $\modsym{a}{p^{n-1}}$ for $a$ prime to $p$ between $1$ and $p^n$. This is the most time consuming part of the calculation. In the supersingular case, $\alpha$ is in a quadratic extension of $\Qp$ and $\mua$ has $p$'s in its denominators on order about $p^\frac{n}{2}$. In this case $L_n(T)$ will not have $\Zp$ coefficients. Nonetheless, no essential information is lost if one computes $\{a\}$ and $(1+T)^j$ only mod $p^n$. The end result should be that $L_n(T) = G_n(T) + H_n(T) \cdot \alpha$ with $G_n,H_n \in \Qp[T]$ and scaling by $p^\frac{n+2}{2}$ will put both of them in $\Zp[[T]]$. \end{document} From rpollack@math.harvard.edu Mon Jan 29 01:26:54 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T6Rgi17175 for ; Mon, 29 Jan 2001 01:27:42 -0500 Received: from localhost (rpollack@localhost) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0T6QsY02179 for ; Mon, 29 Jan 2001 01:26:54 -0500 (EST) Date: Mon, 29 Jan 2001 01:26:54 -0500 (EST) From: Robert Pollack To: "William A. Stein" Subject: Lseries code Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O R:=PolynomialRing(Rationals()); R:=PolynomialRing(Rationals()); //This function cuts out the subspace of H_1(X_0(N),Z) where //T_2 acts as a_2, T_3 acts as a_3, T_5 acts as a_5 and T_7 acts as a_7 //Input: N - conductor // a_2 - eigenvalue of T_2 // a_3 - eigenvalue of T_3 // a_5 - eigenvalue of T_5 // a_7 - eigenvalue of T_7 // //Output: space of modular symbols of level N and specified eigenvalues function form_curve(N,a_2,a_3,a_5,a_7) R:=PolynomialRing(Rationals()); M:=ModularSymbols(N,2,+1); I:=[<2,x-a_2>,<3,x-a_3>,<5,x-a_5>,<7,x-a_7>]; V:=Kernel(I,M); ModularSymbolEven(V,1/2); return(V); end function; function table_of_modular_symbols(V,p,n,D) local q,table,temp,k,scale,q1,mid,temp2,e; if (D mod p eq 0) then print "Twisting by something not relatively prime to p!"; end if; if (KroneckerSymbol(D,-1) eq -1) then print "Twisting by an odd character!"; end if; q1 := p^(n-1); q := q1*p; k:=1; while (ModularSymbolEven(V,k/p^10)[1] eq 0) do k := k+1; end while; scale := Denominator(ModularSymbolEven(V,k/q)[1]); table := []; for a in [0..(q-1) div 2] do if (Valuation(a,p) le 2) or (a eq 0) then temp2 :=0; for b in [1..D] do e := KroneckerSymbol(D,b); if (e ne 0) then temp := ModularSymbolEven(V,(D*a-q*b)/(D*q))[1] * scale; if Denominator(temp) ne 1 then print "PROBLEMS! There are still denominators in the modsyms."; end if; temp2 := temp2 + KroneckerSymbol(D,b)*temp; end if; end for; table := table cat [Numerator(temp2)]; else table := table cat [0]; end if; end for; //USING the fact that [r/s]^+ equals [-r/s]^+ mid := ((q-1) div 2) + 1; for a in [mid..q-1] do table := table cat [table[q-a+1]]; end for; table := table cat [scale]; return(table); end function; //Multiplies a and b interpreted as a[1] + sqrt(-p)*a[2]. function multiply(a,b,p) return([a[1]*b[1] - p*a[2]*b[2],a[1]*b[2]+a[2]*b[1]]); end function; //Here answer[a] = [alpha^-a] //n+1 is the maxl power and m is the accuracy function powers_of_alpha(p,a_p,n,m) local temp,answer,t,q; if a_p mod p ne 0 then temp := a_p mod p; for k in [2..m] do t := -(temp^2-a_p*temp+p) div p^(k-1) mod p * InverseMod(2*temp - a_p,p); temp := temp + t*p^(k-1); end for; q := p^m; temp := InverseMod(temp,q); answer := [[temp]]; for k in [2..n+1] do Append(~answer,[answer[k-1][1]*answer[1][1] mod q]); end for; return(answer); else if a_p eq 0 then temp := [0,-1/p]; else if a_p eq -3 then temp := [-1/2,-1/6]; else temp := [1/2,-1/6]; end if; end if; answer := [temp]; for k in [2..n+1] do Append(~answer,multiply(answer[k-1],answer[1],p)); end for; return(answer); end if; end function; //Here answer = {a} mod p^m function one_teich(a,p,m) local answer,t; answer := a mod p; for k in [2..m] do t := 0; while (answer + t*p^(k-1))^(p-1) mod p^k ne 1 do t := t+1; end while; answer := answer + t*p^(k-1); end for; return(answer); end function; //Here answer[a] = {a} (mod p^m) function list_of_teich(p,m) local answer; answer := []; for a in [1..p-1] do Append(~answer,one_teich(a,p,m)); end for; return(answer); end function; //Here answer[a] = gamma^(a-1) //SO THERE IS A SHIFT //n is the maxl power and m is the accuracy function powers_of_gamma(gamma,p,n) local answer; answer := [1]; for k in [2..p^(n-1)] do Append(~answer,answer[k-1]*gamma mod (p^n)); end for; return(answer); end function; function lift_polynomial(f) return(PolynomialRing(Rationals()) ! (PolynomialRing(Integers()) ! f)); end function; //Here answer[a]=(1+T)^(a-1) //SO THERE IS A SHIFT //n is the maxl power and m is the accuracy function powers_of_oneplusT(p,n,m) local R,S,answer,q; q := p^m; R := ResidueClassRing(q); S := PolynomialRing(R); answer := [1]; for k in [2..p^(n-1)] do answer := answer cat [answer[k-1]*(1+T)] ; end for; return(answer); end function; function alpha(n,ALPHA) return(ALPHA[n]); end function; function modsym(a,MODSYM) return(MODSYM[a+1]); end function; function oneplusT(j,POWER) return(POWER[j+1]); end function; function teich(a,TEICH) return(TEICH[a]); end function; function gamma(j,GAMMA) return(GAMMA[j+1]); end function; function scale(list,c) local k,answer; answer:=[]; for k in [1..#list] do answer := answer cat [list[k]*c]; end for; return(answer); end function; function add(list1,list2) local k,answer; if #list1 ne #list2 then print "Error in add"; end if; answer:=[]; for k in [1..#list1] do answer := answer cat [list1[k]+list2[k]]; end for; return(answer); end function; function subtract(list1,list2) local k,answer; if #list1 ne #list2 then print "Error in subtract"; end if; answer:=[]; for k in [1..#list1] do answer := answer cat [list1[k]-list2[k]]; end for; return(answer); end function; function measure(a,p,n,ALPHA,MODSYM) local q,q1; q := p^n; q1 := q div p; return( subtract(scale(alpha(n,ALPHA),modsym(a,MODSYM)), scale(alpha(n+1,ALPHA),modsym((a mod q1) * p ,MODSYM)))); end function; //This procedure returns a polynomial approximation to the p-adic //L-series of the elliptic curve corresponding to V twisted by a quadratic //character of discriminant D. The degree of the polynomial is p^n-1. //If a_p is prime to p (ordinary) then the answer is simply a polynomial. //Otherwise (in the supersingular case) the answer is in the form [G,H] //where the L-series is approximated by G + sqrt(-p) * H. // //Input: V - space of modular symbols // p - prime // a_p - eigenvalue of T_p acting on V // n - relates to degree of approximation // D - value to twist by function Lseries(V,p,a_p,n,D) local answer,q; ALPHA := powers_of_alpha(p,a_p,n,n+1); MODSYM := table_of_modular_symbols(V,p,n,D); POWER := powers_of_oneplusT(p,n,n+1); TEICH := list_of_teich(p,n+1); GAMMA := powers_of_gamma(1+p,p,n+1); q1 := p^(n-1); q := q1*p; if a_p mod p ne 0 then answer := [0]; else answer := [0,0]; end if; for a in [1..p-1] do for j in [0..q1-1] do answer := add(answer, scale(measure(teich(a,TEICH)*gamma(j,GAMMA) mod q,p,n,ALPHA,MODSYM),lift_polynomial(oneplusT(j,POWER)))); end for; end for; answer:=scale(answer,1/D); answer:=scale(answer,1/MODSYM[#MODSYM]); return(answer); end function; function highest_power(n,p) local k,temp; k:=0; temp:=1; while (n ge temp) do temp := temp*p; k := k+1; end while; return(k-1); end function; function valuation_of_coefficients(f,p,n) local answer,j,temp,okay,bound,list_bad; answer := []; okay:=true; list_bad:=[]; for j in [1..Degree(f)+1] do if Coefficient(f,j-1) ne 0 then answer := answer cat [Valuation(Coefficient(f,j-1),p)]; else answer := answer cat [10000]; end if; bound := n-1-highest_power(j-1,p); if (Valuation(Coefficient(f,j-1),p) ge bound) and (j ne 1) then list_bad := list_bad cat [j]; end if; end for; if f eq 0 then answer := []; answer := answer cat [10000] cat [10000] cat [10000]; end if; return(); end function; function newton_polygon(C) local k,here,next,min_slope,answer,okay,prev_slope; answer:= []; okay:=true; here:=1; next:=1; while here lt #C do min_slope:=100000; for k in [here+1..#C] do if (C[k]-C[here])*(1.0)/(k-here) le 1.0*min_slope then min_slope := (C[k]-C[here])/(k-here); next := k; end if; end for; answer := answer cat [[next-here,-min_slope]]; here := next; end while; return(answer); end function; function find_last_nontriv(NP,p,sign) local slope,place; place := 0; if sign mod 2 eq 0 then slope := 1/(p*(p-1)); else slope := 1/(p-1); end if; for k in [1..#NP] do if (NP[k][2] gt 0) then if (NP[k][2] ne slope) or (NP[k][1] ne 1/slope) then place:=k; else slope:=slope/(p^2); end if; end if; end for; return(place); end function; function refine_newton_polygon(NP,p,a_p,sign) local answer,k,last; answer := []; if a_p mod p ne 0 then for k in [1..#NP] do if NP[k][2] gt 0 then answer := answer cat [NP[k]]; end if; end for; else last := find_last_nontriv(NP,p,sign); for k in [1..last] do answer := answer cat [NP[k]]; end for; end if; return(answer); end function; function minimize_ord(C,p,n) local k,start; start:=1; if C[1] eq 10000 then start := 2; end if; for k in [start..#C] do if C[k] ge n-1-highest_power(k-1,p) then C[k] := n-1-highest_power(k-1,p); end if; end for; return(C); end function; function in_list(list,num) local k,flag; flag:=false; for k in [1..#list] do if list[k] eq num then flag := true; end if; end for; return(flag); end function; function compute_rank(C,N,r,D) local rank; if C[1] eq 10000 then rank := 1; else rank :=0; end if; if (r eq 0) and (rank eq 1) and (KroneckerSymbol(D,N) eq 1) then //it has even rank >= 2. rank:=2; C[2]:=10000; end if; if (r eq 1) and (rank eq 1) and (KroneckerSymbol(D,N) eq -1) then //it has even rank >= 2. rank:=2; C[2]:=10000; end if; return(); end function; function check_NP(NP,C,list_bad,rank,p,a_p,n,sign) local temp,val,k,good; //checks if break points are accurate. good:=true; temp := refine_newton_polygon(NP,p,a_p,sign); val := 1; for k in [1..#temp] do val := val + temp[k][1]; if (in_list(list_bad,val)) then good := false; print "Failed at a breakpoint,",val; end if; end for; //checks if lowering the valuation of any unknown coefficient to its //minimal possible value affects the newton polygon. C_min := minimize_ord(C,p,n); //lowers all unknown coefficients to lowest possible valuation if rank eq 2 then C_min[2] := 10000; end if; NP_min := newton_polygon(C_min); if refine_newton_polygon(NP_min,p,a_p,sign) ne refine_newton_polygon(NP,p,a_p,sign) then good := false; print "Failed while lowering."; end if; return(good); end function; function lambda(NP) local k,lam; k:=1; lam := 0; while (k le #NP) and (NP[k][2] gt 0) do lam := lam + NP[k][1]; k := k+1; end while; return(lam); end function; function mu(NP,C,rank) local k,val,m,start; k:=1; val := 0; if (rank ne 2) then while (k le #NP) and (NP[k][2] gt 0) do if NP[k][2] lt 1000 then val := val + NP[k][1]*NP[k][2]; end if; k := k+1; end while; else if (#NP gt 0) and (NP[1][1] eq 1) then start := 3; else if #NP eq 0 then start := 1; else start := 2; end if; end if; k:=start; while (k le #NP) and (NP[k][2] gt 0) do if NP[k][2] lt 1000 then val := val + NP[k][1]*NP[k][2]; end if; k := k+1; end while; end if; if rank eq 0 then m := C[1]-val; else if rank eq 1 then m := C[2] - val; else m := C[3] - val; end if; end if; return(m); end function; function remove_two(list) local answer,k; answer := []; for k in [1..#list] do if list[k] ne 2 then answer := answer cat [list[k]]; end if; end for; return(answer); end function; function leading_term(C,rank) if rank eq 0 then return(C[1]); else if rank eq 1 then return(C[2]); else if rank eq 2 then return(C[3]); end if; end if; end if; end function; function compute_untwisted_rank(V,p,a_p) local L,C; L:=Lseries(V,p,a_p,2,1); L:=L[1]; C := valuation_of_coefficients(L,p,1); if C[1][1] ne 10000 then return(0); else return(1); end if; end function; function remove_unit_zeroes(NP) local k,answer; answer := []; for k in [1..#NP] do if NP[k][2] gt 0 then answer := answer cat [NP[k]]; end if; end for; return(answer); end function; //Has the space of modular symbols already pre-computed function iwasawa_invariants_w_curve_ord(N,V,p,a_p,n,D) local C,NP,k,lam,m,rank,j,list_bad,list_breapoints,C_min,NP_min,val,good,m2,lam2,temp,leading; L:=Lseries(V,p,a_p,n,D)[1]; C := valuation_of_coefficients(L,p,n); list_bad:=C[2]; C:=C[1]; //FIRST COMPUTE RANK rank := compute_untwisted_rank(V,p,a_p); C := compute_rank(C,N,rank,D); rank := C[2]; //changes valuation of a_1 to 10000 if guesses rank 2. C := C[1]; if rank eq 2 then list_bad := remove_two(list_bad); end if; NP := newton_polygon(C); good := check_NP(NP,C,list_bad,rank,p,a_p,n,0); //0 represents nothing. NP := remove_unit_zeroes(NP); leading := leading_term(C,rank); if (leading eq 0) and (in_list(list_bad,rank+1) eq false) then good:=true; end if; lam:=lambda(NP); m := mu(NP,C,rank); return(); end function; function scale_by_p(C) local k; for k in [1..#C] do if C[k] ne 10000 then C[k]:=C[k]+1; end if; end for; return(C); end function; function remove_forced_zeroes(NP,p,init_slope) local k,slope,temp; temp := []; k := 1; slope:=init_slope; while (k le #NP) and (NP[k][2] gt 0) do if NP[k][2] ne slope then temp := temp cat [NP[k]]; else if NP[k][1] - 1/slope ne 0 then temp := temp cat [[NP[k][1] - 1/slope,slope]]; end if; slope := slope/(p^2); end if; k := k+1; end while; return(temp); end function; //Has the space of modular symbols already pre-computed function iwasawa_invariants_w_curve_ss(N,V,p,a_p,n,D) local L,G,H,k,CG,CH,mg,mh,lamg,lamh,NPG,NPH,rank,rankg,rankh, list_badg,list_badh,denomg,denomh,goodg,goodh,list_breakpointsg,list_breakpointsh, leadingg,leadingh; L := Lseries(V,p,a_p,n,D); G := L[1]; H := L[2]; // denomg and denomh represent the number of p's in the denominator of the measure if n mod 2 eq 0 then denomg:=n div 2; denomh:=(n+2) div 2; else denomg:=(n+1) div 2; denomh:=(n+1) div 2; end if; CG := valuation_of_coefficients(G,p,n-denomg); CH := valuation_of_coefficients(H,p,n-denomh); list_badg := CG[2]; list_badh := CH[2]; CG:=CG[1]; CH:=CH[1]; //Computing the rank rank:=compute_untwisted_rank(V,p,a_p); CG := compute_rank(CG,N,rank,D); //changes a_1 if guesses rank 2 rankg := CG[2]; CG := CG[1]; CH := compute_rank(CH,N,rank,D); rankh := CH[2]; CH := CH[1]; if rankg ne rankh then print "WEIRD..ranks different in G and H"; PrintFile("notes","ranks different in G and H"); PrintFile("notes",D); end if; rank := rankg; if rank eq 2 then list_badg := remove_two(list_badg); end if; if rank eq 2 then list_badh := remove_two(list_badh); end if; NPG := newton_polygon(CG); NPH := newton_polygon(CH); goodg := check_NP(NPG,CG,list_badg,rank,p,a_p,n,0); //0 represents g goodh := check_NP(NPH,CH,list_badh,rank,p,a_p,n,1); //1 represents h leadingg := leading_term(CG,rank); leadingh := leading_term(CH,rank); if (a_p eq 0) then if (leadingg eq -1) and (in_list(list_badg,rank+1) eq false) then goodg:=true; end if; if (leadingh eq -1) and (in_list(list_badh,rank+1) eq false) then goodh:=true; end if; end if; if a_p eq 0 then NPG := remove_forced_zeroes(NPG,p,1/((p-1)*p)); NPH := remove_forced_zeroes(NPH,p,1/(p-1)); else if a_p eq -3 then NPG := remove_forced_zeroes(NPG,p,1/((p-1)*p)); NPH := remove_forced_zeroes(NPH,p,1/(p-1)*p^2); end if; end if; lamg := lambda(NPG); lamh := lambda(NPH); mg := mu(NPG,CG,rank)+1; mh := mu(NPH,CH,rank)+1; return(); end function; //This procedure returns many invariants of the curve corresponding //to V twisted by D. // //In the ordinary case the data returned is: // // //The newton polygon is returned as [n_1,s_1],[n_2,s_2],... where there are //n_1 roots of slope s_1, etc. // //In the supersingular case the data returned is similar. The first entry is //the rank, then the next five are the data for my "g" iwasawa function and //then the next five for my "h" iwasawa function. // function iwasawa_invariants_w_curve(N,V,p,a_p,n,D) if a_p mod p ne 0 then return(iwasawa_invariants_w_curve_ord(N,V,p,a_p,n,D)); else return(iwasawa_invariants_w_curve_ss(N,V,p,a_p,n,D)); end if; end function; From marlene@infomagic.com Mon Jan 29 08:30:53 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0THZLg18415 for ; Mon, 29 Jan 2001 12:35:21 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0TDEEm06422 for ; Mon, 29 Jan 2001 08:14:14 -0500 (EST) Received: from infomagic.com (MAX5-Port44.Downtown.InfoMagic.NET [63.229.95.244]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f0TDEFd32525 for ; Mon, 29 Jan 2001 06:14:15 -0700 Message-ID: <3A75708D.60D8F522@infomagic.com> Date: Mon, 29 Jan 2001 06:30:53 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: was@math.harvard.edu Subject: Re: more pics References: <01012802124101.02934@form> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: RO X-Status: A Good Morning from an Arizona Mom: Just looked at Jan. 27th photos again...the b & w are great, reminds of a little of my Dad's WWII photos, the architecture in Bordeaux is beautiful, like Cinderella's castle and the lamposts are quite a site. The tree in Photo 1390 looks cool in front of the building. I like the building in Photo 1383. Is 1387 a DOG in a car? Mom "William A. Stein" wrote: > http://modular.fas.harvard.edu/pics/ascent3/26Jan01-bordeaux/ > http://modular.fas.harvard.edu/pics/ascent3/27Jan01-bordeaux/ From coleman@math.berkeley.edu Tue Jan 30 01:58:46 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0U6xag19422 for ; Tue, 30 Jan 2001 01:59:36 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0U6wom16863 for ; Tue, 30 Jan 2001 01:58:50 -0500 (EST) Received: from [136.152.193.221] (as3-1-178.HIP.Berkeley.EDU [136.152.194.100]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id WAA24353 for ; Mon, 29 Jan 2001 22:58:49 -0800 (PST) Mime-Version: 1.0 Message-Id: Date: Mon, 29 Jan 2001 22:58:46 -0800 To: was@math.harvard.edu From: "Robert F. Coleman" Subject: some more work Content-Type: text/plain; charset="us-ascii" ; format="flowed" Status: RO X-Status: O I am connecting up our paper w2ith Mazur papers on deformations, in particular with the hypothetical dense weave. This leads to more computations involving Lawren table of slope zero forms. By eyesight it seems that the 3-deprivation of the slope 2 form is congruent to the slope 52 form modulo 3^5. It also seems that the twist of this form by Teichmuller is congruent to another such form. Couild you check this and search for similar phenomenon. I assume you have Lawren's tables. I'll send them to you anyway. -- Robert http://math.berkeley.edu/~coleman From coleman@math.berkeley.edu Tue Jan 30 01:59:33 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0U70Og19428 for ; Tue, 30 Jan 2001 02:00:24 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0U6xZm16880 for ; Tue, 30 Jan 2001 01:59:35 -0500 (EST) Received: from blue2.math.Berkeley.EDU (blue2.math.berkeley.edu [169.229.58.60]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id WAA24363 for ; Mon, 29 Jan 2001 22:59:34 -0800 (PST) Received: (from coleman@localhost) by blue2.math.Berkeley.EDU (8.9.3/8.9.3) id WAA01858; Mon, 29 Jan 2001 22:59:33 -0800 (PST) Date: Mon, 29 Jan 2001 22:59:33 -0800 (PST) Message-Id: <200101300659.WAA01858@blue2.math.Berkeley.EDU> X-Authentication-Warning: blue2.math.Berkeley.EDU: coleman set sender to coleman@robert-1.math.berkeley.edu using -f From: "Robert F. Coleman" To: was@math.harvard.edu Subject: lawren's table Status: RO X-Status: O I, 2. 3 + 3^2 + 3^5 + 3^6 + 2*3^7 + 3^8 + 2*3^9 + 3^11 + 3^13 + 2*3^14 + 2*3^15 + O(3^16)*q^2 3^3 + 3^4 + 3^5 + 3^6 + 2*3^8 + 2*3^10 + 3^11 + 2*3^14 + 3^15 + 2*3^16 + 2*3^17 + O(3^18)*q^6 3^2 + 3^4 + 2*3^6 + 3^7 + 3^8 + 3^12 + 3^13 + 2*3^15 + 3^16 + O(3^17)*q^3 3^4 + 2*3^6 + 2*3^8 + 3^10 + 3^11 + 3^12 + 3^15 + 2*3^16 + 3^17 + 2*3^18 + O(3^19)*q^9 2*3 + 3^2 + 3^3 + 2*3^4 + 2*3^5 + 3^6 + 3^8 + 2*3^9 + 2*3^12 + O(3^16)*q^5 2*3^3 + 3^4 + 3^6 + 2*3^7 + 3^9 + 2*3^10 + 2*3^12 + 3^13 + 2*3^14 + 3^15 + O(3^18)*q^15 2 + 3 + 2*3^3 + 2*3^4 + 2*3^5 + 3^6 + 3^7 + 2*3^8 + 2*3^9 + 3^10 + 3^12 + 2*3^13 + 3^14 + O(3^15)*q^7 2*3^2 + 3^3 + 2*3^4 + 3^6 + 3^7 + 2*3^9 + 3^12 + 2*3^14 + 3^15 + 3^16 + O(3^17)*q^21 3 + 2*3^2 + 2*3^3 + 2*3^5 + 3^6 + 2*3^7 + 3^8 + 2*3^9 + 3^10 + 3^14 + O(3^16)*q^11 3^3 + 2*3^4 + 3^7 + 2*3^8 + 3^9 + 3^10 + 3^11 + 3^12 + 3^13 + 3^14 + O(3^18)*q^33 2 + 2*3 + 2*3^2 + 3^5 + 2*3^6 + 2*3^8 + 3^9 + 3^11 + 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2*3^192 + 3^195 + 2*3^196 + 2*3^198 + 3^200 + 2*3^201 + 3^202 + 2*3^203 + O(3^204)*q^129 3 + 3^3 + 3^6 + 3^8 + 3^9 + 2*3^10 + 2*3^12 + 2*3^13 + 2*3^14 + 3^15 + 3^16 + 3^17 + 2*3^20 + 2*3^21 + 2*3^23 + 2*3^24 + 2*3^26 + 2*3^27 + 2*3^28 + 2*3^29 + 2*3^30 + 3^31 + 3^33 + 2*3^34 + 3^35 + 3^37 + 3^38 + 3^39 + 3^40 + 3^41 + 3^42 + 2*3^44 + 3^45 + 3^47 + 2*3^48 + 3^50 + 2*3^52 + 3^53 + 3^54 + 3^55 + 3^56 + 3^58 + 2*3^61 + 2*3^62 + 2*3^63 + 3^64 + 2*3^65 + 2*3^66 + 2*3^67 + 2*3^68 + 3^69 + 2*3^70 + 2*3^71 + 2*3^72 + 3^73 + 3^74 + 3^76 + 2*3^77 + 2*3^80 + 2*3^81 + 2*3^82 + 2*3^84 + 3^86 + 2*3^87 + 3^ 9 + 2*3^90 + 3^91 + 2*3^92 + 3^93 + 3^94 + 2*3^95 + 2*3^96 + 3^98 + 2*3^100 + 3^102 + 3^104 + 3^106 + 3^107 + 2*3^109 + 2*3^110 + 3^112 + 2*3^113 + 2*3^114 + 3^115 + 2*3^118 + 3^124 + 2*3^125 + 2*3^126 + 3^128 + 3^129 + 2*3^131 + 3^132 + 3^133 + 2*3^142 + 3^143 + 3^145 + 3^149 + 3^151 + 2*3^152 + 3^153 + 3^154 + 2*3^156 + 2*3^157 + 3^159 + 2*3^160 + 3^161 + 2*3^163 + 2*3^164 + 3^165 + 3^166 + 3^169 + 3^170 + 3^171 + 2*3^172 + 2*3^174 + 3^175 + 3^177 + 2*3^179 + 2*3^183 + 2*3! ^1! ! 84 + 2*3^186 + 3^187 + 2*3^188 + 3^189 + 2*3^191 + 3^192 + 2*3^194 + 3^196 + 3^197 + 3^198 + 2*3^200 + 3^201 + 2*3^202 + O(3^203)*q^47 3^99 + 2*3^100 + 3^101 + 3^102 + 2*3^103 + 3^104 + 3^106 + 2*3^107 + 3^108 + 3^109 + 3^111 + 3^112 + 2*3^114 + 2*3^116 + 2*3^117 + 2*3^119 + 2*3^120 + 2*3^122 + 2*3^123 + 2*3^124 + 3^125 + 3^127 + 2*3^128 + 2*3^130 + 2*3^133 + 2*3^134 + 3^139 + 3^140 + 3^142 + 2*3^146 + 3^147 + 2*3^149 + 3^152 + 3^153 + 2*3^154 + 2*3^155 + 3^157 + 2*3^159 + 2*3^160 + 3^161 + 2*3^162 + 3^163 + 2*3^165 + 3^168 + 3^170 + 2*3^171 + 3^172 + 2*3^176 + 3^178 + 3^181 + 3^183 + 2*3^184 + 2*3^186 + 3^187 + 2*3^188 + 2*3^189 + 3^190 + 2*3^191 + 3^193 + 3^194 + 2*3^196 + 3^199 + 3^201 + 3^203 + O(3^205)*q^141 From whitney@math.berkeley.edu Tue Jan 30 12:44:26 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UHavg19972 for ; Tue, 30 Jan 2001 12:36:57 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UHaBm07107 for ; Tue, 30 Jan 2001 12:36:11 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0UHiQ015483 for ; Tue, 30 Jan 2001 09:44:26 -0800 Date: Tue, 30 Jan 2001 09:44:26 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: Re: nice In-Reply-To: <01013017230303.16570@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O On Tue, 30 Jan 2001, William A. Stein wrote: > I will have to get to the bottom of this when I get back to Cambridge. First try kernel 2.4.1 (sure to be out by the time you get back), then try kernel 2.2.19-pre if 2.4.1 doesn't work. Note that 2.2.19-pre does not support high memory, so to use all 2GB of RAM on modular, you'll reduce the user address space to 2GB, so you'll probably need this patch I hacked up to make mmap() to grow downward. Cheers, Wayne From whitney@math.berkeley.edu Tue Jan 30 15:28:45 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UKLEg20251 for ; Tue, 30 Jan 2001 15:21:15 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UKKTm19650 for ; Tue, 30 Jan 2001 15:20:29 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0UKSjg15844 for ; Tue, 30 Jan 2001 12:28:45 -0800 Date: Tue, 30 Jan 2001 12:28:45 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: silly questions Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A silly question 1: why do you run 4 magma's at once on a machine? silly question 2: how do we feel about galen running eyecandy that uses 25% of a CPU? Cheers, Wayne From whitney@math.berkeley.edu Tue Jan 30 16:03:52 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UKuKg20356 for ; Tue, 30 Jan 2001 15:56:20 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UKtYm22301 for ; Tue, 30 Jan 2001 15:55:35 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0UL3qJ15896 for ; Tue, 30 Jan 2001 13:03:52 -0800 Date: Tue, 30 Jan 2001 13:03:52 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: Re: silly questions In-Reply-To: <0101302138160G.16570@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A On Tue, 30 Jan 2001, William A. Stein wrote: > So, in summary, I have 4 magma running at once because it is easier to > manage. OK, but it sounds like from your explanation it would make more sense for them to be at different nicenesses, since I think I understood that one pair was a short term set you wanted to finish quickly. > A tiny bit annoyed. In fact, I logged in a few minutes ago and > reniced "stars". I would think that renicing something suggests more than a 'tiny bit' of annoyance. I'm not sure if renicing stars is even effective, as stars only use 5% of a CPU, but it causes X to use 20% of CPU. In general resource allocation with regard to X is difficult, as a program can easily cause X to use lots of CPU or lots of memory. [e.g. netscape can load lots of images, passing them to X, X gets charged with the memory usage, so if OOM killer kicks in, it may kill X.] Anyway, galen seems to have suspended the job, as I had sent him email about it. > On the subject of me renicing other people's jobs, I wonder if it > would be sensible to explicitly say in the motd that "William reserves > the right to reduce the priority of other user's jobs if he is running > a computation." Well, perhaps a more blanket warning of 'non-number theory jobs maybe be reniced by the administrators without warning' would be more appropriate. On the other hand, the only real issue is with Galen, so I think it is better that I just reach a private understanding with him. > When you asked Hendrik for cluster money, did you suggest that the > cluster was > > a) A tool for William to do computations > > b) A tool for number theorists somehow associated with William to > do computations. I would say that the cluster is now (b). Clearly you are central, though, e.g. you are the only number theorist with the root password. > The original cluster that was purchased using the Chancellor Research > Grant was clearly a). True, but that has been more or less subsumed in the new cluster. In fact, officially the old cluster was "supplies" and the new cluster is "equipment"--the old cluster supplies were used in assembling the new cluster. So I will make sure that 2 of the 3 nodes are officially the property of Hendrik's grant, and as you and I discussed in September, he may arrange to send them to Harvard when I leave (May or August or December 2001). The node constructed from the former wish-bone will remain the property of the department. Cheers, Wayne From whitney@math.berkeley.edu Tue Jan 30 16:29:44 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0ULMDg20401 for ; Tue, 30 Jan 2001 16:22:13 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0ULLRm23432 for ; Tue, 30 Jan 2001 16:21:27 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0ULTil15947 for ; Tue, 30 Jan 2001 13:29:44 -0800 Date: Tue, 30 Jan 2001 13:29:44 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: Re: silly questions In-Reply-To: <0101302208220H.16570@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A On Tue, 30 Jan 2001, William A. Stein wrote: > By the way, I just added an acknowledgement that mentions you to the > bottom of > > http://modular.fas.harvard.edu/Tables/index.html > > Is this OK with you? That's fine, although I would say that I "assembled" it rather than "built" it. Didn't I assemble the whole thing, as opposed to "much"? As for the grants, perhaps you should credit UCB (or even the Vice-Chancellor for Research) instead of "my grant" and specify that Hendrik's grant is from the NSF. Cheers, Wayne From whitney@math.berkeley.edu Tue Jan 30 17:51:07 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UMhYg20854 for ; Tue, 30 Jan 2001 17:43:34 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UMgnm27630 for ; Tue, 30 Jan 2001 17:42:49 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0UMp7B16014 for ; Tue, 30 Jan 2001 14:51:07 -0800 Date: Tue, 30 Jan 2001 14:51:07 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: Re: Acknowledgement In-Reply-To: <01013023210400.17899@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O On Tue, 30 Jan 2001, William A. Stein wrote: > I'll change "built" to "chosen and assembled". The high-faluting term is "spec'ed" or "specified", but spec'ed sounds better :-) And if modular is part of our cluster, our cluster isn't much of a cluster, but I guess that's true. Neither of us has done much to unify the separate machines except installing ruptime :-) The wording does't really matter, but I couldn't resist a couple more comments. :-) Cheers, Wayne From mbaker@math.harvard.edu Tue Jan 30 17:55:13 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UMtwg20860 for ; Tue, 30 Jan 2001 17:55:58 -0500 Received: from dirichlet.math.harvard.edu (dirichlet.math.harvard.edu [140.247.28.18]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UMtDm28177 for ; Tue, 30 Jan 2001 17:55:13 -0500 (EST) Received: from localhost (mbaker@localhost) by dirichlet.math.harvard.edu (8.11.1/8.11.1) with ESMTP id f0UMtDb08822 for ; Tue, 30 Jan 2001 17:55:13 -0500 (EST) X-Authentication-Warning: dirichlet.math.harvard.edu: mbaker owned process doing -bs Date: Tue, 30 Jan 2001 17:55:13 -0500 (EST) From: Matt Baker To: "William A. Stein" Subject: Re: generating Hecke algebra In-Reply-To: <01013015414200.16570@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A Hi William I know what the problem is: it's that Edixhoven and Coleman are lying at the end of their article. Here's the best result I could dig out of the literature: in Atkin's article "Weierstrass Points at cusps of Gamma_0(n)", he proves the following: Theorem: Suppose that $n$ is not squarefree. Then infinity is a Weierstrass point on Gamma_0(n) except in case (a) below and possibly in cases (b) - (d) below: (a) n = 4,8,9,16,25,27,32,36,49,81 (b) n=8p,16p (c) n=(p^2)(q) and not both p and q are 1 mod 12 (d) n = (p^2)qr and neither x^2 + 1 = 0 (mod pqr) nor x^2 - x + 1 = 0 (mod pqr) is solvable. Note that case (c) includes n=28. -- Matt On Tue, 30 Jan 2001, William A. Stein wrote: > On Monday 29 January 2001 21:05, you wrote: > > > Cool. Evidently, "sufficiently composite" N are > 200. > > > > Hmmm, speaking of Coleman-Edixhoven, they say at the end of their paper > > that if N is divisible by 4 or 9, then infinity _is_ a Weierstrass point > > on X_0(N). I'll have to think about why this does not contradict the > > experimental evidence you're telling me. > > That's interesting. Have we mistranslated the criterion? Before > going further, I better tell you what I think a Weierstrass point is. > A point $P$ on a Riemann surface $X$ of genus $g$ is NOT a Weierstrass > point if: > > "The space of holomorphic differentials vanishing at $P$ with order > at least $i$ has dimension exactly equal to $g-i$, for $i=0,...,g$." > > First, this is an intrinsic property. It is not a relative property > involving a covering of curves. Thus, for example, curves of genus > $0$ have no Weierstrass points. I think elliptic curves can't have > Weierstrass points either because the (up to nonzero scalars) > holomorphic differential has no zeros. So, people probably only > consider Weierstrass points for curves genus at least 2. > > The curve X_0(28) is a curve of genus 2, and 28 is divisible by 4. > However, the q-expansions of holomorphic functions at infinity are: > > q - 2*q^3 + q^7 + O(q^8) > q^2 - q^4 - 2*q^6 + O(q^8) > > To see this, note that the elliptic curve of conductor 14 has q-expansion > q - q^2 - 2*q^3 + q^4 + 2*q^6 + q^7 + O(q^8) > so the forms > > q - q^2 - 2*q^3 + q^4 + 2*q^6 + q^7 + O(q^8), and > q^2 - q^4 - 2*q^6 + O(q^8) > > form a basis for level 28 cusp forms. Thus infinity is not a > Weierstrass point. > > Hmmm. In case we don't resolve this by the day after tomorrow, I'll > just ask Edixhoven, as I'm going to be visiting him in Rennes starting > Thursday. > > -- William > > > > > > > > > > From whitney@math.berkeley.edu Tue Jan 30 19:02:14 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UNsgg20919 for ; Tue, 30 Jan 2001 18:54:42 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0UNrvm29816 for ; Tue, 30 Jan 2001 18:53:57 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0V02FS16079 for ; Tue, 30 Jan 2001 16:02:15 -0800 Date: Tue, 30 Jan 2001 16:02:14 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: kill -STOP 9684 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A I did 'kill -STOP 9684' on mf3, as process 9684 (magma) had grown to 710372K in size. mf3 was still fairly responsive, but kswapd was using more CPU than some of the magma jobs, so that didn't seem very efficient. I hope this is OK. Maybe this one needs to run on mf2? Maybe you should ulimit your jobs to 512MB by default, and increase it only if necessary? BTW, is it the case that in situations like this you will always prefer one of {suspend,kill} for the job? If so, let me know, but I imagine that it may depend (e.g., once the other jobs are finished, you could 'kill -CONT 9684', so that speaks for kill -STOP, but if you were running a script that did several levels in a row, then you'd probably want to kill the job, as then your script would proceed.) Cheers, Wayne From whitney@math.berkeley.edu Tue Jan 30 19:31:39 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0V0OAg20931 for ; Tue, 30 Jan 2001 19:24:10 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0V0NOm00486 for ; Tue, 30 Jan 2001 19:23:24 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0V0Vg216135 for ; Tue, 30 Jan 2001 16:31:43 -0800 Date: Tue, 30 Jan 2001 16:31:39 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: job 17964 on mf2 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O .. . . got to be 1000M in size, and kswapd was using 50% of a CPU. Again, mf2 was responsive enough, but this didn't seem efficient, so I reniced 17964 from 5 to 10. This caused the other three magma jobs to get most of the CPU, kswapd to need only about 10%, and 17964 to get about 5%. If 17964 is the important one, then I guess you should kill -STOP the others until it finishes. Cheers, Wayne From marlene@infomagic.com Tue Jan 30 21:01:30 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0V1jZg20955 for ; Tue, 30 Jan 2001 20:45:35 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0V1inm02708 for ; Tue, 30 Jan 2001 20:44:49 -0500 (EST) Received: from infomagic.com (MAX2-Port31.Downtown.InfoMagic.NET [63.229.94.101]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f0V1inK17550 for ; Tue, 30 Jan 2001 18:44:50 -0700 Message-ID: <3A7771F9.41509DED@infomagic.com> Date: Tue, 30 Jan 2001 19:01:30 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: was@math.harvard.edu Subject: Re: 3 new days References: <01013019542808.16570@form> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: RO X-Status: O Thanks, I'll check later.....dinnertime, brought home Dragon Bowl Express on Milton Street....good food...you can get a vegetarian combo for about $5.00, a lot of food. Mom "William A. Stein" wrote: > Hi, > > I added three new sets of photos. Most of these were taken in Bonn. > In the second set, you can see a butterfly made out of a carrot. The > third day contains a Jewish memorial, among other things. > > http://modular.fas.harvard.edu/ascent3/pics/29Jan01-bonn/ > http://modular.fas.harvard.edu/ascent3/pics/30jan01-bonn/ > http://modular.fas.harvard.edu/ascent3/pics/30jan01-last_day_in_bonn/ > > -- William From texasborg@hotmail.com Tue Jan 30 22:04:33 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0V35Og20984 for ; Tue, 30 Jan 2001 22:05:24 -0500 Received: from hotmail.com (f108.law10.hotmail.com [64.4.15.108]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0V34dm04792 for ; Tue, 30 Jan 2001 22:04:39 -0500 (EST) Received: from mail pickup service by hotmail.com with Microsoft SMTPSVC; Tue, 30 Jan 2001 19:04:33 -0800 Received: from 64.157.117.164 by lw10fd.law10.hotmail.msn.com with HTTP; Wed, 31 Jan 2001 03:04:33 GMT X-Originating-IP: [64.157.117.164] From: "rebecca colborg" To: was@math.harvard.edu Subject: help bill! Date: Tue, 30 Jan 2001 21:04:33 -0600 Mime-Version: 1.0 Content-Type: text/html Message-ID: X-OriginalArrivalTime: 31 Jan 2001 03:04:33.0517 (UTC) FILETIME=[8956D9D0:01C08B32] Status: RO X-Status: A
Hey Bill,
       How was your summer touring?  Mine was great.  I got to stay with a family in Holland, and they gave the group I was in a huge party and we gave them a big concert.  My favorite place was Chamonix, France in the Alps because it was so beautiful.  While I was over in Europe, it rained all the time for the first half of the trip.
     This year I'm applying to TAMS (Texas Accademy of Math and Science), a program up at UNT in Denton where I'll finnish my jounior and senior year of high school while taking my first two years of college.  I've taken the SAT already but had to raise my math score 10-15 points to get in.  I had an 1130 and needed an 1100, but the math score has to be at least a 600.  
     I have a math question for you, no one in my class can answer.                                        A playhouse has a base of (2x + 4) by (?), a hight of (4x), and the roof has an altitude of       (x + 1), with a total volume of  9 ^3 + 46x^2 + 59x + 6 cubic feet.

I've tried and this is as far as I can get.

(2x +4)(4x)(?) + .5(x + 1)(2x + 4)(?) = 9x^3 + 46x^2 + 59x + 6                                      

(?)(8x^2 + 16x) + (?)(x^2 + 3x + 2) = 9x^3 +...+ 6                                                          

(?)(8x(x + 2)(x - 2)(x +2)) = 9x^3 +...+ 6                                                               

 synthetic division of x = -2                                                                                                    [-2]      9     46     59     6                                        &n! bsp;                                                    

                   -18   -56   -6

           9x^2 + 28x + 3 = 0             

quadratic to get  (-1/9) & -3 for x.

I just can't figure out which one is the width of the base (?)   

-Rebecca Colborg


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From whitney@math.berkeley.edu Tue Jan 30 22:43:36 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0V3a0g20995 for ; Tue, 30 Jan 2001 22:36:00 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0V3ZFm05562 for ; Tue, 30 Jan 2001 22:35:15 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0V3hal16228 for ; Tue, 30 Jan 2001 19:43:36 -0800 Date: Tue, 30 Jan 2001 19:43:36 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: Re: kill -STOP 9684 In-Reply-To: Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O On Tue, 30 Jan 2001, Wayne Whitney wrote: > I did 'kill -STOP 9684' on mf3, as process 9684 (magma) had grown to > 710372K in size. I have now done 'kill 9684; kill -CONT 9684' (you have to continue the job so that it will receive the kill signal and exit), as mf3 had run out of swap. In case you are wondering how it is that I am paying attention to all this, it is really easy: I do ruptime, and if the short time average is integral, then I consider that all is happy. If it's far from integral (fractional part near 0.5), then I look to see what is going on: usually a magma job has gotten too large and is swapping excessively, or somebody is running obnoxious eyecandy. As to the latter, I think Galen and I have reached an understanding, so this shouldn't happen anymore. For one thing GNOME runs xscreensaver automatically, and he was unaware of this, as the monitor was blanking before xscreensaver started (xscreensaver is not supposed to run while the monitor is blanked, but this seems to be broken). Cheers, Wayne From rpollack@math.harvard.edu Wed Jan 31 12:53:03 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VHtkW21912 for ; Wed, 31 Jan 2001 12:55:46 -0500 Received: from hum9.math.harvard.edu (hum9.math.harvard.edu [140.247.29.207]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VHrLm28644 for ; Wed, 31 Jan 2001 12:53:24 -0500 (EST) Received: from localhost (rpollack@localhost) by hum9.math.harvard.edu (8.11.1/8.11.1) with ESMTP id f0VHr3L10124 for ; Wed, 31 Jan 2001 12:53:08 -0500 (EST) X-Authentication-Warning: hum9.math.harvard.edu: rpollack owned process doing -bs Date: Wed, 31 Jan 2001 12:53:03 -0500 (EST) From: Robert Pollack To: "William A. Stein" Subject: modularsymbolodd Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O Hi William, I just tried using your ModularSymbolOdd procedure on X_0(11) and it seems to always yield 0. Could you try it to see if I'm doing something wrong? Thanks, Robert From gaer@math.harvard.edu Wed Jan 31 12:40:39 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VHuvW21916 for ; Wed, 31 Jan 2001 12:56:57 -0500 Received: from localhost (gaer@localhost) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VHedK27355 for ; Wed, 31 Jan 2001 12:40:44 -0500 (EST) Date: Wed, 31 Jan 2001 12:40:39 -0500 (EST) From: Arthur Gaer To: everyone@math.harvard.edu Subject: Unscheduled Network Outage (01/31/01) (fwd) Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O It's not just you: Harvard's connectivity to most of the Internet is currently broken. Arthur ---------- Forwarded message ---------- Date: Wed, 31 Jan 2001 12:21:36 -0500 (EST) From: FAS Computer Services To: FAS Network Operations Newsgroups: harvard.hascs Subject: Unscheduled Network Outage (01/31/01) Currently, connectivity between the FAS Network and parts of the Internet is unavailable. Our service provider (UIS), has been notified and will be working to fix the problem. We will post updates as they become available. Connectivity to Internet II sites initially appears to be unaffected by this outage. - FAS Network Operations - From whitney@math.berkeley.edu Wed Jan 31 13:20:53 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VIEuW21926 for ; Wed, 31 Jan 2001 13:14:56 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VICem29512 for ; Wed, 31 Jan 2001 13:12:44 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0VIKrM16663 for ; Wed, 31 Jan 2001 10:20:53 -0800 Date: Wed, 31 Jan 2001 10:20:53 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: mf2, mf3 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O Each of these machines ran out of swap (mf2 has 2GB of swap!), so rather than let them thrash until the OOM killer kicked in (I don't know if the 2.2.19pre OOM killer is clever or not, certainly earlier 2.2.x OOM killers were random!), I killed the biggest 'size' magma on each. Cheers, Wayne From andrew@sabre.la.asu.edu Wed Jan 31 14:43:27 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VJhZW21956 for ; Wed, 31 Jan 2001 14:43:35 -0500 Received: from sabre.la.asu.edu (sabre.la.asu.edu [129.219.44.34]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VJgom05311 for ; Wed, 31 Jan 2001 14:42:50 -0500 (EST) Received: from sabre.la.asu.edu (sabre.la.asu.edu [129.219.44.34]) by sabre.la.asu.edu (8.9.3+Sun/8.9.3) with SMTP id MAA25159; Wed, 31 Jan 2001 12:43:27 -0700 (MST) Message-Id: <200101311943.MAA25159@sabre.la.asu.edu> Date: Wed, 31 Jan 2001 12:43:27 -0700 (MST) From: Andrew Bremner Reply-To: Andrew Bremner Subject: Re: Accommodation To: was@math.harvard.edu Cc: andrew@sabre.la.asu.edu MIME-Version: 1.0 Content-Type: TEXT/plain; charset=us-ascii Content-MD5: DxyZ3VoD2jXfqu2oG94zKw== X-Mailer: dtmail 1.3.0 @(#)CDE Version 1.4 SunOS 5.8 sun4u sparc Status: RO X-Status: O William, You're booked for the nights of Wednesday, Thursday 7, 8 March at the Super8 Motel: confirmation number 21656. This should be direct billed to the Department. Address is 1020 E. Apache Blvd, Tempe. I'm putting a map in the post. A. From fcale@math.berkeley.edu Wed Jan 31 15:44:28 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VKjDW22055 for ; Wed, 31 Jan 2001 15:45:13 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VKiTm09443 for ; Wed, 31 Jan 2001 15:44:29 -0500 (EST) Received: from pub-708c-14.math.Berkeley.EDU (pub-708c-14.Math.Berkeley.EDU [169.229.58.109]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id MAA16332 for ; Wed, 31 Jan 2001 12:44:28 -0800 (PST) Received: from localhost (fcale@localhost) by pub-708c-14.math.Berkeley.EDU (8.9.3/8.9.3) with ESMTP id MAA25360 for ; Wed, 31 Jan 2001 12:44:28 -0800 (PST) X-Authentication-Warning: pub-708c-14.math.berkeley.edu: fcale owned process doing -bs Date: Wed, 31 Jan 2001 12:44:28 -0800 (PST) From: Frank Calegari To: William A Stein Subject: Re: RFC: SGA on the web In-Reply-To: <0010292012080I.01129@modular> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O Dave Savitt asked me about "the project". I didn't think I was supposed to release any information, so I will let you do it instead... Cheers, Frank. From whitney@math.berkeley.edu Wed Jan 31 16:12:40 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VL4tW22074 for ; Wed, 31 Jan 2001 16:04:55 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VL4Bm10459 for ; Wed, 31 Jan 2001 16:04:11 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f0VLCeF17060; Wed, 31 Jan 2001 13:12:40 -0800 Date: Wed, 31 Jan 2001 13:12:40 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" cc: Galen Huntington Subject: VM on mf3 Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O Well, you've managed to trip the OOM killer on mf3: Jan 30 19:21:38 mf3 kernel: VM: killing process magma.exe Jan 31 13:01:14 mf3 kernel: VM: killing process xload Jan 31 13:01:15 mf3 kernel: VM: killing process panel Jan 31 13:01:47 mf3 kernel: VM: killing process gnome-session This last time around it was not so clever, it did not kill a magma. I happened to notice it (the load briefly hit 12 on mf3 before galen got logged out, presumably by the last line), so I killed a magma (the largest 'size' one) Cheers, Wayne From DianBlu@cdw.com Wed Jan 31 17:51:59 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VMr0W22123 for ; Wed, 31 Jan 2001 17:53:00 -0500 Received: from [12.32.90.183] (falcon.cdw.com [12.32.90.183]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VMqFm17216 for ; Wed, 31 Jan 2001 17:52:16 -0500 (EST) Message-ID: <8189F8D452FDDE428137D3D89BBE71345C4BFD@pow.corp.cdw.com> From: DianBlu@cdw.com To: was@math.harvard.edu Subject: Your Epson Stylus Photo Printer... Date: Wed, 31 Jan 2001 16:51:59 -0600 MIME-Version: 1.0 X-Mailer: Internet Mail Service (5.5.2650.21) Content-Type: text/plain; charset="iso-8859-1" Status: RO X-Status: O Hi William- Your printer has been waiting to be drop shipped to you straight from the manufacturer. Meanwhile, the manufacturer has discontinued the product. ....Technology... lovely how it changes constantly, isn't it? I wanted to let you know about the situation, and someone at your house told me email was the best way to get to you. (Hope you are having a good time in Europe, by the way) I am looking at comperable products for you right now and will send you a couple of alternatives via email. You can weigh your options and get back to me. Sorry for the inconvenience-- more info on the way. > Diane Bluett > Account Manager CDW*G, Inc Computing Solutions Built for Government & Education > CDW-G is Your Direct Solutions Provider > Toll Free: 877-530-8482 > Direct Phone: 847-968-9800 > Direct Fax: 847-968-1800 > Email: dianblu@cdwg.com > Web Address: www.cdwg.com > From dianblu@cdw.com Wed Jan 31 18:04:30 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VN5PW22131 for ; Wed, 31 Jan 2001 18:05:25 -0500 Received: from outmail.cdw.com (outmail.cdw.com [12.32.91.121]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f0VN4fm17636 for ; Wed, 31 Jan 2001 18:04:41 -0500 (EST) Received: from as400.cdw.com ([10.1.1.164]) by outmail.cdw.com with Microsoft SMTPSVC(5.0.2195.1600); Wed, 31 Jan 2001 17:04:30 -0600 From: To: Subject: CDW Quote/Order DS59613 Message-ID: X-OriginalArrivalTime: 31 Jan 2001 23:04:30.0986 (UTC) FILETIME=[2B2CC6A0:01C08BDA] Date: 31 Jan 2001 17:04:30 -0600 Status: RO X-Status: A Here you go William- The Epson I quoted here is the printer that replaced the one you originally wanted, but is quite a bit higher priced. The Canon is a comperable price and will still work with PC or MAC and work fine for you. I also quoted on an HP that is about the same price because many people trust the HP brand over Canon-- it is up to you. All of them will print photo quality and work with either PC or MAC. Please let me know what you'd like to do. Thanks! --------------------------------------------------------------------------------- This e-mail best viewed in a fixed font such as Courier. WILLIAM STEIN Thank you for considering CDW for your computing needs. Following are the details of your quote/order. Quote Date and Time: 01-31-01 , 16:53:07 Quote Number: DS59613 P.O. Number: PRINTERS QUOTE Terms: Request Terms Shipped Via: UPS COMMERCIAL GROUND ================================================================================ QTY. CDW NO. DESCRIPTION/MFG. PART NUMBER UNIT PRICE EXT. PRICE 1 207473 EPS STYLUS PHOTO 870 INKJET 230.00 230.00 EPS-C304101 1 193408 CANON BJC8200 PHOTO 1200X1200 DPI 188.00 188.00 CAN-Q30-3201-US1 1 215370 HP DESKJET 935C INKJET PRINTER MAC 191.00 191.00 H-P-C6427E#A2L Subtotal 609.00 Sales Tax .00 Freight 28.77 Total 637.77 ================================================================================ For future reference, you may also use your extranet to view your order history and the status of all of your CDW orders. You can also use your extranet to view your quote history or retrieve new quotes. Ship To: WILLIAM STEIN 345 HARVARD ST HARVARD UNIVERSITY- MATH CAMBRIDGE MA 02138-4248 Bill To: WILLIAM STEIN 345 HARVARD ST HARVARD UNIVERSITY- MATH CAMBRIDGE MA 02138-4248 Once again, thank you for choosing CDW. We stand ready to serve you. DIANE BLUETT Direct Line: 877-530-8482 Fax: 847-968-1800 dianblu@cdw.com ================================================================================ From verrill@math.ku.dk Wed Jan 31 17:58:45 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f1102EW22192 for ; Wed, 31 Jan 2001 19:02:14 -0500 Received: from localhost.localdomain (IDENT:root@ip206.opnxr2.ras.tele.dk [195.41.168.206]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f1101Rm19285 for ; Wed, 31 Jan 2001 19:01:29 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id XAA17824 for ; Wed, 31 Jan 2001 23:58:46 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Wed, 31 Jan 2001 23:58:45 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: Another improvement for finding generators In-Reply-To: <01013018074204.16570@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O Hi William, Here is another improvement for your intersection paring program, I think this should cut the time in half (so if you've also changed to the new version of the "MatrixToWord" that will be 8 times faster than we originally had it). I finally read through your method of finding generators properly, and realised it is exactly the same as the method I'm using to find generators - except by choosing a good connected domain I am trying to maximize the number of times that what you call x*S*s(x)^{-1} and x*T*t(x)^{-1} are equal to 1 or -1. Anyway, one remark on your method is that if s(x) = y (Recall that this means x and y are coset reps, and xS \sim y) then also s(y) = x, so, we have x*S*s(x)^(-1) = (y*S*s(y)^(-1))^(-1) So, you don't need to count both of these as generators. And looking at your output of GeneratorsForGamma0(N), you'll see you do get pairs g and -1/g occuring, so that's twice as many as you need. In general, perhaps you can add a list, and check off ends of S pairs so you don't in effect count any pair twice. In some cases this will nearly half the number of generators you get. OK, there are also the T relations, but for N=p prime I expect that your program only gives a few of these; it's not difficult to arange that it only gives 2 of these, just choose coset reps I and ST^i for i=0..p-1. In fact, I just realised that from this observation, you can very easily see how to prove the statement Merel makes at the end of his paper about how ST^{k}.S.T^{k0}S give generators for Gamma_0(p). (in your notation, just use that s(ST^{k}) = ST^{-k0} and just write down what the condition on k0 and k is. You'll get the condition Merel gives) (I wonder if Merel noticed he is counting g and -1/g, ie, double what's needed?) The T relations are just T, (since t(I) = I, so ITt(I)^(-1) = T) and ST^pS (since t(ST^(p-1))=S, so ST^{p-1}TS^{-1} = -ST^pS (this is the inverse of what Merel writes)) (Hey, thanks again for the image of the paper, I'm looking at it on my screen right now!) Helena From verrill@math.ku.dk Thu Feb 1 08:07:48 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f11EB9W22699 for ; Thu, 1 Feb 2001 09:11:09 -0500 Received: from localhost.localdomain (IDENT:root@verrill.math.ku.dk [130.225.103.24]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f11EAPm06134 for ; Thu, 1 Feb 2001 09:10:26 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id OAA29556 for ; Thu, 1 Feb 2001 14:07:49 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Thu, 1 Feb 2001 14:07:48 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: Arizona In-Reply-To: <01013018074204.16570@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O By the way, I asked you about Tuscon vs Pheonix, but I decided I better just get the ticket anyway, which is to Tuscon, since things are selling out so fast. Even the one I asked about yesterday had gone today, and to go to Pheonix I have to arrive a day earlier cos things are sold out. Arizona must be a really popular destination! Helena From bgv@pacific.net.sg Thu Feb 1 10:38:54 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f11FMLW22795 for ; Thu, 1 Feb 2001 10:22:21 -0500 Received: from seed.pacific.net.sg (seed.pacific.net.sg [203.120.90.77]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f11FLam09509 for ; Thu, 1 Feb 2001 10:21:37 -0500 (EST) Received: from pop1.pacific.net.sg (pop1.pacific.net.sg [203.120.90.85]) by seed.pacific.net.sg with ESMTP id f11FLT307893 for ; Thu, 1 Feb 2001 23:21:29 +0800 (SGT) Received: from pc27 (ppp67.dyn67.pacific.net.sg [210.24.67.67]) by pop1.pacific.net.sg with SMTP id XAA18833 for ; Thu, 1 Feb 2001 23:21:23 +0800 (SGT) Message-ID: <000a01c08c65$16c88740$107ffea9@mandelbaum.usyd.edu.au> From: "Beatrice G. Venturi" To: References: <002801c07e48$77893640$0f7afea9@mandelbaum.usyd.edu.au> <01012816450009.10286@form> Subject: :o) Date: Fri, 2 Feb 2001 02:38:54 +1100 MIME-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: 7bit X-Priority: 3 X-MSMail-Priority: Normal X-Mailer: Microsoft Outlook Express 5.00.2014.211 X-MimeOLE: Produced By Microsoft MimeOLE V5.00.2014.211 Status: RO X-Status: O Aloha there!!!... Well, a 3-night airfare and hotel package to Sydney from Singapore (rtn) is about $1100. Each additional nights accomodation is $150 per room. ...that's the rate we are paying to ship ourselves to Sydney for graduation (leaving on the 28th of March, graduation is on the 30th and we will return on 3 April.). I am not sure what the rate is like in Sydney. I know that return fare from Sydney to Singapore is between $600-$850 depending on airline and season. My suggestion is to look into a package (airfare + hotel) deal. You may also want to look into flying home direct from Singapore. Have sent my resume to Singapore Airlines and am now in the ' fingers crossed' mode!!!.... I'll get the uni to sort out the GPA thingie if anyone asks for it...and I agree with you about the varying standards at universities worldwide!!!... Ah....Bordeaux...; wine, grapes and wine???... You are too lucky!!!... so, when can I call you Dr. Stein???... Am taking it slow at present but I am gonna sign up for GMAT courses as soon as I get back from graduation...might as well prepare to do the exam for when I want to embark on 'Operation MBA'....probably in a couple of years time. Am bent on going to the States for that!!!... So how's the going apart from the conference circuit???!!!... Miss Sydney???... Well, I better go catch my zzzzz....I am sleeping an average of 8 hours for the first time in a loooong time!!!... Ciao for now, Take care and Write soon!!! Hugs, Beatrice :o) ----- Original Message ----- From: William A. Stein To: Beatrice G. Venturi Sent: Monday, January 29, 2001 8:45 AM Subject: Re: :o) > On Sunday 14 January 2001 11:38, you wrote: > > Cool!!!...Australia again!!!...AND Singapore on June 1-3???!!!... > > I think I will be soooo here!!!... > > The only problem is that it might cost too much for me to fly between > Australia and Singapore. Do you have any idea how much it should > cost? Is it a thousand dollars or a few hundred or what? > > > They want a GPA of at least 3.75. Any idea how to calculate this??... I > > I know how it works in the United States. You get letter grades A, B, > C, and D. An A counts as a 4, a B as a 3, a C as a 2, and a D as a 1. > You add up the numerical equivalents and divide by the number of > grades. It seems bizarre to require "a GPA of at least 3.75" because > the way grades are given out various drastically from University to > University. At some places, e.g., Caltech, a GPA of 3.75 would be > extremely difficult to obtain, but at others (Stanford?) it wouldn't > be so hard. > > > have a distinction average so I can't be that far off... Am getting my > > resume/application ready and will probably send it in after or just before > > graduation (the date is still to be announced!!!)... > > My advice: Let several people you trust look at your resume and ask > them to be open in their criticism. Doing this helped me immensely > when I last applied for jobs. > > I spent the last few days in Bordeaux, which was quite beautiful. I > took tons of pictures, which you can see at > > http://modular.fas.harvard.edu/pics/ascent3/25Jan01-bordeaux > http://modular.fas.harvard.edu/pics/ascent3/26Jan01-bordeaux > http://modular.fas.harvard.edu/pics/ascent3/27Jan01-bordeaux > > -- William From k.buzzard@ic.ac.uk Fri Feb 2 05:19:30 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12ALFW23862 for ; Fri, 2 Feb 2001 05:21:15 -0500 Received: from judy.ic.ac.uk (judy.ic.ac.uk [155.198.5.28]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12AKWm21693 for ; Fri, 2 Feb 2001 05:20:33 -0500 (EST) Received: from juliet.ic.ac.uk ([155.198.5.4]) by judy.ic.ac.uk with esmtp (Exim 2.12 #1) id 14OdKS-00025K-00 for was@math.harvard.edu; Fri, 2 Feb 2001 10:20:32 +0000 Received: from geometry.ma.ic.ac.uk ([155.198.192.7]) by juliet.ic.ac.uk with esmtp (Exim 2.12 #1) id 14OdKh-0006rk-00 for was@math.harvard.edu; Fri, 2 Feb 2001 10:20:47 +0000 Received: from kbuzzard ([155.198.192.134] helo=kbuzzard.ma.ic.ac.uk ident=buzzard) by geometry.ma.ic.ac.uk with esmtp (Exim 1.90 #1) id 14OdKQ-0005BX-00; Fri, 2 Feb 2001 10:20:30 +0000 Date: Fri, 2 Feb 2001 10:19:30 +0000 (GMT) From: Kevin Buzzard X-Sender: Reply-To: Kevin Buzzard To: "William A. Stein" Subject: Re: XFree 4 In-Reply-To: <01020117231800.22620@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O > I finally had that resume-from-hibernation problem that you have, where > the screen memory after resume is corrupted. Hooray! Thanks for the X tips. I don't have the time to upgrade anything really so I definitely won't be doing it :-) I have much less teaching this term but I am still catching up on all the jobs which should have been done months ago. Today the main task is to review Shkimura's new book for the LMS. They sent it me 4 months ago and the deadline is today. So I guess I should think about opening it soon...*doh*...actually, I have read the introduction. I might just waffle about how interesting the zeta function is and then copy bits out of the intro and have done with it. > I'm on my way > to Rennes at the moment. The TGV and Thalys are frightenly fast. Aren't they just! Say hi to Bas and Reinie and Luc and the other one for me. Kevin From verrill@math.ku.dk Fri Feb 2 04:57:14 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12B0hW23875 for ; Fri, 2 Feb 2001 06:00:44 -0500 Received: from localhost.localdomain (IDENT:root@ip89.opnxr1.ras.tele.dk [195.215.236.89]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12Axtm22214 for ; Fri, 2 Feb 2001 06:00:00 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id KAA06911 for ; Fri, 2 Feb 2001 10:57:15 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Fri, 2 Feb 2001 10:57:14 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: Re: usna In-Reply-To: <01020117412405.22620@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O > What do you mean "his BP job"? It is "a BP job". I think I meant "this". Typo, like when "not" means "now". Thanks for explaining it. > will be at Harvard a total of five years. I think this will be better > for me than having several postdoc positions over five years. In the > meantime, I'll keep my eyes open for a tenure-track position. Yep, I think it will be a lot better. Good for you! Helena From verrill@math.ku.dk Fri Feb 2 05:02:26 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12B5nW23884 for ; Fri, 2 Feb 2001 06:05:49 -0500 Received: from localhost.localdomain (IDENT:root@ip89.opnxr1.ras.tele.dk [195.215.236.89]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12B56m22359 for ; Fri, 2 Feb 2001 06:05:06 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id LAA07529 for ; Fri, 2 Feb 2001 11:02:26 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Fri, 2 Feb 2001 11:02:26 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: Re: ... In-Reply-To: <01020117341903.22620@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O > I think we should submit our paper to "Noriko's Journal". > I don't think it belongs in the AMS J. of Computation because > it isn't very computational. What do you think? Yep, I think that seems like a reasonable place to submit it. > Before submitting it, we could add in some computations of > j-invariants of CM and non-CM elliptic curves of weight 4 > as a third example, if you want. Yep, this is a good idea. Noriko would probably like that. We could make a note about why the curve is real, and why CM in the form gives CM in the curve (if we can work this out; I think we talked about this before but didn't get it done properly.) We already had most of this data in a previous version of the paper, so we just have to check through the computations, maybe improve the degree of acuracy - remember Noam Elkies and John Cremona both pointed out places where c_4 and c_6 were out by a fairly large amount (I think one was 8.something instead of 0) Helena From verrill@math.ku.dk Fri Feb 2 05:05:15 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12B8dW23888 for ; Fri, 2 Feb 2001 06:08:39 -0500 Received: from localhost.localdomain (IDENT:root@ip89.opnxr1.ras.tele.dk [195.215.236.89]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12B7um22389 for ; Fri, 2 Feb 2001 06:07:56 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id LAA07868 for ; Fri, 2 Feb 2001 11:05:15 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Fri, 2 Feb 2001 11:05:15 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: Re: Arizona In-Reply-To: <01020115594300.21564@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: O > So... if you have a car, then going to Phoenix is fine. If you won't > have a car, I don't recommend going to Phoenix. OK, guess it is just as well I got the Tucson tickets. Thanks for the Advice. I might take my drivers liscence just in case... if I can find it. (Not used it for several years). I think a UK liscence is OK in US for a short trip, or I could get an international one. Helena From goins@mpim-bonn.mpg.de Fri Feb 2 07:07:45 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12C8UW23908 for ; Fri, 2 Feb 2001 07:08:30 -0500 Received: from ismene.mpim-bonn.mpg.de (ismene.mpim-bonn.mpg.de [192.68.254.38]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12C7lm23306 for ; Fri, 2 Feb 2001 07:07:47 -0500 (EST) Received: from [192.68.254.81] (nb_goins [192.68.254.81]) by ismene.mpim-bonn.mpg.de (8.8.6/8.8.6) with ESMTP id NAA17843 for ; Fri, 2 Feb 2001 13:07:49 +0100 (MET) User-Agent: Microsoft-Outlook-Express-Macintosh-Edition/5.02.2022 Date: Fri, 02 Feb 2001 13:07:45 +0100 Subject: Re: Benjamin Peirce From: Edray Herber Goins To: "William A. Stein" Message-ID: In-Reply-To: <01020117375204.22620@form> Mime-version: 1.0 Content-type: text/plain; charset="ISO-8859-1" Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by modular.fas.harvard.edu id f12C8UW23908 Status: RO X-Status: O William, Yes, Paris was wonderful! I have to rate it as my all-times favorite city. I will definitely come back before I leave for the States in June. There was indeed a strike on Thursday. It took me all day to figure this out since I didn't know how to say "strike" in French! The METRO was delayed up to 30 min in most places. I finally finished calculating the coefficients I asked for your help with. It took Mathematica just over 6 hours to invert the 3500 x 3500 matrix. Unfortunately, this is just the first stage of a larger process; I need to use these coefficients to figure out how to find an elliptic curve. When I write up the details, I'll explain more about what I'm trying to do. Edray ----------------------------------------------- Max-Planck-Institut f黵 Mathematik Vivatsgasse 7 / 53111 Bonn Germany Office 49-228-402-266 / Mobile: (323) 791-4940 > On Monday 29 January 2001 03:52, you wrote: >> I was in Cologne pretty late, so I didn't get your e-mail until this > > No problem. I'm sure I'll see you at a conference sometime in the > near future. Good luck in decided where you'll be employed next year. > >> morning. Anyhow, I'm off to Paris! So take care, and I'll talk with you >> sometime soon. > > I hope you enjoyed Paris. I'm writing this on February 1, and I was > in Paris 30 minutes ago. There were national police everywhere in the > Metro and train station stopping people. The gates in the Metro were > all open, so the Metro was free. I think this might be because of > some sort of strikes. From marlene@infomagic.com Fri Feb 2 08:45:35 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12DTYW23933 for ; Fri, 2 Feb 2001 08:29:34 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12DSpm25147 for ; Fri, 2 Feb 2001 08:28:51 -0500 (EST) Received: from infomagic.com (MAX5-Port17.Downtown.InfoMagic.NET [63.229.95.217]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f12DSqT09442 for ; Fri, 2 Feb 2001 06:28:53 -0700 Message-ID: <3A7AB9FF.6658AB1A@infomagic.com> Date: Fri, 02 Feb 2001 06:45:35 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: was@math.harvard.edu Subject: Re: AHCCCS ADDRESS FOR COMPUTER References: <0101281745030B.10286@form> <3A746BB1.B5A35C1D@infomagic.com> <01020117593209.22620@form> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: RO X-Status: A I want to come for sure Sept. 14-16 because I want to go to the Squeeze-In so we may come either the week before or stay the week after. If I go any other time, will miss this fun weekend. We may not see much of you, just a little? We're thinking of just going for a week instead of two weeks. I haven't looked at any of your movies that you have sent yet....I would love to see a video of you teaching. Nancy asked if you are getting a "tenure track position"??? Busy weekend for us, not much time except for music and dancing, not much time for sleeping...so will look at photos later. Weather is improving, we need it, tired of snow. Mom Mom "William A. Stein" wrote: > On Sunday 28 January 2001 13:57, you wrote: > > I told you that we want to fly backeast but I guess you have a lot on your > > mind as we all do. > > Yes, but you didn't tell me when. > > Is it necessary for you to come in September, or could you come during > a time that is more convenient for me? What about sometime around the > second week of August, say? If we arrange it right, Tish could also > visit at the same time. Another alternative is the second week of June. > > > We would LOVE to come to a class you are teaching. > > I will be teaching starting September 12. However, I won't be teaching > on the weekend, which is when you'll be in Boston, it seems. > > I bet I can get somebody to video me while I teach, then I'll send the > video to you, or let you watch it on the web. > > -- William From watkins@math.psu.edu Fri Feb 2 08:34:20 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12DZ3W23939 for ; Fri, 2 Feb 2001 08:35:07 -0500 Received: from math.psu.edu (leibniz.math.psu.edu [146.186.130.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12DYKm25288 for ; Fri, 2 Feb 2001 08:34:20 -0500 (EST) Received: from jensen.math.psu.edu (jensen.math.psu.edu [146.186.131.84]) by math.psu.edu (8.9.3/8.9.3) with ESMTP id IAA08023 for ; Fri, 2 Feb 2001 08:34:20 -0500 (EST) From: Mark J Watkins Received: (from watkins@localhost) by jensen.math.psu.edu (8.9.3/8.9.3) id IAA11720 for was@math.harvard.edu; Fri, 2 Feb 2001 08:34:20 -0500 (EST) Message-Id: <200102021334.IAA11720@jensen.math.psu.edu> Subject: Re: "Special values of newforms of weight > 2" To: was@math.harvard.edu Date: Fri, 2 Feb 2001 08:34:20 -0500 (EST) In-Reply-To: <01020118241101.22693@form> from "William A. Stein" at Feb 01, 2001 06:24:11 PM X-Mailer: ELM [version 2.5 PL2] MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: RO X-Status: O William A. Stein wrote: > > > I recently refereed a paper of Dummigan's on constructing visible Sha > > elements in symmetric square motives associated with level 1 rational > > newforms (k=12,16,18,20,22,26), but I don't know how relevant it is here. > > Can you send me this paper, or, failing that, the abstract for this paper? > Or, should I ask Dummigan? I don't know if his work is relevant without > looking at it myself. Here's the abstract Silvio Levy sent me when asking me to referee it: Abstract: We use Zagier's method to compute the critical values of the symmetric square L-functions of six cuspidal eigenforms of level one with rational coefficients. According to the Bloch-kato conjecture, certain large primes dividing these critical values must be the orders of elements in generalized Shafarevich-Tate groups. We give some conditional constructions of these elements. One uses Heegner cycles and Ramanujan-style congruences. The other uses Kurokawa's congruences for Siegel modular forms of degree two. The first construction also applies to the tensor product L-functions attached to a pair of eigenforms of level one. Here the critical values can be both caluclated and analyzed theoretically using a formula of Shimura. The Zagier's method part is rather trivial and straightforward; the main meat of the paper is giving the construction of elements in Sha(Sym^2(M),3k/2) via Heegner cycles (with some cohomology I didn't quite understand). The part with Kurokawa congruences almost seemed to be an after-thought, giving elements in Sha(Sym^2(M),2k-1) (or maybe I have the 3k/2 and 2k-1 mixed up). I can send you the paper if you like. === Mark Watkins watkins@math.psu.edu From merel@math.jussieu.fr Fri Feb 2 09:15:17 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12EGQW23986 for ; Fri, 2 Feb 2001 09:16:26 -0500 Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12EFim26915 for ; Fri, 2 Feb 2001 09:15:44 -0500 (EST) Received: from cantor.math.jussieu.fr (cantor.math.jussieu.fr [134.157.13.106]) by shiva.jussieu.fr (8.10.0/jtpda-5.3.3) with ESMTP id f12EFhq83367 for ; Fri, 2 Feb 2001 15:15:43 +0100 (CET) Received: from [134.157.61.16] (macmerel.institut.math.jussieu.fr [134.157.61.16]) by cantor.math.jussieu.fr (8.9.3/jtpda-5.3.3) with ESMTP id PAA05406 for ; Fri, 2 Feb 2001 15:15:42 +0100 (CET) Mime-Version: 1.0 X-Sender: merel@pophost.math.jussieu.fr Message-Id: In-Reply-To: <01020118212300.22693@form> References: <01020118212300.22693@form> X-Mailer: Eudora Pro F4.2.2 Date: Fri, 2 Feb 2001 15:15:17 +0100 To: was@math.harvard.edu From: =?iso-8859-1?Q?Lo=EFc?= Merel Subject: Re: spurious torsion Content-Type: text/plain; charset="iso-8859-1" ; format="flowed" Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by modular.fas.harvard.edu id f12EGQW23986 Status: RO X-Status: A Dear William, After looking at our past correspondence, I realized that I let you down on our paper. I am sorry. I am glad that you are in France and that you are contacting me. >Loic, > >In your paper in Springer Lecture Notes 1585, you define a space >M_k(N;Z) of modular symbols. It is the Z-module generated by Manin >symbols [X^iY^{k-2-i},(c,d)] modulo certain relations AND modulo >torsion. > >I was in Leuven, Belgium yesterday and I talked with a student of >Oesterle who is studying Serre's conjectures. He began describing his >work to me by letting L_k(N) denote the space like your M_k(N;Z) >except he did *NOT* mod out by torsion. Thus L_k(N)/tor = M_k(N;Z). > >Have you ever studied L_k(N), especially in the case when k>2? I did >some computations which suggest that this is a very bad space to >consider, in general. For example, I have reason to believe that the >Hecke operators (computed using Heilbronn matrices acting on Manin >symbols) on L_8(5) don't even commute! I am considering writing a >short paper about this, but I don't want to do so if you have already >thoroughly studied this problem. Please let me know, so I can decide >how to proceed. I am not sure to understand what you mean. It seems to me that if you consider the module L_k(N) (=some SL_2(Z)-module modulo the relations x+x\sigma=0 et x+x\tau+x\tau^2=0, where \sigma and \tau are of ordre 4 and 6 in SL_2(Z)) you have only 2 and 3 torsion there (and not much of it by the way). I used to believe that you had a Hecke action on L_k(N). Maybe we could discuss that when we will meet. >Mazur is refereeing our short IMRN paper. He has a number of new >comments, and there are a few points that require further >clarification. I can email you about them, or I could visit you in >Paris. Please email them to me, if you wish so. >I am visiting Edixhoven in Rennes, France right now, and I'll >be here until February 15. You can come fairly easily from Rennes to Paris for the day (There are many trains and the trip last only two hours). You could come on Monday. There is a seminar that day. I'll be available from 9 am to 17 pm (I have to take care of my daughters after 17). Of course I'll attend the seminar at 2:15. If you do not wish to come on Monday, I'll be available on Tuesday. You have to know that the mathematicians of Jussieu are not anymore in Jussieu, but at 175 rue du Chevaleret. Here is how to reach the place: You'll get out of the train at gare Montparnasse. From there you can take the metro line number 6 (direction Nation), and get out of the Metro at the station Chevaleret, the 175 rue du Chevaleret is about 100 meters away from the metro station. We are in a big building. My office is 7A31 (7th floor, part A). My phone numbers are 01 44 32 85 24 (office) and 01 43 31 41 07 (home). >Perhaps we can arrange one day before then >for me to visit you in Paris, so that we can finish our paper? Hopefully we will! > Maybe I also need to talk with your student about my >special value computation on X_1(13)? Marusia is not anymore in Paris. She comes from time to time to discuss with me. The next time is February 12. I'll let her know that you are here. See you soon, Lo颿 From mbaker@math.harvard.edu Fri Feb 2 09:28:12 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12EStW23997 for ; Fri, 2 Feb 2001 09:28:55 -0500 Received: from dirichlet.math.harvard.edu (dirichlet.math.harvard.edu [140.247.28.18]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12ESDm27530 for ; Fri, 2 Feb 2001 09:28:13 -0500 (EST) Received: from localhost (mbaker@localhost) by dirichlet.math.harvard.edu (8.11.1/8.11.1) with ESMTP id f12ESDY09358 for ; Fri, 2 Feb 2001 09:28:13 -0500 (EST) X-Authentication-Warning: dirichlet.math.harvard.edu: mbaker owned process doing -bs Date: Fri, 2 Feb 2001 09:28:12 -0500 (EST) From: Matt Baker To: "William A. Stein" Subject: Re: Weierstrass points: I'm still not satisfied. In-Reply-To: <01020116070901.21564@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: RO X-Status: A Hi William, The first sentence of the theorem of Atkin which I quoted is "Suppose that $n$ is *not* squarefree". So the theorem says nothing one way or another about Gamma_0(30). Still, it is curious that it remains an open problem whether or not infinity is ever a Weierstrass point on X_0(n) for n squarefree. We should think about trying to solve this problem some time. How far do you think you could check this experimentally? Maybe infinity *is* a Weierstrass point for some squarefree n but nobody's bothered to look at high enough levels... -- Matt p.s. When will you be back in Harvard? On Thu, 1 Feb 2001, William A. Stein wrote: > On Tuesday 30 January 2001 17:55, you wrote: > > I know what the problem is: it's that Edixhoven and Coleman are lying at > > the end of their article. Here's the best result I could dig out of the > > Oops! > > > literature: in Atkin's article "Weierstrass Points at cusps of > > Gamma_0(n)", he proves the following: > > > > Theorem: Suppose that $n$ is not squarefree. Then infinity is a > > Weierstrass point on Gamma_0(n) except in case (a) below and possibly in > > cases (b) - (d) below: > > > > (a) n = 4,8,9,16,25,27,32,36,49,81 > > (b) n=8p,16p > > (c) n=(p^2)(q) and not both p and q are 1 mod 12 > > (d) n = (p^2)qr and neither x^2 + 1 = 0 (mod pqr) nor x^2 - x + 1 = 0 (mod > > pqr) is solvable. > > > > Note that case (c) includes n=28. > > I'm still not satisfied. I find that when n=30, then infinity is NOT > a Weierstrass point. However, I don't see 30 amongs any of (a)-(d). > It's clearly not in 1 or 2, assuming "p" means "a prime". It's not in > (c) or (d) either, because 30 is square-free. > > A basis of q-expansions for S_2(Gamma_0(30)) is: > > q - q^4 - q^6 - 2*q^7 + q^9 + q^10 - 2*q^11 + 2*q^12 + 2*q^14 - q^15 + > 3*q^16 + O(q^17) > q^2 - q^4 - q^6 - q^8 + q^10 + q^12 + 3*q^16 + O(q^17) > q^3 + q^4 - q^5 - q^6 - 2*q^7 - 2*q^8 + q^10 + 2*q^11 + 2*q^13 + > 2*q^14 + q^16 + O(q^17) > > Infinity is not a Weierstrass point here. > > -- William > From DianBlu@cdw.com Fri Feb 2 11:33:30 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12GYIW24423 for ; Fri, 2 Feb 2001 11:34:18 -0500 Received: from [12.32.90.183] (falcon.cdw.com [12.32.90.183]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12GXam06533 for ; Fri, 2 Feb 2001 11:33:36 -0500 (EST) Message-ID: <8189F8D452FDDE428137D3D89BBE71345C4C1C@pow.corp.cdw.com> From: DianBlu@cdw.com To: was@math.harvard.edu Subject: RE: CDW Quote/Order DS59613 Date: Fri, 2 Feb 2001 10:33:30 -0600 MIME-Version: 1.0 X-Mailer: Internet Mail Service (5.5.2650.21) Content-Type: text/plain; charset="iso-8859-1" Status: R X-Status: N Consider it done. Have a good weekend, and please let me know if I can help. > Diane Bluett > Account Manager CDW*G, Inc Computing Solutions Built for Government & Education > CDW-G is Your Direct Solutions Provider > Toll Free: 877-530-8482 > Direct Phone: 847-968-9800 > Direct Fax: 847-968-1800 > Email: dianblu@cdwg.com > Web Address: www.cdwg.com -----Original Message----- From: William A. Stein [mailto:was@math.harvard.edu] Sent: Friday, February 02, 2001 2:05 PM To: dianblu@cdw.com Subject: Re: CDW Quote/Order DS59613 On Wednesday 31 January 2001 18:04, you wrote: > Please let me know what you'd like to do. Thanks! Please cancel my order. I will shop around on the web sometime in the next few days when I get a chance to look for printers again, and re-evaluate my options. Thanks, William From DStein@MBE.COM Fri Feb 2 14:41:54 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12JkQW24644 for ; Fri, 2 Feb 2001 14:46:29 -0500 Received: from mail.mbe.com ([63.145.223.100]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12Jjhm17185 for ; Fri, 2 Feb 2001 14:45:43 -0500 (EST) Subject: Article in Today's Union-Tribune To: fvaldivieso@delhaize.be, was@math.harvard.edu, marlene@infomagic.com X-Mailer: Lotus Notes Release 5.0.2b December 16, 1999 Message-ID: From: DStein@MBE.COM Date: Fri, 2 Feb 2001 11:41:54 -0800 X-MIMETrack: Serialize by Router on domino1/MBE(Release 5.0.4 |June 8, 2000) at 02/02/2001 11:41:59 AM MIME-Version: 1.0 Content-type: multipart/mixed; Boundary="0__=882569E7006C15E28f9e8a93df938690918c882569E7006C15E2" Content-Disposition: inline Status: R X-Status: N --0__=882569E7006C15E28f9e8a93df938690918c882569E7006C15E2 Content-type: text/plain; charset=us-ascii This appeared in the San Diego newspaper today. Dennis MBE set to double its global presence (Embedded image moved to file: pic22549.gif) By Frank Green UNION-TRIBUNE STAFF WRITER February 2, 2001 Mail Boxes Etc. yesterday announced a massive plan for international expansion, even as it prepares to be delivered to new owners by its troubled parent company. The San Diego-based firm said it will add 1,000 stores outside the United States in the next five years, more than doubling its global presence. "We will eventually have more stores in other countries than we do here," said Alfonso Vilches, MBE's coordinator of international operations. MBE has 3,400 franchise outlets in the United States and 900 shops in 30 other countries, including Canada, Spain, France and Italy. And within the next two months, it will add at least 100 stores in Kuwait, Norway, Finland and elsewhere. MBE said it recently signed a master license agreement with Grupo PYV, which is owned by the Pellerano family of Santo Domingo, to roll out dozens of stores in the Dominican Republic and Haiti. And in May, Caleb Enterprises will open a pilot store in Nassau, the Bahamas, before expanding MBE centers to Jamaica, St. Lucia, Bermuda, the Cayman Islands and Turks and Caicos Islands. "I've been traveling a lot" overseeing construction and development of new sites, Vilches said. The expansion comes as MBE's owner, U.S. Office Products, is about to divest the company -- the crown jewel of its portfolio -- to raise desperately needed cash. USOP has been floundering recently in the office products and services market, battered by Office Depot, Staples and kindred competitors. Indeed, the Washington, D.C.-based company reported a net loss in the second quarter ended Oct. 28 of $261 million. Yesterday, shares of USOP closed at 15 cents. As recently as May 1998, USOP stock was selling at about $40 a share. But USOP's quarterly report stressed that MBE sales were up 9 percent, compared to the same period a year earlier. Company officials said yesterday they had nothing to announce regarding the pending sale of MBE, but several franchisees said that a major investment group is in negotiations with USOP to buy the business. "I've heard it has already been consummated," said John Boyd, president of the Independent Association of Mail Box Center Owners Inc. The dissident group, which boasts a membership of about 500 MBE franchisees, tried to acquire MBE from USOP last year but was rebuffed. In the past, the group has accused MBE of not dealing with numerous franchisees' concerns, including allegations that the company infringed on the exclusive territory of some franchises by opening small retail outlets in nearby hotels and convention centers. MBE was founded in 1980 by Tony DeSio and several partners, who opened the firm's first outlet in La Costa. The company's strategy was to offer basic postal services as an alternative to the U.S. Postal Service, which at the time did not have a wide network of retail outlets. MBE was quickly pressed to add services to accommodate home-based businesses that needed access to computers, teleconferencing equipment, fax machines and similar business tools. The company opened its 1,000th center in October 1989. Five years later, its U.S. number had doubled. --0__=882569E7006C15E28f9e8a93df938690918c882569E7006C15E2 Content-type: image/gif; name="pic22549.gif" Content-Disposition: attachment; filename="pic22549.gif" Content-transfer-encoding: base64 Content-Description: Compuserve GIF R0lGODlhAQABAIAAAAAAAP///ywAAAAAAQABAEACAkQBADs= --0__=882569E7006C15E28f9e8a93df938690918c882569E7006C15E2-- From grigorov@math.harvard.edu Fri Feb 2 15:05:25 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12K68W24654 for ; Fri, 2 Feb 2001 15:06:08 -0500 Received: from hum1.math.harvard.edu (hum1.math.harvard.edu [140.247.28.157]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12K5Qm18382 for ; Fri, 2 Feb 2001 15:05:26 -0500 (EST) Received: from localhost (grigorov@localhost) by hum1.math.harvard.edu (8.11.1/8.11.1) with ESMTP id f12K5Qe19470 for ; Fri, 2 Feb 2001 15:05:26 -0500 (EST) X-Authentication-Warning: hum1.math.harvard.edu: grigorov owned process doing -bs Date: Fri, 2 Feb 2001 15:05:25 -0500 (EST) From: Grigor Grigorov To: "William A. Stein" Subject: mailbox etc. In-Reply-To: <01011617151603.00866@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N Hi William, I just noticed that there are 3 packages for you and god knows how many letters in your mailbox. When are you coming back? Today was the organizational meeting for Marty's seminar and we decided it to be every Monday and Wednesday. Finally I am going to the Arizona Winter School. The department will pay half of my travel expenses and the other half will be payed by the winter school. I am ill now for more than a week and it's not getting much better so I don't do much work. Is anything interesting going around you? Best, Grigor From verrill@math.ku.dk Fri Feb 2 14:17:17 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12KKkW24665 for ; Fri, 2 Feb 2001 15:20:46 -0500 Received: from localhost.localdomain (IDENT:root@ip126.opnxr1.ras.tele.dk [195.215.236.126]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12KJxm20591 for ; Fri, 2 Feb 2001 15:19:59 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id UAA24473 for ; Fri, 2 Feb 2001 20:17:18 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Fri, 2 Feb 2001 20:17:17 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: drawing hyperbolic triangles In-Reply-To: <01020116101103.21564@form> Message-ID: MIME-Version: 1.0 Content-Type: MULTIPART/MIXED; BOUNDARY="8323328-1295270616-981141437=:6592" Status: R X-Status: N This message is in MIME format. The first part should be readable text, while the remaining parts are likely unreadable without MIME-aware tools. Send mail to mime@docserver.cac.washington.edu for more info. --8323328-1295270616-981141437=:6592 Content-Type: TEXT/PLAIN; charset=US-ASCII Hi William, Well, I've now written a simple program for drawing pictures of hyperbolic triangles, images of the "standard" fundamental domain for SL_2(Z), under the action of SL_2(Z), eg, you type: addTriangle(matrix,colour,filename); (in magma, after loading the package) and it will add a triangle, of the given colour, corresponding to the given matrix, to the ps file given by "filename". (you have to start the ps file by typing clearpsfile(filename); which deletes what's in it and adds a new ps header) I also added a function which uses the System command to bring up the ps file in "watch" mode, so it renews when more triangles are added. Anyway, this makes it easy to draw pictures of fundmental domains, for either your output of cosets, using the P1classes and LiftToCosetRep functions, or else using the program I wrote to draw a connected domain. If you're interested I can send you the premininary version (which does the above stuff), so you could comment on what you think it ought to do/how it ought to work. Otherwise I can send it later after I've added more features, like letting the user change the scale of the image etc. I've not decided what features it should have yet. At the moment I just attached a ps file created using these new magma functions. I guess it's really an eps file. You should be able to view it using ghostview or gv anyway. 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coleman@math.berkeley.edu Fri Feb 2 15:52:13 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f12Ks3W24680 for ; Fri, 2 Feb 2001 15:54:03 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f12KrKm23788; Fri, 2 Feb 2001 15:53:20 -0500 (EST) Received: from gold (localhost [127.0.0.1]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id MAA00278; Fri, 2 Feb 2001 12:53:03 -0800 (PST) Received: from [169.229.58.143] (robert-1.Math.Berkeley.EDU [169.229.58.143]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id MAA00214 for ; Fri, 2 Feb 2001 12:52:18 -0800 (PST) Mime-Version: 1.0 Message-Id: To: ntsfor@math.berkeley.edu From: Robert Coleman Content-Type: text/plain; charset="us-ascii" ; format="flowed" Subject: [nts] next week Sender: nts-admin@math.berkeley.edu Errors-To: nts-admin@math.berkeley.edu X-BeenThere: nts@math.berkeley.edu X-Mailman-Version: 2.0.1 Precedence: bulk List-Help: List-Post: List-Subscribe: , List-Id: UCB number theory seminar (locals) List-Unsubscribe: , List-Archive: Date: Fri, 2 Feb 2001 12:52:13 -0800 Status: R X-Status: N On Weds at 3 (room to be announced. Matt Emerton will speak on Towards a notion of p-adic automorphic representation In this talk I will report on work in progress, whose object is to apply the representation-theoretic point of view on automorphic forms to the problem of p-adically interpolating automorphic forms. I will explain a generalization of Hida's theory of the ordinary part of the cohomology of arithmetic subgroups of GL_n to arbitrary reductive groups over Q. I will then explain how I hope to extend this to find an analogue for an arbitrary reductive group over Q of Coleman's theory of the interpolation of overconvergent Hecke eigenforms on GL_2. and on Friday at 4:10 Richard Groenewegen, Universiteit Leiden, The Netherlands}, will speak on, Bounds for computing the tame kernel \bigskip\noindent Let $F$ be a number field. We define $$ K_2F = (F^* \otimes F^*) / \langle a\otimes b : a+b=1 \rangle $$ and we denote the class of $a\otimes b$ by $\{a,b\}$. For every non-archimedean valuation~$v$ we have a map $t_v\colon K_2F\to k_v^*$, given by $\{a,b\}\mapsto (-1)^{v(a)v(b)} a^{v(b)}/b^{v(a)}\;{\rm mod}\;v$. The tame kernel is defined as the kernel of the map $$ (t_v)_{v<\infty}\colon K_2F \to \bigoplus_{v<\infty}k_v^*. $$ We know the tame kernel is a finite group, but in general there is no way to determine this group explicitly. It is possible, however, to give a set of generators for the tame kernel. The key ingredient is a theorem that implies we do not have to consider the maps $t_v$, where $Nv$ is greater than some bound. There are articles which give an explicit bound in the quadratic imaginary case and use this bound for explicit calculations. In this talk I will show how to give a bound for general number fields that is better than the ones currently found in the literature. We will also see what obstructs us from giving a bound on the relations of the tame kernel. \bye ----:---F1 Groenewegen.html 12:46PM 3.06 Mail (HTML Fill)--L1--All----------- _______________________________________________ nts mailing list nts@math.berkeley.edu http://www.math.berkeley.edu/mailman/listinfo/nts From partner-news1@amazon.com Thu Feb 1 04:33:58 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f130ANW24793 for ; Fri, 2 Feb 2001 19:10:23 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f1309fm05249 for ; Fri, 2 Feb 2001 19:09:41 -0500 (EST) Received: from mm-outgoing-6.amazon.com (mm-outgoing-6.amazon.com [208.33.217.123]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id QAA16710 for ; Fri, 2 Feb 2001 16:09:40 -0800 (PST) Received: from mail-ems-1.amazon.com by mm-outgoing-6.amazon.com with ESMTP (crosscheck: mail-ems-1.amazon.com [10.16.42.116]) id PAA-58256-19749; Fri, 2 Feb 2001 15:29:28 -0800 (PST) Received: by mail-ems-1.amazon.com id AAA-58256-00364,3622; 1 Feb 2001 01:33:58 -0800 Date: 1 Feb 2001 01:33:58 -0800 Message-ID: <.AAA-58256-00364,3622.981020038@mail-ems-1.amazon.com> X-AMAZON-TRACK: 58256 To: was@gold.math.berkeley.edu From: "Amazon.com" Subject: Save $50 to $150 on Your Valentine Shopping at Ashford.com Bounces-to: ems+FTJ5MKBSAG7KAH@bounces.amazon.com MIME-Version: 1.0 Content-Type: Multipart/Alternative;boundary=MuLtIpArT_BoUnDaRy Status: R X-Status: N --MuLtIpArT_BoUnDaRy Content-Type: text/plain Dear Amazon Customer, With Valentine's Day just around the corner, we'd like to introduce you to our friends at Ashford.com. 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OFFER INSTRUCTIONS BELOW Follow these steps to save $50 to $150 and get free overnight shipping: 1. Visit http://www.ashford.com/special/valentine.asp?urlid=5162 2. Select the items you'd like to purchase from Ashford.com. 3. Place your order. (Be sure to enter your special promotional code 86A224774A on the order form.) 4. Relax--$50 will be deducted from the total price of your order of $100 or more, and you will also receive free overnight shipping automatically. Also, if you spend $500 or more, $150 will be deducted from the total price of your order. The fine print: This offer may not be used in conjunction with any other Ashford.com coupons or promotions, including but not limited to gift with purchase promotions and the "special offer" feature. Limit one per customer; promotional code will expire after one use. Not valid toward purchase of gift certificates, loose diamonds, corporate gifts, items in the fragrance department, or some select items in the Ashford.com Designer Jewelry department (see Web site for more details and restrictions). We would like to point out that when you redeem this offer, Ashford.com will know that you are part of a select group of Amazon.com customers and will provide aggregated redemption information to us. Offer expires 11:59 p.m. CST on February 28, 2001. PS: We hope you enjoyed receiving this message. 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    • Live customer service, available 24/7
    • Next-day delivery on in-stock items ordered by 5 p.m. CST
    • Free gift packaging
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    • A world-class return policy that includes 60-day returns on new watches and 30-day returns on all other items

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Delighted Ashford.com Shopper
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1. Visit http://www.ashford.com/special/valentine.asp?urlid=5163.
2. Select the items you'd like to purchase from Ashford.com. 
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4. Relax--$50 will be deducted from the total price of your order of $100 or more, and you will also receive free overnight shipping automatically. Also, if you spend $500 or more, $150 will be deducted from the total price of your order.

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--MuLtIpArT_BoUnDaRy-- From ribet@math.berkeley.edu Fri Feb 2 19:49:50 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f130oXW24808 for ; Fri, 2 Feb 2001 19:50:33 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f130npm06696 for ; Fri, 2 Feb 2001 19:49:51 -0500 (EST) Received: from rancilio.math.Berkeley.EDU (rancilio.Math.Berkeley.EDU [169.229.58.27]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id QAA19481 for ; Fri, 2 Feb 2001 16:49:50 -0800 (PST) From: "Kenneth A. Ribet" Received: (from ribet@localhost) by rancilio.math.Berkeley.EDU (8.9.3/8.9.3) id QAA06267 for was@math.harvard.edu; Fri, 2 Feb 2001 16:49:50 -0800 (PST) Date: Fri, 2 Feb 2001 16:49:50 -0800 (PST) Message-Id: <200102030049.QAA06267@rancilio.math.Berkeley.EDU> To: was@math.harvard.edu Subject: a paper to referee Status: R X-Status: N William, I wonder whether you'd be interested in refereeing the paper that follows. It's by Neil Dummigan; he submitted it to Mathematische Annalen. (I wrote to him just now, though, to suggest Math Research Letters instead.) The title is "Visible elements and congruences of modular forms," so you can see why I thought of you! Best, Ken \documentclass{article} \usepackage{amssymb} \usepackage{amsmath} \usepackage{amscd} \newtheorem{prop}{Proposition}[section] \newtheorem{defi}[prop]{Definition} \newtheorem{conj}[prop]{Conjecture} \newtheorem{lem}[prop]{Lemma} \newtheorem{thm}[prop]{Theorem} \newtheorem{cor}[prop]{Corollary} \newtheorem{examp}[prop]{Example} \newtheorem{remar}[prop]{Remark} \newcommand{\Ker}{\mathrm {Ker}} \newcommand{\Aut}{{\mathrm {Aut}}} \def\id{\mathop{\mathrm{ id}}\nolimits} \renewcommand{\Im}{{\mathrm {Im}}} \newcommand{\ord}{{\mathrm {ord}}} \newcommand{\End}{{\mathrm {End}}} \newcommand{\Hom}{{\mathrm {Hom}}} \newcommand{\Mor}{{\mathrm {Mor}}} \newcommand{\Norm}{{\mathrm {Norm}}} \newcommand{\Nm}{{\mathrm {Nm}}} \newcommand{\tr}{{\mathrm {tr}}} \newcommand{\Tor}{{\mathrm {Tor}}} \newcommand{\Sym}{{\mathrm {Sym}}} \newcommand{\Hol}{{\mathrm {Hol}}} \newcommand{\vol}{{\mathrm {vol}}} \newcommand{\tors}{{\mathrm {tors}}} \newcommand{\cris}{{\mathrm {cris}}} \newcommand{\dR}{{\mathrm {dR}}} \newcommand{\lcm}{{\mathrm {lcm}}} \newcommand{\Frob}{{\mathrm {Frob}}} \def\rank{\mathop{\mathrm{ rank}}\nolimits} \newcommand{\Gal}{\mathrm {Gal}} \newcommand{\Spec}{{\mathrm {Spec}}} \newcommand{\Ext}{{\mathrm {Ext}}} \newcommand{\res}{{\mathrm {res}}} \newcommand{\Cor}{{\mathrm {Cor}}} \newcommand{\AAA}{{\mathbb A}} \newcommand{\CC}{{\mathbb C}} \newcommand{\RR}{{\mathbb R}} \newcommand{\QQ}{{\mathbb Q}} \newcommand{\ZZ}{{\mathbb Z}} \newcommand{\NN}{{\mathbb N}} \newcommand{\EE}{{\mathbb E}} \newcommand{\HHH}{{\mathbb H}} \newcommand{\pp}{{\mathfrak p}} \newcommand{\FF}{{\mathbb F}} \newcommand{\KK}{{\mathbb K}} \newcommand{\GL}{\mathrm {GL}} \newcommand{\SL}{\mathrm {SL}} \newcommand{\Sp}{\mathrm {Sp}} \newcommand{\Br}{\mathrm {Br}} \newcommand{\Qbar}{\overline{\mathbb Q}} \newcommand{\Sha}{\mathrm{\underline{III}}} \newcommand{\HH}{{\mathfrak H}} \newcommand{\aaa}{{\mathfrak a}} \newcommand{\bb}{{\mathfrak b}} \newcommand{\dd}{{\mathfrak d}} \newcommand{\ee}{{\mathbf e}} \newcommand{\Fbar}{\overline{F}} \newcommand{\CH}{\mathrm {CH}} \begin{document} \title{Visible elements and congruences of modular forms} \author{Neil Dummigan} \date{February 2nd, 2001} \maketitle \footnote{2000 Mathematics Subject Classification 11F33, 11F37, 11F67, 11G40. Keywords: modular form, $L$-function, Bloch-Kato conjecture, Shafarevich-Tate group.} \begin{abstract} We show that the congruence modulo $11$ between the normalized cusp form $\Delta$ of weight $12$ and the normalized cusp form of weight $2$ and level $11$ `descends' to a congruence between forms of weights $13/2$ and $3/2$. Combining Waldspurger's theorem with the Bloch-Kato conjecture we predict the existence of elements of order $11$ in Selmer groups for certain quadratic twists of $\Delta$. These are then constructed using rational points on twists of the elliptic curve $X_0(11)$, assuming the Birch and Swinnerton-Dyer conjecture on the rank. Everything generalizes to forms of weights $2+10s$ in an $11$-adic family, to congruences modulo higher powers of $11$, and to other elliptic curves over $\QQ$ of prime conductor $p\equiv 3(\bmod{4})$. \end{abstract} \section{Introduction} The Tamagawa-number conjecture of Bloch and Kato \cite{BK} is a very general conjecture on the special values (at integer points) of the $L$-functions arising from arithmetic geometry. Examples of such $L$-functions are those attached to modular forms. The connection with arithmetic geometry is through modular curves, and gives rise to the Galois representations famously associated with modular forms. In an earlier paper \cite{Du1} we gained some evidence for the conjecture, in the special case of modular forms of level one and weight $k$, by comparing special values at different points in the critical range $1\leq s\leq k-1$. Looking at ratios of critical values avoided the difficulty of determining a canonical period, by making it cancel. In the present paper we look mainly at the central critical point $s=3Dk/2$, but consider ratios of the values at that point of various quadratic twists of the $L$-function attached to the modular form. The particular example on which we concentrate is the celebrated discriminant function $$\Delta(z)=3D\eta(z)^{24}=3Dq\prod_{n=3D1}^{\infty}(1-q^n)^{24}=3D\sum_{= n=3D1}^{\infty}\tau(n)q^n,$$ where $q=3De^{2\pi iz}$ and $\eta(z)=3Dq^{1/24}\prod(1-q^n)$ is Dedekind's eta function. $\Delta$ is the unique normalized cusp form of weight $12$ and level one. Let $E$ be the elliptic curve with minimal Weierstrass equation $y^2+y=3Dx^3-x^2-10x-20$. This is in fact the modular curve $X_0(11)$, so $E$ is trivially `modular'. The associated modular form, the unique normalized cusp form of weight $2$ on $\Gamma_0(11)$, is $$F(z)=3D\eta(z)^2\eta(11z)^2=3Dq\prod(1-q^n)^2\prod(1-q^{11n})^2.$$ It is evident from the infinite products that the $q$-expansions of $\Delta$ and $F$ are congruent modulo $11$. It is this congruence which acts as a lever for us to prise some information out of the motives attached to quadratic twists of $\Delta$. A question raised by Hida is whether congruences between integer weight newforms descend to congruences between half-integral weight Hecke eigenforms which correspond to them under the Shimura map \cite{Shim},\cite{K1},\cite{K2}. Maeda \cite{Ma} produced an example with congruences modulo $433$, using newforms of weight $k=3D8$ and level $N=3D52$. Further examples involving congruences between cusp forms and Eisenstein series were considered in \cite{Kob} and \cite{G}. In our case, we can associate with $F$ and $\Delta$ explicit Hecke eigenforms $g$ and $h$, of weights $3/2$ and $13/2$, on $\Gamma_0(44)$ and $\Gamma_0(4)$ respectively. The forms $F$ and $\Delta$ have different weights, and we find that the congruence modulo $11$ between $F$ and $\Delta$ descends to a congruence modulo $11$ between the Fourier coefficients not quite of $g$ and $h$, but of $g$ and $h|T_{11}$. Put another way, if $g=3D\sum a_nq^n$ and $h=3D\sum b_nq^n$ then $a_n\equiv b_{11n}(\bmod{11})$ for all $n$. We prove the congruence using a theorem of Sturm \cite{Stu}, which shows in our case that it suffices to check the congruence for $n\leq 156$. A corollary is Theorem 10 of \cite{BDO}, that $11|b_{11n}$ whenever $n$ is a quadratic non-residue modulo $11$. Their proof also uses Sturm's theorem, but necessitates checking the divisibility for $n\leq 95832$. Let $d$, positive and coprime to $11$, be such that $-d$ is the discriminant of a quadratic field. Let $E_{-d}$ be the quadratic twist of $E$ associated to the quadratic character $({-d\over n})$. According to a theorem of Waldspurger \cite{Wa}, the central critical value $L(E_{-d},1)$ is (more or less) proportional to the square of the $d^{th}$ Fourier coefficient of $g$. If we restrict now to $d$ which are quadratic residues modulo $11$, the sign in the functional equation of $L(E_{-d},s)$ is positive, so $L(E_{-d},s)$ vanishes to even order at $s=3D1$. If the $d^{th}$ Fourier coefficient of $g$ is zero then $L(E_{-d},s)$ vanishes to order at least two at $s=3D1$. According to the Birch and Swinnerton-Dyer conjecture, the group of rational points $E_{-d}(\QQ)$ then has rank at least $2$. I am grateful to J. Cremona for checking in the first few cases that the rank really is $2$. Waldspurger's theorem implies that the square of $b_{11d}$ is more or less proportional to the central critical value of the twisted $L$-function, $L(\Delta,\chi_{11d},6)$. In those cases where $a_d=3D0$, the aforementioned congruence implies that $11|b_{11d}$. Take two different $d$ and $d'$ as above, both quadratic residues modulo $11$. Suppose that $a_d=3D0$ but that $a_{d'}\neq 0(\bmod{11})$. Then the ratio\newline $\sqrt{d'/d}L(\Delta,\chi_{11d},6)/L(\Delta,\chi_{11d'},6)$ is a rational number divisible by $11$. If it is non-zero, then the Bloch-Kato conjecture, applied to these critical values, predicts that some Shafarevich-Tate group for $\Delta_{11d}$ has order divisible by $11$, even by $11^2$. (We already mentioned the cancellation of periods. Similarly, although we do not know the local fudge factors $c_p$ appearing in the Bloch-Kato formula, we can show that their $11$-parts do not change from one twist to another.) Now, to provide some evidence for the Bloch-Kato conjecture, all we need to do is to produce an element of order $11$ in the appropriate Selmer group for the motive attached to $\Delta_{11d}$. Recall that since $a_d=3D0$, $L(E_{-d},s)$ vanishes to order at least two at $s=3D1$. The Birch and Swinnerton-Dyer conjecture would then imply that $E_{-d}(\QQ)$ has rank at least two. Assuming this is true (it can be checked in particular cases), we get two independent elements of order $11$ in a Selmer group for $E_{-d}$. The congruence of $F$ and $\Delta$ implies that the mod $11$ Galois representations attached to $E$ and $\Delta$ are isomorphic. This allows us to transfer these elements of order $11$ to some Galois cohomology group attached to $\Delta_{11d}$. To show that they are in the Selmer group, as required, we would have to show that they satisfy the necessary local conditions at every prime $p$. For $p\neq 11$ this is quite straightforward. For $p=3D11$ we might be tempted to try to apply the theory of Fontaine and Lafaille, as in \cite{Du2}. However, this does not apply when $p0}d^3$) and $\theta(z)=3D1+2\sum_{n=3D1}^{\infty}q^{n^2}$. This formula is very suitable for the purpose of computing the Fourier coefficients $b_n$. Example 3.5 of \cite{Shin} gives a useful formula for the Fourier coefficients of $g\,$: $a_n=3Dc_{4n}$ where $$\sum c_nq^n=3D\theta(11z)\eta(2z)\eta(22z).$$ \section{The proof of the congruence} \begin{thm} Let $g=3D\sum a_nq^n\in S^+_{3/2}(\Gamma_0(44))$ and $h=3D\sum b_nq^n\in S^+_{13/2}(\Gamma_0(4))$ be the forms described above. Then $a_n\equiv b_{11n}(\bmod{11})$ for all $n$. \end{thm} {\em Proof. } We need to show that the coefficients of $g$ and $h|T_{11}$ are congruent modulo $11$. (Note that although $h$ has level $4$, we consider it at level $44$, so that $11$ divides the level and the Hecke operator $T_{11}$ exists. Also, though $h$ has trivial character, $h|T_{11}$ has character $({11\over {\cdot}})$. ) It is clearly equivalent to show the same thing for the coefficients of $g^2$ and $(h|T_{11})^2$, or even for $g^4$ and $(h|T_{11})^4$, which are modular forms of weights $6$ and $26$ respectively. For even $r>2$ let $E_r=3D1-{2r\over B_r}\sum_{n=3D1}^{\infty}\sigma_{r-1}(n)q^n$ be the normalized Eisenstein series of weight $r$. A well known consequence of the Clausen-Von Staudt theorem is that for an odd prime number $p>3$ the $q$-expansion of $E_{p-1}$ is congruent to $1$ mod $p$. Letting $p=3D11$, multiplying $g^4$ by $E_{10}^2$ will not make any difference to its Fourier coefficients modulo $11$. Now the forms $g^4E_{10}^2$ and $(h|T_{11})^4$ have the same weight $k=3D26$, and are both forms on $\Gamma_0(M)$ with $M=3D44$. This allows us to apply the theorem of Sturm \cite{Stu} which guarantees that the congruence holds for all $n$ as long as it holds for all $n\leq {kM\over 12}\prod_{p|M}(1+{1\over p})$. In our case the bound is only $156$, so the condition is easily checked using Maple.\newline $\blacksquare$ \section{Waldspurger's theorem} The Shimura correspondence tells us relations among the Fourier coefficients $a_{dt^2}$ of an eigenform $g$ of half-integral weight, for a fixed square-free $d$, in terms of the coefficients of the Shimura lift $f$. It does not say anything about the relation between the $a_d$ for different square-free values of $d$ (or $d/4$). Waldspurger \cite{Wa} proved a theorem to the effect that the ratios of the squares of these coefficients are more or less the ratios of the central critical values of the $L$-functions associated to $f$ with quadratic twists. We state a refined version in a special case which covers what we need. This is Corollary 1 to Theorem 3 in \cite{K3}. \begin{thm}\label{wald} Suppose that $N$ is an odd, square-free positive integer and $k$ is a positive even integer. Temporarily abandoning earlier names, let $g=3D\sum b_nq^n\in S^+_{(k+1)/2}(\Gamma_0(4N))$ correspond to $f=3D\sum c_nq^n\in S_k(\Gamma_0(N)$ under Kohnen's isomorphism, where $f$ is a newform. Let $d$ be a positive integer such that $D:=3D(-1)^{k/2}d$ is the discriminant of a quadratic field. Let $L(f,\chi_D,s)$ be the analytic continuation of $\sum_{n=3D1}^{\infty}\chi_D(n)c_n n^{-s}$, where $\chi_D(n)=3D({D\over n})$. Let $\langle f,f \rangle$ be the Petersson norm and let $\nu(N)$ be the number of distinct prime divisors of $N$. Suppose that for every prime $l|N$, $({D\over l})=3Dw_l,$ the eigenvalue for $f$ with respect to the Atkin-Lehner involution $W_l$. Then $${b_d^2\over {\langle g,g \rangle}}=3D2^{\nu(N)}{({k\over 2}-1)!\over \pi^{k/2}}d^{(k-1)/2}{L(f,\chi_D,k/2)\over {\langle f,f \rangle}}.$$\end{thm} \begin{cor} For two different $d'$ and $d$ as above, supposing the conditions about $({D\over l})$ and $({D'\over l})$ are satisfied, and that $L(f,\chi_D,k/2)\neq 0$, $$L(f,\chi_{D'},k/2)/L(f,\chi_D,k/2)=3D(d/d')^{(k-2)/2}\sqrt{d/d'}b_{d'}^= 2/b_d^2.$$ \end{cor} In the example with $\Delta$ of weight $k=3D12$ and $h$ of weight $(k+1)/2=3D13/2$, $\,N=3D1$ so there are no conditions to check. In the example with $F$ of weight $k=3D2$ and $g$ of weight $(k+1)/2=3D3/2$, $\,N=3D11$ and $w_{11}=3D-1$ so we require $({-d\over 11})=3D-1$, which is equivalent to $({d\over 11})=3D1$. Actually, we shall only be using the above corollary in the example where $k=3D12$. \section{Motives and Galois representations} Let $f=3D\sum e_nq^n$ be a newform of weight $k$ for $\Gamma_0(N)$ (and let's say trivial character and rational coefficients, since that is all we need). A special case of a theorem of Deligne \cite{De1} implies the existence, for each prime $l$, of a continuous representation $$\rho_l:\Gal(\Qbar/\QQ)\rightarrow \Aut(V_l)$$ ($V_l$ is a two-dimensional vector space over $\QQ_l$), such that \begin{enumerate} \item $\rho_l$ is unramified at $p$ for all primes $p$ not dividing $lN$; \item if $\Frob_p$ is an arithmetic Frobenius element at such a $p$ then = the characteristic polynomial of $\Frob_p^{-1}$ acting on $V_l$ is $x^2-e_px+p^{k-1}$. \end{enumerate} Following Scholl\cite{Sc}, $V_l$ may be constructed as the $l$-adic realisation of a Grothendieck motive $M$. There are also Betti and de Rham realisations $V_B$ and $V_{\dR}$, both $2$-dimensional $\QQ$-vector spaces. For details of the construction see \cite{Sc}. The de Rham realisation has a Hodge filtration $V_{\dR}=3DF^0\supset F^1=3D\ldots =3DF^{k-1}\supset F^k=3D\{0\}$. The Betti realisation $V_B$ comes from singular cohomology, while $V_l$ comes from \'etale $l$-adic cohomology. There are natural isomorphisms $V_B\otimes \QQ_l\simeq V_l$. Using a basis for singular cohomology with $\ZZ$-coefficients, we get $\Gal(\Qbar/\QQ)$-stable $\ZZ_l$-modules $T_l$ inside each $V_l$. Define $A_l=3DV_l/T_l$. There are two kinds of twist we shall have to consider. There is the Tate twist $V_l(j)$ (for an integer $j$), which amounts to multiplying the action of $\Frob_p$ by $p^j$. Then, for $D$ the discriminant of a quadratic field, there is the quadratic twist $V_l(\chi_D)$, where $V_l$ has been tensored with a one-dimensional space on which $\Gal(\Qbar/\QQ)$ acts via the quadratic character $\chi_D$. Following \cite{BK}, for $p\neq l$ let $$H^1_f(\QQ_p,V_l(\chi_D,j))=3D\ker (H^1(D_p,V_l(\chi_D,j))\rightarrow H^1(I_p,V_l(\chi_D,j))).$$ Here $D_p$ is a decomposition subgroup at a prime above $p$, $I_p$ is the inertia subgroup, and the cohomology is for continuous cocycles and coboundaries. For $p=3Dl$ let $$H^1_f(\QQ_l,V_l(\chi_D,j))=3D\ker (H^1(D_l,V_l(\chi_D,j))\rightarrow H^1(D_l,V_l(\chi_D,j)\otimes B_{\cris}))$$ (see \cite{BK} for a definition of Fontaine's ring $B_{\cris}$). The subscript $f$ stands for ``finite part''. Let $H^1_f(\QQ,V_l(\chi_D,j))$ be the subspace of elements of $H^1(\QQ,V_l(\chi_D,j))$ whose local restrictions lie in $H^1_f(\QQ_p,V_l(\chi_D,j))$ for all primes $p$. There is a natural exact sequence $$\begin{CD}0@>>>T_l(\chi_D,j)@>>>V_l(\chi_D,j)@>\pi>>A_l(\chi_D,j)@>>>0\= end{CD}.$$ Let $H^1_f(\QQ_p,A_l(\chi_D,j))=3D\pi_*H^1_f(\QQ_p,V_l(\chi_D,j))$. Define the $l$-Selmer group $H^1_f(\QQ,A_l(\chi_D,j))$ to be the subgroup of elements of $H^1(\QQ,A_l(\chi_D,j))$ whose local restrictions lie in $H^1_f(\QQ_p,A_l(\chi_D,j))$ for all primes $p$. Define the Shafarevich-Tate group $$\Sha(\chi_D,j)=3D\oplus_lH^1_f(\QQ,A_l(\chi_D,j))/\pi_*H^1_f(\QQ,V_l(\c= hi_D,j)).$$ For $1\leq j\leq k-1$ define the set of global points $$\Gamma_{\QQ}(\chi_D,j)=3D\oplus_lH^0(\QQ,A_l(\chi_D,j)).$$ \section{The Bloch-Kato conjecture} In this section we assume further that $f$ has level $N=3D1$. Let $j$ be an integer such that $k/2\leq j\leq k-1$. (Certainly then $L(f,\chi_D,j)$ is a critical value in the sense of \cite{De2}.) If $j=3Dk/2$ we suppose we are in the case that $L(f,\chi_D,k/2)\neq 0$. The Bloch-Kato conjecture for the motive $M(\chi_D,j)$ predicts that $$L(f,\chi_D,j)=3D(\prod_pc_p(\chi_D,j))\vol_{\infty}(\chi_D,j)\#\Sha(\ch= i_D,j)/(\#\Gamma_{\QQ}(\chi_D,j) \#\Gamma_{\QQ}(\chi_D,k-j)).$$ The local factors $\vol_{\infty}(\chi_D,j)$ and $c_p(\chi_D,j)$ depend on the choice of a basis for $V_{\dR}$, but their product does not. A careful treatment of real periods such as $\vol_{\infty}(\chi_D,j)$ is given in \cite{De2}. Note that the definition of $\vol_{\infty}(j)$ given in Section 4 of \cite{Du1} is mistaken, what is defined there actually being $1/\vol_{\infty}(j)$. The calculation of the periods of Artin motives in terms of generalized Gauss sums, in Section 6 of \cite{De2}, yields the following lemma. \begin{lem} $\vol_{\infty}(\chi_D,j)=3D\vol_{\infty}(j)/\sqrt{D}.$ \end{lem} Define $H^1_f(\QQ_p,T_l(\chi_D,j))$ to be the inverse image of $H^1_f(\QQ_p,V_l(\chi_D,j))$ under the natural map from $H^1(\QQ_p,T_l(\chi_D,j))$ to $H^1(\QQ_p,V_l(\chi_D,j))$. The factor $c_p(\chi_D,j)$, which is $1$ for all but finitely many $p$, is defined to be $\vol_p(\chi_D,j)$ divided by the Euler factor $(1-\chi_D(p)a_pp^{-j}+p^{k-1-2j})$. Here, $\vol_p(\chi_d,j)$ is the product of the size of $\oplus_{l\neq p}H^1_f(\QQ_p,T_l(\chi_D,j))$ and a certain measure of $H^1_f(\QQ_p,T_p(\chi_D,j))$. We need to look a bit more carefully at the definition of the $p$-part of $\vol_p(\chi_d,j)$. Let $D_{\dR}(V_p)=3DH^0(\QQ_p,V_p\otimes B_{\dR})$, where $B_{\dR}$ is one of Fontaine's rings \cite{Fo}. This $D_{\dR}(V_p)$ is a filtered $\QQ_p$-vector space which, according to Fontaine's de Rham conjecture, should be naturally isomorphic to $V_{\dR}\otimes \QQ_p$ with its Hodge filtration. Since $p$ is a prime of good reduction for the motive $M$, $\,V_p$ is a crystalline representation of $\Gal(\Qbar_p/\QQ_p)$, meaning $D_{\cris}(V_p)$ and $V_p$ have the same dimension, where $D_{\cris}(V_p):=3DH^0(\QQ_p,V_p\otimes B_{\cris})$. (This is a consequence of Theorem 5.6 of \cite{Fa1}.) It follows from Theorem 4.1(ii) of \cite{BK} that $$D_{\dR}(V_p)/F^jD_{\dR}(V_p)\simeq H^1_f(\QQ_p,V_p(j)),$$ via their exponential map. Now if we admit the de Rham conjecture, then a choice of basis for $V_{\dR}$ will give a measure on the right-hand side, with respect to which $|\vol_p(j)|_p^{-1}$ is defined to be the volume of $H^1_f(\QQ_p,T_p(j))$. Fortunately, since $p$ is a prime of good reduction, the de Rham conjecture is a consequence of the crystalline conjecture, which follows from Theorem 5.6 of \cite{Fa1}. In Section 7 of \cite{Du1}, we had a condition that $p>k$, because we needed a comparison theorem with $\ZZ_p$ coefficients rather than just $\QQ_p$ coefficients. That condition is not necessary here, which is good, given that $11<12$. \begin{lem} Let $q$ be an odd prime and $D$ a quadratic discriminant. For any prime $p\neq q$, we have $\ord_q(c_p(\chi_{D},j))=3D0$. \end{lem} {\em Proof. } As on p. 30 of \cite{Fl1}, for $p\neq q$, $$|c_p(\chi_D,j)|_q^{-1}=3D\#H^0(\QQ_p,H^1(I_p,T_q(\chi_D,j))_{\tors})),$= $ where $I_p$ is the inertia group at $p$. If $p$ does not divide $D$ then $T_q(\chi_D,j)$ is a trivial $I_p$-module, so $H^1(I_p,T_q(\chi_D,j))$ is torsion-free. In general, since $q\neq p$,$$H^1(I_p,T_q(\chi_{D},j))\simeq T_q(\chi_{D},j)/(1-\tau)T_q(\chi_{D},j),$$ where $\tau$ is a topological generator of the tame quotient of $I_p$. If $p|D$ then thanks to the quadratic twist ramified at $p$, $\,\tau$ acts as multiplication by $-1$ on $T_q(\chi_{D},j)$, hence $H^1(I_p,T_q(\chi_{D},j))$ is trivial (remember $q\neq 2$).\newline $\blacksquare$ \begin{lem} Suppose that $D$ and $D'$ are two quadratic discriminants, both divisible by $q$, with $\chi_{D/\epsilon q}(q)=3D\chi_{D'/\epsilon q}(q)$, where $\epsilon=3D\chi_{-4}(q)$. Then $c_q(\chi_D,j)=3Dc_q(\chi_{D'},j)$. \end{lem} {\em Proof. }For all primes $l,$$\,T_l(\chi_D,j)$ and $T_l(\chi_{D'},j)$ are exactly the same as modules for $\Gal(\Qbar_q/\QQ_q)$. Furthermore, the Euler factors $(1-\chi_D(q)a_qq^{-j}+q^{k-1-2j})$ and $(1-\chi_{D'}(q)a_qq^{-j}+q^{k-1-2j})$ are the same. For primes $l\neq q$ we have $\ord_l(\vol_q(\chi_D,j))=3D\#H^1_f(\QQ_q,T_l(\chi_D,j))$, so it is a triviality that \newline $\ord_l(\vol_q(\chi_D,j))=3D\ord_l(\vol_q(\chi_{D'},j))$. For $l=3Dq$, $$(V_{\dR}(\chi_D)\otimes\QQ_q)/F^j\simeq H^1_f(\QQ_q,V_q(\chi_D,j)),$$ and a choice of basis for $V_{\dR}$ determines the $q$-part of $c_q(\chi_D,j)$. If $\chi_{D/\epsilon q}(q)=3D\chi_{D'/\epsilon q}(q)$ then replacing $D$ by $D'$ introduces no essential change on either side, so $c_q(\chi_D,j)=3Dc_q(\chi_{D'},j)$.\newline $\blacksquare$ \begin{remar} We will use only the `$q$-part' of the above lemma. \end{remar} \begin{lem} $\ord_l(\#\Gamma_{\QQ}(\chi_D,j))=3D0$ unless the coefficients of $f=3D\sum e_nq^n$ satisfy the congruence $e_p\equiv \chi_D(p)(p^j+p^{k-1-j})(\bmod{l})$ for all primes $p\neq l$. \end{lem} This follows from the interpretation of $e_p$ as a trace of Frobenius. Note that in the case of interest to us, where $f=3D\Delta, \,j=3Dk/2=3D6$ and $l=3D11$, the congruence is not = satisfied for any $D$. See \cite{SwD}. Now let us concentrate on the case where $f=3D\Delta$ and $j=3Dk/2=3D6$. Recall the forms $F$, $g=3D\sum a_nq^n$ and $h=3D\sum b_nq^n$ of weights $2,3/2$ and $13/2$ respectively. Now $b_{33}=3D-8968320\equiv 2(\bmod{11})$ so we can take $d'=3D3$ and have $11$ not dividing $b_{11d'}$. When $a_d=3D0$, the congruence $a_n\equiv b_{11n}(\bmod{11})$ implies that $11|b_{11d}$. Combining the above lemmas with Theorem \ref{wald}, we find the following. \begin{prop} Consider the case $k=3D12,\,f=3D\Delta$. Suppose $d$ is a positive integer such that $11d$ is the discriminant of a quadratic field, $d$ is a quadratic residue modulo $11$,$\,a_d=3D0$ and $L(\Delta,\chi_{11d},6)\neq 0$. If the Bloch-Kato conjecture is true then $11^2|\#\Sha(\chi_{11d},6)$, so $H^1_f(\QQ,A_{11}(\chi_{11d},6))$ contains elements of order $11$. \end{prop} \begin{remar} Via Flach's generalization of the Cassels-Tate pairing \cite{Fl2}, if \newline $11|\#\Sha(\chi_{11d},6)$ then $11^2|\#\Sha(\chi_{11d},6)$. \end{remar} Even if $L(\Delta,\chi_{11d},6)=3D0$, as explained in the introduction, we still expect that $H^1_f(\QQ,A_{11}(\chi_{11d},6))$ contains an element of order $11$. Using Maple we checked that $L(\Delta,\chi_{11d},6)\neq 0$, for all relevant $d$ with $1\leq d\leq 10,000.$ See \cite{OS} for what can be proved about non-vanishing. As noted in the next section, the sign in the functional equation of \newline $L(\Delta,\chi_{11d},s)$ is positive, so the order of vanishing at $s=3D6$ is always even. Our task in the next section is to construct an element of order $11$ in the Selmer group $H^1_f(\QQ,A_{11}(\chi_{11d},6))$. \section{Local conditions} Given a prime $l$, we have defined $V_l,T_l$ and $A_l$ for a modular form $f$ of even weight $k$, and retained this notation for the special case $f=3D\Delta$,$\,k=3D12$. For the modular form $F$ of weight $2$ and level $11$ associated to the elliptic curve $E=3DX_0(11)$, we shall use the notation $V_l',T_l'$ and $A_l'$. The $l$-adic Tate module of $E$ is then $T_l'(1)$. We use the notation $A'[l]$ for the $l$-torsion subgroup of $A_l'$. \begin{thm}\label{local} Suppose that $d>0$ and $11d$ is the = discriminant of a quadratic field. Let $r$ be the rank of the group of rational points $E_{-d}(\QQ)$. If $\,r>1$, the $11$-torsion subgroup of the Selmer group $H^1_f(\QQ,A_{11}(\chi_{11d},6))$ has $\FF_{11}$-rank at least $r-1$. \end{thm} {\em Proof. } As $\Gal(\Qbar/\QQ)$-modules, $E_{-d}[11]\simeq A'(\chi_{-d},1)[11]\simeq A(\chi_{-d},1)[11]$, the second isomorphism being thanks to the congruence between $\Delta$ and $F$, and the irreducibility of these modules. The fifth power of the $11$-cyclotomic character is congruent mod $11$ to the quadratic character of conductor $11$, so this in turn is isomorphic to $A(\chi_{11d},6)[11]$. The injection from $E_{-d}(\QQ)/11E_{-d}(\QQ)$ to $H^1(\QQ, E_{-d}[11])$ then gives us an $\FF_{11}$-subspace of dimension $r$ in $H^1(\QQ,A(\chi_{11d},6)[11])$, which injects into \newline $H^1(\QQ,A_{11}(\chi_{11d},6))$ because, as already noted, $H^0(\QQ,A_{11}(\chi_{11d},6))$ is trivial. Choose some element $c\in H^1(\QQ,A(\chi_{11d},6)[11])$, coming from $E_{-d}(\QQ)/11E_{-d}(\QQ)$, mapping to (say) $\gamma\in H^1(\QQ,A_{11}(\chi_{11d},6))$. First we show that $\res_p(\gamma)\in H^1_f(\QQ_p,A_{11}(\chi_{11d},6))$, for all primes $p\neq 11$, i.e. that $\res_p(\gamma)$ in \newline $H^1(I_p,A_{11}(\chi_{11d},6))$ is zero. Looking at $c\in H^1(\QQ, A'(\chi_{-d},1)[11])$ it maps to $\gamma'\in H^1_f(\QQ,A'_{11}(\chi_{-d},1))$, so $\res_p(\gamma')$ in $H^1(I_p,A'_{11}(\chi_{-d},1))$ is zero. Now $A'_{11}(1)$ is unramified at $p\neq 11$, so $H^0(I_p,A'_{11}(\chi_{-d},1))$ is $11$-divisible and $H^1(I_p,A'(\chi_{-d},1)[11])$ injects into $H^1(I_p,A'_{11}(\chi_{-d},1))$. Hence $\res_p(c)$ in $H^1(I_p,A'(\chi_{-d},1)[11])$\newline $=3DH^1(I_p,A(\chi_{11d},6)[11])$ is zero, and its image $\res_p(\gamma)$ in $H^1(I_p,A_{11}(\chi_{11d},6))$ is zero, as required. It remains to deal with the local condition at $p=3D11$. The exponential map of Bloch and Kato gives an isomorphism $$V_{\dR}(\chi_{11d})\otimes\QQ_{11}/F^6\simeq = H^1_f(\QQ_{11},V_{11}(\chi_{11d},6)).$$ Meanwhile, Section 3 of \cite{BK} tells us that $H^1_f(\QQ_{11},V_{11}(\chi_{11d},6))$ is its own annihilator in $H^1(\QQ_{11},V_{11}(\chi_{11d},6))$ with respect to the Tate duality pairing. Hence $H^1_f(\QQ_{11},V_{11}(\chi_{11d},6))$ has codimension one in $H^1(\QQ_{11},V_{11}(\chi_{11d},6))$. It follows that our $r$-dimensional $\FF_{11}$-subspace of the $11$-torsion in $H^1(\QQ,A_{11}(\chi_{11d},6))$ has at least an $(r-1)$-dimensional subspace landing in $H^1_f((\QQ_{11},A_{11}(\chi_{11d},6))$.\newline $\blacksquare$ For a newform of weight $k$ on $\Gamma_0(N)$, and $D$ the discriminant of a quadratic field, the sign in the functional equation of $L(f,\chi_D,s)$ is $(-1)^{k/2}w_N\chi_D(-N)$. For $L(\Delta,11d,s)$ as above, the sign is always positive, since $\chi_D(-1)=3D1$ for the character associated to a real quadratic field. For our $F$ of weight $2$ and level $11$ we have $w_{11}=3D-1$, so the sign in the functional equation of $L(f,s)$ is positive. When we twist by $-d$ as above, the sign becomes $\chi_{-d}(-11)$. Hence when $({d\over 11})=3D1$ the sign is positive and the $L$-function vanishes to even order at $s=3D1$. If also the coefficient $a_d=3D0$ then Waldspurger's theorem implies that $L(E_{-d},s)$ vanishes at $s=3D1$, and we now know it must do so to order at least $2$. According to the Birch and Swinnerton-Dyer conjecture, in this case $r\geq 2$, so the above theorem provides the required element of order $11$ in the Selmer group $H^1_f(\QQ,A_{11}(\chi_{11d},6))$. The squarefree $d<550$ such that $d\equiv 3(\bmod{4}),\,$$({d\over 11})=3D1$ and $a_d=3D0$ are $d=3D47, 103, 119, 159$ and $455$. Cremona has checked, for all these values of $d$, that the order of vanishing of $L(E_{-d},s)$ at $s=3D1$ is precisely two, and that $E_{-d}(\QQ)$ does have rank $2$. Guerzhoy and Panchishkin \cite{GP} found the simple rational points $(-1,1)$,$\,(-3,1)$ and $(1,1)$, of infinite order on $E_{-47}$,$\,E_{-103}$ and $E_{-119}$ respectively, with reference to Weierstrass equations $-dy^2=3D4x^3-4x^2-40x-79$. The proof of Theorem \ref{local} is easily adapted to show that if the $11$-torsion in \newline $H^1_f(\QQ,A_{11}(\chi_{11d},6))$ has dimension $t$ then the $11$-torsion in $H^1_f(\QQ,A'_{11}(\chi_{-d},1))$ has dimension at least $t-1$. In fact, we made the first part of the proof slightly longer than necessary just to ensure this reversibility. As noted in the introduction, since $11<12$ we are unable to apply the theory of Fontaine and Lafaille to show that the $11$-torsion in $H^1_f(\QQ,A_{11}(\chi_{11d},6))$ and the $11$-torsion in $H^1_f(\QQ,A'_{11}(\chi_{-d},1))$ are isomorphic. This is actually a good thing, since if $({d\over 11})=3D-1$, so that the order of vanishing of $L(E_{-d},s)$ at $s=3D1$ is odd, we would expect the dimensions of these $\FF_{11}$-vector spaces to differ in parity. \section{More general results} Let $E$ be an elliptic curve of prime conductor $p$. Let $f_2$ be the normalized Hecke eigenform of weight $2$ and level $p$ attached to $E$. Since $E$ has multiplicative reduction at $p$, the form $f_2$ is ordinary at $p$. In other words, $p$ does not divide the $p^{th}$ Fourier coefficient. Hence, according to \cite{H}, if $k>2$ is an integer such that $k\equiv 2(\bmod{p^{t-1}(p-1)})$ then, if $t$ is sufficiently large, there exists a normalized eigenform $f_k$, an oldform of weight $k$ on $\Gamma_0(p)$, such that $f_k\equiv f_2(\bmod{p^t})$ as $q$-expansions. (In the example with $p=3D11$, it happens that $t=3D1$ is large enough.) There is a normalized newform $F_k$ of level one such that $f_k(z)=3DF_k(z)-\beta F_k(pz)$, where $\beta$ is the non-unit root of Frobenius. Let $\epsilon=3D\pm 1$, $\,\nu=3D(-1)^{(p-1)/2}$ and let $d$ be a positive integer such that $\epsilon d$ is the discriminant of a quadratic field. The sign in the functional equation of $L(E_{\epsilon d},s)$ is $-w_p\chi_{\epsilon d}(-p)$. We shall be interested in those $d$ such that this sign is positive, so that if $L(E_{\epsilon d},s)$ vanishes at $s=3D1$, it does so to order at least two. Consider the motive attached to $F_k$, with its $V_{\lambda},\,T_{\lambda}$ and $A_{\lambda}$, for prime ideals $\lambda$ in the ring of integers of the field generated by the coefficients of $F_k$. These coefficients are naturally regarded as elements of $\ZZ_p$, so there is a favoured split prime $\frak p$ dividing $p$ in the coefficient field. \begin{thm} Suppose that $d>0$ is coprime to $p$, and that $\epsilon d$ is the discriminant of a quadratic field. Let $r$ be the rank of the group of rational points $E_{\epsilon d}(\QQ)$, and suppose that $\,r>1$. Suppose that $k\equiv 2(\bmod{p^{t-1}(p-1)})$. If $k-2$ is an odd multiple of $\phi(p^t)$, let $S=3DH^1_f(\QQ,A_{\frak p}(\chi_{\epsilon\nu pd},k/2))$. Otherwise, let $S=3DH^1_f(\QQ,A_{\frak p}(\chi_{\epsilon d},k/2))$. Then $S$ has a subgroup isomorphic to $(\ZZ/p^t\ZZ)^{r-1}$. \end{thm} This may be proved in essentially the same way as Theorem \ref{local}. Note that it follows from Hida's construction that $E[p^t](-1)$ and $A[p^t]$ are isomorphic as $\Gal(\Qbar/\QQ)$-modules, not just that the traces of Frobenius (i.e. the Fourier coefficients modulo $p^t$) are equal. If $L(E_{\epsilon d},s)$ vanishes to order at least two at $s=3D1$, then the conjecture of Birch and Swinnerton-Dyer predicts that $r\geq 2$. If $r=3D2$ then the above theorem provides a subgroup isomorphic to $\ZZ/p^t\ZZ$ in a suitable Selmer group. Assuming that the Selmer group equals the Shafarevich-Tate group, it follows from \cite{Fl2} that its order is at least $p^{2t}$. Now we just have to content ourselves that this is as predicted by the Bloch-Kato conjecture, applied to a ratio $\sqrt{d'/d}L(f_k,\chi_{\epsilon d},k/2)/L(f_k,\chi_{\epsilon d'},k/2)$ or $\sqrt{d'/d}L(f_k,\chi_{\epsilon\nu pd},k/2)/L(f_k,\chi_{\epsilon\nu pd'},k/2)$. First, the representation of \newline $\Gal(\Qbar/\QQ)$ on $E[p]$ is irreducible, thanks to Theorem 4 of \cite{Maz1}. Hence factors such as $\Gamma_{\QQ}(\chi_{\epsilon d},k/2)$ make no contribution. Second, it may be shown just as in Section 6 that fudge factors such as $c_q(\chi_{\epsilon d},k/2)$ make no contribution. (There is a minor complication in that we must consider their orders at all the primes dividing $p$.) What we need is then provided by the proposition below, except we must impose the conditions $p\equiv 3(\bmod{4})$ and $\epsilon=3D-1$. If $p\equiv 1(\bmod{4})$ and $\epsilon=3D1$ then the $L$-values in question would automatically be zero, thanks to a negative sign in the functional equation. If $\epsilon\neq \nu$ then the theorems quoted below do not apply. \begin{prop} Suppose that $p\equiv 3(\bmod{4})$, with $E/\QQ$ an = elliptic curve of conductor $p$. Let $d\equiv d'(\bmod{8})$ be positive, squarefree integers, coprime to $p$, with $d\equiv 3(\bmod{4})$ and $-w_p\chi_{-d}(-p)=3D-w_p\chi_{-d'}(-p)=3D1$. Suppose that $L(E_{-d},1)=3D0$ but $L(E_{-d'},1)\neq 0$. Suppose that $k>2$ is an integer with $k\equiv 2(\bmod{\phi(p^t)})$, and $t$ sufficiently large. Let $s=3D(k-2)/\phi(p^t)$. If $s$ is even, then the rational ratio \newline $\sqrt{d'/d}L(f_k,\chi_{-d},k/2)/L(f_k,\chi_{-d'},k/2)$ (if defined) is divisible by $p^{2(t-u)}$. If $s$ is odd, then $\sqrt{d'/d}L(f_k,\chi_{pd},k/2)/L(f_k,\chi_{pd'},k/2)$ (if defined) is divisible by $p^{2(t-u)}$. Here, $u$ is a non-negative integer which depends only on $d'$, not on $k$. \end{prop} {\em Sketch of proof. } Let $\chi$ be the trivial character if $s$ is odd, the Legendre symbol $(\frac{\cdot}{p})$ if $s$ is even. Let $\theta_{\chi}(f_k)\in S_{(k+1)/2}(\Gamma_0(4p^m),\chi^*)$ be Shintani's lifting \cite{Shin}, where $m=3D(1+(-1)^s)/2$ and $\chi^*(n)=3D\chi(n)(\frac{(-1)^{k/2}p}{n})$. Stevens \cite{Ste} constructed a $\Lambda$-adic Shintani lifting, $p$-adically interpolating the Shintani liftings of the various $f_k$. Here we consider only a sufficiently small open neighbourhood of $k=3D2$ in his $X(R)$. It follows from his Theorem 3.3 that, as $q$-expansions, $$\Omega_{k,\chi}\theta_{\chi}(f_k)/\omega(f_k)|T_p^{1-m}\equiv \Omega_2\theta(f_2)/\omega(f_2)(\bmod{p^t}).$$ Here, $\omega(f_k)$ and $\omega(f_2)$ are certain periods chosen so that $\theta_{\chi}(f_k)/\omega(f_k)$ and $\theta(f_2)/\omega(f_2)$ are algebraic integers, or at least algebraic numbers, integral at primes dividing $p$. The factors $\Omega_{k,\chi}$ and $\Omega_2$ are elements of $\QQ_p$. For $\omega(f_2)$ we take $2\pi i$ times the canonical real period of the elliptic curve $E$. We choose $E$ to be a strong Weil curve within its isogeny class. The rational number $\theta(f_2)/\omega(f_2)$ really is integral at $p$. This is because the coefficients of $\theta(f_2)$ are integer linear combinations of integrals of $f_2(\tau)\,d\tau$ over paths joining $\Gamma_0(p)$-equivalent cusps. (Manin's constant is coprime to $p$ (Corollary 4.1 of \cite{Maz1}), so $\frac{1}{2\pi i} f_2(\tau)\,d\tau$ is as good as the pullback of the canonical differential on $E$.) The $\Omega_k$ in \cite{Ste} are the same as those in \cite{GS}, where $\Omega_2=3D1$ for our choice of $\omega(f_2)$ (see \cite{GS} between 0.8 and 0.9). Let $\theta(f_2)/\omega(f_2)=3D\sum a_n q^n$ and $\Omega_{k,\chi}\theta_{\chi}(f_k)/\omega(f_k)=3D\sum b_n q^n$. Then $a_n\equiv b_n(\bmod{p^t})$ if $s$ is even, while $a_n\equiv b_{pn}(\bmod{p^t})$ if $s$ is odd. Let $u=3D\ord_p(a_{d'})$ and make sure $t$ is large enough not only for $f_k$ to exist, but for $t>u$. Since $a_d=3D0$, the congruences imply that $p^{t-u}$ divides $b_d/b_{d'}$, or $b_{pd}/b_{pd'}$ as appropriate. Now the proposition follows directly from Corollaire 2 to Th\'eor\`eme 1 of \cite{Wa}, which, under our hypotheses, states that $$(d/d')^{(k-1)/2}L(f_k,\chi_{-d},k/2)/L(f_k,\chi_{-d'},k/2)=3Db_d^2/b_{d= '}^2$$ or $$(d/d')^{(k-1)/2}L(f_k,\chi_{pd},k/2)/L(f_k,\chi_{pd'},k/2)=3Db_{pd}^2/b= _{pd'}^2,$$ as appropriate, in both cases assuming that the $L$-value in the denominator is non-zero.\newline $\blacksquare$ \begin{remar} In the $p=3D11$ case, we could choose $d'$ so that $u=3D0$. One might expect that this is typical. \end{remar} We should mention the work of Guerzhoy \cite{G2}, who considers certain $p$-adic families of cusp forms and generalizes the `quadratic congruences' of \cite{BDO}. His approach is via two-variable $p$-adic $L$-functions, and can be used to prove congruences of the type we have used. 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Shimura, On modular forms of half-integral weight, {\em Ann. Math. }{\bf 97 }(1973), 440--481. \bibitem[Shin]{Shin} T. Shintani, On construction of holomorphic cusp forms of half integral weight, {\em Nagoya Math. J. }{\bf 58 }(1975), 83--126. \bibitem[Ste]{Ste} G. Stevens, $\Lambda$-adic modular forms of half-integral weight and a $\Lambda$-adic Shintani lifting. Arithmetic geometry (Tempe, AZ, 1993), 129--151, Contemp. Math., 174, Amer. Math. Soc., Providence, RI, 1994. \bibitem[Stu]{Stu} J. Sturm, On the congruence of modular forms, {\em Lect. Notes Math. }{\bf 1240, } Springer, 1984. \bibitem[SwD]{SwD} H. P. F. Swinnerton-Dyer, On $l$-adic representations = and congruences for coefficients of modular forms,{\em Modular functions of one variable} III, Lect. Notes Math. {\bf 350, } Springer, 1973. \bibitem[Wa]{Wa} J.L. Waldspurger, Sur les coefficients de Fourier des formes modulaires de poids demi-entier, {\em J. Math. Pures et Appl. }{\bf 60 }(1981), 375--484. \end{thebibliography} University of Sheffield, Department of Pure Mathematics, Hicks Building, Hounsfield Road, Sheffield, S3 7RH, U.K.\newline E-mail:n.p.dummigan@shef.ac.uk \end{document} ------=_NextPart_000_000A_01C08D37.F8CA5060-- From vmwarenews@vmware.m0.net Fri Feb 2 21:10:23 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f132BaW24835 for ; Fri, 2 Feb 2001 21:11:36 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f132Asm08772 for ; Fri, 2 Feb 2001 21:10:54 -0500 (EST) Received: from fac204.m0.net (ant.m0.net [209.10.46.156]) by math.berkeley.edu (8.9.3/8.9.3) with SMTP id SAA26140 for ; Fri, 2 Feb 2001 18:10:53 -0800 (PST) Message-ID: <3048500939.981166223228@m0.net> Date: Fri, 2 Feb 2001 18:10:23 -0800 (PST) From: VMware Reply-to: vmwarenews@vmware.m0.net To: was@math.berkeley.edu Subject: VMware Newsletter - February 2001 - Issue VIII Errors-to: vmwarenews@vmware.m0.net Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" X-cid: 3048500939 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by modular.fas.harvard.edu id f132BaW24835 Status: R X-Status: N VMWARE NEWSLETTER FEBRUARY 2001 ISSUE VIII Welcome to the February issue of the VMware(TM) Newsletter. This month our GSX Server(TM) joins the growing lineup of VMware products that are now shipping. At the same time, Halliburton and Informatica have joined the group of early adopters that are using this innovative new product. Since there's so much going on at VMware, it's unlikely that you would want to remove yourself from our mailing list. If, however, this is necessary, click to http://vmware1.m0.net/m/u/vmw/v.asp?e=was%40math.berkeley.edu -------- IN THIS ISSUE ----------------------------------------- -- VMware GSX Server Now Shipping -- New Early Adopters for GSX Server: Halliburton and Informatica -- The NSA and VMware Build More Secure Computer Systems -- Jobs at VMware -- Press Coverage -- Events ------------------------------------------------------------------ GSX SERVER NOW SHIPPING The beta is over. On January 23 we began shipping VMware GSX Server, Version 1.0 for Linux. Now you can enjoy mainframe-class control on an Intel server. Whether you are running departmental or workgroup servers, you can use this innovative software to cut down on the number of machines. And manage the others more easily. It doesn't matter if you're dealing with hardware -- or software. VMware GSX Server brings order to any server installation. For the details, check out the pages at http://vmware1.m0.net/m/s.asp?HB3048500939X973933X50537X ------------------------------------- NEW EARLY ADOPTERS FOR GSX SERVER Halliburton Energy Services was one of the earliest GSX Server adopters in our beta program. The company had been faced with growing server proliferation. Now, with GSX Server, it can consolidate up to 10 servers in a single host. What's more, it can lower the cost of high-reliability hardware that will deliver mainframe-level reliability and security for its customers. Over at Informatica, the technical support engineers had been spending a lot of time installing (or reinstalling) operating systems and applications -- so they could duplicate problems their customers see. With GSX Server, the engineers have now set up "potted combinations" of various programs and applications, and they keep them ready to use when a customer calls. And they do it without ever leaving their desks. http://vmware1.m0.net/m/s.asp?HB3048500939X973935X50537X ------------------------------------- VMWARE AND THE NATIONAL SECURITY AGENCY TEAM TO BUILD MORE SECURE COMPUTER SYSTEMS Our company has joined with the U.S. National Security Agency on a cooperative research and development project. The goal is to develop virtual machine technology that provides verifiable security. In this way, government users will be able to use commercial off-the-shelf software for sensitive or classified work. For more details on this project, read our press release at http://vmware1.m0.net/m/s.asp?HB3048500939X973982X50537X ------------------------------------- EXCITING CAREERS Like to work at an innovative and successful company? Perhaps you should drop in on the jobs section of our Web site. You'll find listings that range from business director or account manager to senior applications architect -- and virtually everything in between. http://vmware1.m0.net/m/s.asp?HB3048500939X973998X50537X ------------------------------------- RECENT PRESS Online publication 8wire recently took a long and detailed look at VMware Workstation 2.0.3. And they gave us a four-star rating. In fact, in the area of usability, their opinion hit the five-star level. Over at the National Security Agency, Tech Trend Notes took a detailed look at NetTop. This new project plans to use security enhanced virtual machines as building blocks -- for applications that require separation of information domains. For example, to provide secure remote Internet access for classified computer networks. And VMware technology is a big part of their current thinking. And the folks at InfoWorld took a look at the ways you can use VMware Workstation to help ease multiple OS overload. Also, they have a lot of good things to say about us at Computerwoche. If you're one of the many VMware customers who are fluent in German, take a look at the review. http://vmware1.m0.net/m/s.asp?HB3048500939X974017X50537X ------------------------------------- COMING EVENTS To stay updated on our next shows, visit our Events page at http://vmware1.m0.net/m/s.asp?HB3048500939X974025X50537X CeBIT SuSE booth, Halle 3, Stand E45 Hannover, Germany March 22-28, 2001 ------------------------------------- TO UNSUBSCRIBE from this newsletter, go to http://vmware1.m0.net/m/u/vmw/v.asp?e=was%40math.berkeley.edu FOR MORE INFORMATION ABOUT VMWARE, check out the pages at http://vmware1.m0.net/m/s.asp?HB3048500939X974071X50537X (English) or http://vmware1.m0.net/m/s.asp?HB3048500939X974092X50537X (German) TO UPDATE YOUR USER PROFILE, visit the page at http://vmware1.m0.net/m/s.asp?HB3048500939X974106X50537X FOR TECHNICAL SUPPORT ASSISTANCE, go to http://vmware1.m0.net/m/s.asp?HB3048500939X974123X50537X QUESTIONS OR COMMENTS? Contact vmwareservice@vmware.com This newsletter may link to Internet sites that are owned and operated by third parties. Our company is not responsible for the content of these third-party sites. This message is sent via Digital Impact, Inc. for VMware, Inc. From vmwarenews@vmware.m0.net Fri Feb 2 21:12:08 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f132DLW24839 for ; Fri, 2 Feb 2001 21:13:21 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f132Cdm08805 for ; Fri, 2 Feb 2001 21:12:39 -0500 (EST) Received: from fac206.m0.net (ant.m0.net [209.10.46.156]) by math.berkeley.edu (8.9.3/8.9.3) with SMTP id SAA26233 for ; Fri, 2 Feb 2001 18:12:38 -0800 (PST) Message-ID: <3048519883.981166328393@m0.net> Date: Fri, 2 Feb 2001 18:12:08 -0800 (PST) From: VMware Reply-to: vmwarenews@vmware.m0.net To: was@gold.math.berkeley.edu Subject: VMware Newsletter - February 2001 - Issue VIII Errors-to: vmwarenews@vmware.m0.net Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" X-cid: 3048519883 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by modular.fas.harvard.edu id f132DLW24839 Status: R X-Status: N VMWARE NEWSLETTER FEBRUARY 2001 ISSUE VIII Welcome to the February issue of the VMware(TM) Newsletter. This month our GSX Server(TM) joins the growing lineup of VMware products that are now shipping. At the same time, Halliburton and Informatica have joined the group of early adopters that are using this innovative new product. Since there's so much going on at VMware, it's unlikely that you would want to remove yourself from our mailing list. If, however, this is necessary, click to http://vmware1.m0.net/m/u/vmw/v.asp?e=was%40mail.math.berkeley.edu -------- IN THIS ISSUE ----------------------------------------- -- VMware GSX Server Now Shipping -- New Early Adopters for GSX Server: Halliburton and Informatica -- The NSA and VMware Build More Secure Computer Systems -- Jobs at VMware -- Press Coverage -- Events ------------------------------------------------------------------ GSX SERVER NOW SHIPPING The beta is over. On January 23 we began shipping VMware GSX Server, Version 1.0 for Linux. Now you can enjoy mainframe-class control on an Intel server. Whether you are running departmental or workgroup servers, you can use this innovative software to cut down on the number of machines. And manage the others more easily. It doesn't matter if you're dealing with hardware -- or software. VMware GSX Server brings order to any server installation. For the details, check out the pages at http://vmware1.m0.net/m/s.asp?HB3048519883X973933X50537X ------------------------------------- NEW EARLY ADOPTERS FOR GSX SERVER Halliburton Energy Services was one of the earliest GSX Server adopters in our beta program. The company had been faced with growing server proliferation. Now, with GSX Server, it can consolidate up to 10 servers in a single host. What's more, it can lower the cost of high-reliability hardware that will deliver mainframe-level reliability and security for its customers. Over at Informatica, the technical support engineers had been spending a lot of time installing (or reinstalling) operating systems and applications -- so they could duplicate problems their customers see. With GSX Server, the engineers have now set up "potted combinations" of various programs and applications, and they keep them ready to use when a customer calls. And they do it without ever leaving their desks. http://vmware1.m0.net/m/s.asp?HB3048519883X973935X50537X ------------------------------------- VMWARE AND THE NATIONAL SECURITY AGENCY TEAM TO BUILD MORE SECURE COMPUTER SYSTEMS Our company has joined with the U.S. National Security Agency on a cooperative research and development project. The goal is to develop virtual machine technology that provides verifiable security. In this way, government users will be able to use commercial off-the-shelf software for sensitive or classified work. For more details on this project, read our press release at http://vmware1.m0.net/m/s.asp?HB3048519883X973982X50537X ------------------------------------- EXCITING CAREERS Like to work at an innovative and successful company? Perhaps you should drop in on the jobs section of our Web site. You'll find listings that range from business director or account manager to senior applications architect -- and virtually everything in between. http://vmware1.m0.net/m/s.asp?HB3048519883X973998X50537X ------------------------------------- RECENT PRESS Online publication 8wire recently took a long and detailed look at VMware Workstation 2.0.3. And they gave us a four-star rating. In fact, in the area of usability, their opinion hit the five-star level. Over at the National Security Agency, Tech Trend Notes took a detailed look at NetTop. This new project plans to use security enhanced virtual machines as building blocks -- for applications that require separation of information domains. For example, to provide secure remote Internet access for classified computer networks. And VMware technology is a big part of their current thinking. And the folks at InfoWorld took a look at the ways you can use VMware Workstation to help ease multiple OS overload. Also, they have a lot of good things to say about us at Computerwoche. If you're one of the many VMware customers who are fluent in German, take a look at the review. http://vmware1.m0.net/m/s.asp?HB3048519883X974017X50537X ------------------------------------- COMING EVENTS To stay updated on our next shows, visit our Events page at http://vmware1.m0.net/m/s.asp?HB3048519883X974025X50537X CeBIT SuSE booth, Halle 3, Stand E45 Hannover, Germany March 22-28, 2001 ------------------------------------- TO UNSUBSCRIBE from this newsletter, go to http://vmware1.m0.net/m/u/vmw/v.asp?e=was%40mail.math.berkeley.edu FOR MORE INFORMATION ABOUT VMWARE, check out the pages at http://vmware1.m0.net/m/s.asp?HB3048519883X974071X50537X (English) or http://vmware1.m0.net/m/s.asp?HB3048519883X974092X50537X (German) TO UPDATE YOUR USER PROFILE, visit the page at http://vmware1.m0.net/m/s.asp?HB3048519883X974106X50537X FOR TECHNICAL SUPPORT ASSISTANCE, go to http://vmware1.m0.net/m/s.asp?HB3048519883X974123X50537X QUESTIONS OR COMMENTS? Contact vmwareservice@vmware.com This newsletter may link to Internet sites that are owned and operated by third parties. Our company is not responsible for the content of these third-party sites. This message is sent via Digital Impact, Inc. for VMware, Inc. From vmwarenews@vmware.m0.net Fri Feb 2 21:21:51 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f132N5W24846 for ; Fri, 2 Feb 2001 21:23:05 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f132MNm09040 for ; Fri, 2 Feb 2001 21:22:23 -0500 (EST) Received: from fac208.m0.net (ant.m0.net [209.10.46.156]) by math.berkeley.edu (8.9.3/8.9.3) with SMTP id SAA26673 for ; Fri, 2 Feb 2001 18:22:22 -0800 (PST) Message-ID: <3048612352.981166911837@m0.net> Date: Fri, 2 Feb 2001 18:21:51 -0800 (PST) From: VMware Reply-to: vmwarenews@vmware.m0.net To: was@blue1.math.berkeley.edu Subject: VMware Newsletter - February 2001 - Issue VIII Errors-to: vmwarenews@vmware.m0.net Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" X-cid: 3048612352 Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by modular.fas.harvard.edu id f132N5W24846 Status: R X-Status: N VMWARE NEWSLETTER FEBRUARY 2001 ISSUE VIII Welcome to the February issue of the VMware(TM) Newsletter. This month our GSX Server(TM) joins the growing lineup of VMware products that are now shipping. At the same time, Halliburton and Informatica have joined the group of early adopters that are using this innovative new product. Since there's so much going on at VMware, it's unlikely that you would want to remove yourself from our mailing list. If, however, this is necessary, click to http://vmware1.m0.net/m/u/vmw/v.asp?e=was%40yuban.math.berkeley.edu -------- IN THIS ISSUE ----------------------------------------- -- VMware GSX Server Now Shipping -- New Early Adopters for GSX Server: Halliburton and Informatica -- The NSA and VMware Build More Secure Computer Systems -- Jobs at VMware -- Press Coverage -- Events ------------------------------------------------------------------ GSX SERVER NOW SHIPPING The beta is over. On January 23 we began shipping VMware GSX Server, Version 1.0 for Linux. Now you can enjoy mainframe-class control on an Intel server. Whether you are running departmental or workgroup servers, you can use this innovative software to cut down on the number of machines. And manage the others more easily. It doesn't matter if you're dealing with hardware -- or software. VMware GSX Server brings order to any server installation. For the details, check out the pages at http://vmware1.m0.net/m/s.asp?HB3048612352X973933X50537X ------------------------------------- NEW EARLY ADOPTERS FOR GSX SERVER Halliburton Energy Services was one of the earliest GSX Server adopters in our beta program. The company had been faced with growing server proliferation. Now, with GSX Server, it can consolidate up to 10 servers in a single host. What's more, it can lower the cost of high-reliability hardware that will deliver mainframe-level reliability and security for its customers. Over at Informatica, the technical support engineers had been spending a lot of time installing (or reinstalling) operating systems and applications -- so they could duplicate problems their customers see. With GSX Server, the engineers have now set up "potted combinations" of various programs and applications, and they keep them ready to use when a customer calls. And they do it without ever leaving their desks. http://vmware1.m0.net/m/s.asp?HB3048612352X973935X50537X ------------------------------------- VMWARE AND THE NATIONAL SECURITY AGENCY TEAM TO BUILD MORE SECURE COMPUTER SYSTEMS Our company has joined with the U.S. National Security Agency on a cooperative research and development project. The goal is to develop virtual machine technology that provides verifiable security. In this way, government users will be able to use commercial off-the-shelf software for sensitive or classified work. For more details on this project, read our press release at http://vmware1.m0.net/m/s.asp?HB3048612352X973982X50537X ------------------------------------- EXCITING CAREERS Like to work at an innovative and successful company? Perhaps you should drop in on the jobs section of our Web site. You'll find listings that range from business director or account manager to senior applications architect -- and virtually everything in between. http://vmware1.m0.net/m/s.asp?HB3048612352X973998X50537X ------------------------------------- RECENT PRESS Online publication 8wire recently took a long and detailed look at VMware Workstation 2.0.3. And they gave us a four-star rating. In fact, in the area of usability, their opinion hit the five-star level. Over at the National Security Agency, Tech Trend Notes took a detailed look at NetTop. This new project plans to use security enhanced virtual machines as building blocks -- for applications that require separation of information domains. For example, to provide secure remote Internet access for classified computer networks. And VMware technology is a big part of their current thinking. And the folks at InfoWorld took a look at the ways you can use VMware Workstation to help ease multiple OS overload. Also, they have a lot of good things to say about us at Computerwoche. If you're one of the many VMware customers who are fluent in German, take a look at the review. http://vmware1.m0.net/m/s.asp?HB3048612352X974017X50537X ------------------------------------- COMING EVENTS To stay updated on our next shows, visit our Events page at http://vmware1.m0.net/m/s.asp?HB3048612352X974025X50537X CeBIT SuSE booth, Halle 3, Stand E45 Hannover, Germany March 22-28, 2001 ------------------------------------- TO UNSUBSCRIBE from this newsletter, go to http://vmware1.m0.net/m/u/vmw/v.asp?e=was%40yuban.math.berkeley.edu FOR MORE INFORMATION ABOUT VMWARE, check out the pages at http://vmware1.m0.net/m/s.asp?HB3048612352X974071X50537X (English) or http://vmware1.m0.net/m/s.asp?HB3048612352X974092X50537X (German) TO UPDATE YOUR USER PROFILE, visit the page at http://vmware1.m0.net/m/s.asp?HB3048612352X974106X50537X FOR TECHNICAL SUPPORT ASSISTANCE, go to http://vmware1.m0.net/m/s.asp?HB3048612352X974123X50537X QUESTIONS OR COMMENTS? Contact vmwareservice@vmware.com This newsletter may link to Internet sites that are owned and operated by third parties. Our company is not responsible for the content of these third-party sites. This message is sent via Digital Impact, Inc. for VMware, Inc. From xlw@math.berkeley.edu Sat Feb 3 07:34:45 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f13DPoW28262 for ; Sat, 3 Feb 2001 08:25:50 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f13DP8m17967 for ; Sat, 3 Feb 2001 08:25:09 -0500 (EST) Received: from elmart.co.kr (elmart.co.kr [210.96.243.74]) by math.berkeley.edu (8.9.3/8.9.3) with SMTP id FAA13995 for ; Sat, 3 Feb 2001 05:25:06 -0800 (PST) Received: from Domain (unverified [211.100.90.156]) by elmart.co.kr (EMWAC SMTPRS 0.83) with SMTP id ; Sat, 03 Feb 2001 22:35:17 +0900 Message-ID: From: "xlw" Subject: CHIC 2001中国国际服装服饰网上博览会召开在即 To: anny@math.berkeley.edu X-Mailer: V3,1,6,1 (W95/NT) (Build: Oct 18 1999) Mime-Version: 1.0 Date: Sat, 03 Feb 2001 20:34:45 +0800 Status: R X-Status: N 新年好! 这封信如果打扰了您,请您删除，本人并向您表示深深的歉意。 不过我还是希望您看完此信，也许能找到对您有用的东西。 CHIC 2001中国国际服装服饰网上博览会 搜狐网站与风格在线强强联手 风格在线是由中国服装服饰博览会主办的大型服装 时尚专业站点，旨在整合行业资源，推广服装服饰文化， 传播时尚风格；搜狐网站是国内最著名的门户网站，其 现代时尚风格被社会各界所认可。本次博览会最新的构 思是地面博览会与网上展览会同步举行。为了让世界了 解中国、了解中国服装业的现状并同时让世界各地的人 都能观看到此次博览会的盛况和参展企业的品牌风格， 风格在线与搜狐公司联手， 举办"CHIC 2001中国服装服饰网上博览会"，为此次博览 会的参展企业搭建网上展台。 主 办：搜狐网站、风格在线网站 展出时间：2001年3月15日-2001年5月15日 展出地点：搜狐网站 "网上展览中心"、风格在线网站 详情请浏览http://www.chicshow.com From marlene@infomagic.com Sat Feb 3 11:27:16 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f13GBFW28349 for ; Sat, 3 Feb 2001 11:11:15 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f13GAXm06563 for ; Sat, 3 Feb 2001 11:10:33 -0500 (EST) Received: from infomagic.com (MAX6-Port74.Downtown.InfoMagic.NET [63.229.95.174]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f13GAZB19630 for ; Sat, 3 Feb 2001 09:10:35 -0700 Message-ID: <3A7C3164.78EF9FF7@infomagic.com> Date: Sat, 03 Feb 2001 09:27:16 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: was@math.harvard.edu Subject: Re: AHCCCS ADDRESS FOR COMPUTER References: <01020117593209.22620@form> <3A7AB9FF.6658AB1A@infomagic.com> <01020214421101.01014@tx-irmar-48> Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: R X-Status: N I'll tell Nancy. My x-boss, Irene, who has the professor of Economics husband also wondered. He was excited for you when he heard the good news, what an honor no matter what :)..... I'll tell both of them what you told me. Busy day, making cookies for the dance tonight....brownies, laundry, leaving at noon and will be in town til after 11:00p.m. The weather will be in the 50's, a heatwave. THE DANCE CALLER for our dance is a guy from Washington STate and a friend of Sharon and Dave's. Marlene "William A. Stein" wrote: > On Friday 02 February 2001 08:45, you wrote: > > I haven't looked at any of your movies that you have sent yet....I would > > love to see a video of you teaching. Nancy asked if you are getting a > > "tenure track position"??? > > Tell her that Harvard doesn't have "tenure-track positions". All > Assistant Prof. jobs at Harvard are temporary. > > -- William From clarital@yahoo.com Sat Feb 3 16:25:22 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f13LQ4W28470 for ; Sat, 3 Feb 2001 16:26:04 -0500 Received: from web10708.mail.yahoo.com (web10708.mail.yahoo.com [216.136.130.216]) by abel.math.harvard.edu (8.11.0/8.11.0) with SMTP id f13LPNm13944 for ; Sat, 3 Feb 2001 16:25:23 -0500 (EST) Message-ID: <20010203212522.22943.qmail@web10708.mail.yahoo.com> Received: from [150.135.118.160] by web10708.mail.yahoo.com; Sat, 03 Feb 2001 13:25:22 PST Date: Sat, 3 Feb 2001 13:25:22 -0800 (PST) From: Clarita Lefthand Subject: Re: skiing To: was@math.harvard.edu In-Reply-To: <0102011807300B.22620@form> MIME-Version: 1.0 Content-Type: text/plain; charset=us-ascii Status: R X-Status: N Ya'ah'teeh William, > I am begining to doubt that I will have the time to > ski, because I'm > going to be so busy setting up my parent's computer > and teaching them > how to use it. We will see. At least the opportunity will be available. > How are your classes going? My classes are fun for the most part. In my History and Philosophy of Dine' class I'm going to read about Oral Tradition. Included is a story about how we were taught how to weave, and the symbolisms behind weaving. Since seeing so many beautiful rugs at Harvard my interest in weaving has increased. I'm thinking about researching the topic in depth. I share with you what I learn. Did you try out that > integration > web page for fun? I've looked at a few integration web sites, but not for fun. There are many examples given on line of how to integrate, which is helpful in studying integration. I have a test on Monday, so wish me well! I'm on my way to the grocery store. Afterwards, I would like like to call you. I hope that I'm able to reach you, if not I'll try again tomorrow. Tish __________________________________________________ Get personalized email addresses from Yahoo! Mail - only $35 a year! http://personal.mail.yahoo.com/ From kohel@maths.usyd.edu.au Sun Feb 4 02:25:33 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f147QmW28658 for ; Sun, 4 Feb 2001 02:26:48 -0500 Received: from siv.maths.usyd.edu.au (talus.maths.usyd.edu.au [129.78.68.1]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f147Q5m25569 for ; Sun, 4 Feb 2001 02:26:06 -0500 (EST) Received: from smap@localhost by siv.maths.usyd.edu.au (8.8.8/5.7 to SMTP) id SAA12600 for was@math.harvard.edu; Sun, 4 Feb 2001 18:25:31 +1100 (EST) Received: from bizet.maths.usyd.edu.au (root@bizet.maths.usyd.edu.au) [129.78.68.109] by talus.maths.usyd.edu.au via smtpdoor V12.3 id 12277 for was@math.harvard.edu; Sun, 4 Feb 2001 18:25:31 +1100 Message-Id: <200102040725.SAA19826@bizet> Received: from maths.usyd.edu.au by bizet with ESMTP (8.8.8+Sun/11.6 to SMTP) id SAA19826; Sun, 4 Feb 2001 18:25:33 +1100 (EST) To: was@math.harvard.edu Subject: Re: Flashy magma graphics? From: "David R. Kohel" Reply-To: "David R. Kohel" In-reply-to: <01012905292010.10286@form> References: <01012905292010.10286@form> Comments: In-reply-to "William A. Stein" message dated "Mon, 29 Jan 2001 05:29:20 -0500." Date: Sun, 04 Feb 2001 18:25:33 +1100 Sender: kohel@maths.usyd.edu.au Status: R X-Status: N Hi William, Sorry I didn't reply early. As far as I know there has never been any project of the magma group to add graphics of any sort. This is a major distinguishing characteristic between magma and commercial products like mathematica or maple, and I might have commented on this distinction before. There would be absolutely no grant money to develop this sort of thing, so there is not likely to be any such undertaking in Sydney. So if you want to devote a section of your magma-user page to this, then you're welcome to take the initiative :-). I know that Lancelot Pecquet was interested in graphics and used pov-ray (http://www.povray.org/) as a backend for image rendering. One could also imagine using the postscript language as a backend. --David From schoof@wins.uva.nl Sun Feb 4 10:31:49 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f14FWWW03284 for ; Sun, 4 Feb 2001 10:32:32 -0500 Received: from mbox.wins.uva.nl (root@mbox.wins.uva.nl [146.50.16.20]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f14FVqm04182 for ; Sun, 4 Feb 2001 10:31:52 -0500 (EST) Received: from turing.wins.uva.nl [146.50.15.20] by mbox.wins.uva.nl with ESMTP (sendmail 8.10.2/config 10.1). id f14FVog23056; Sun, 4 Feb 2001 16:31:51 +0100 (MET) Received: from localhost by turing.wins.uva.nl (sendmail 8.10.2/config 9.4). id f14FVnk08014; Sun, 4 Feb 2001 16:31:49 +0100 (MET) Message-Id: <200102041531.f14FVnk08014@turing.wins.uva.nl> Date: Sun, 4 Feb 2001 16:31:49 +0100 (MET) From: schoof@wins.uva.nl (Rene Schoof) X-Organisation: Faculty of Science University of Amsterdam The Netherlands X-Address: See http://www.science.uva.nl/location To: was@math.harvard.edu Status: R X-Status: N Hi, enjoying Rennes? Give my regards to Bas. In my next email I'll be sending you the paper with Lario. It's plain TeX. As you'll see the paper has no serious pretentions. It's just a worked out example that may be useful for students studying these deformation rings. Please, see if you'd like to add an appendix and if so, write it soon! We would appreciate any comments that you may have on our paper, of course. Best Rene' From schoof@wins.uva.nl Sun Feb 4 10:32:03 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f14FWjW03288 for ; Sun, 4 Feb 2001 10:32:45 -0500 Received: from mbox.wins.uva.nl (root@mbox.wins.uva.nl [146.50.16.20]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f14FW5m04189 for ; Sun, 4 Feb 2001 10:32:05 -0500 (EST) Received: from turing.wins.uva.nl [146.50.15.20] by mbox.wins.uva.nl with ESMTP (sendmail 8.10.2/config 10.1). id f14FW4g23060; Sun, 4 Feb 2001 16:32:04 +0100 (MET) Received: from localhost by turing.wins.uva.nl (sendmail 8.10.2/config 9.4). id f14FW3M08022; Sun, 4 Feb 2001 16:32:03 +0100 (MET) Message-Id: <200102041532.f14FW3M08022@turing.wins.uva.nl> Date: Sun, 4 Feb 2001 16:32:03 +0100 (MET) From: schoof@wins.uva.nl (Rene Schoof) X-Organisation: Faculty of Science University of Amsterdam The Netherlands X-Address: See http://www.science.uva.nl/location To: was@math.harvard.edu Status: R X-Status: N \magnification\magstep1 \font\tengoth=eufm10 \font\ninegoth=eufm9 \font\eightgoth=eufm8 \font\sevengoth=eufm7 \font\sixgoth=eufm6 \font\fivegoth=eufm5 \newfam\gothfam \def\goth{\fam\gothfam\tengoth} \textfont\gothfam=\tengoth \scriptfont\gothfam=\sevengoth \scriptscriptfont\gothfam=\fivegoth \font\ninerm=cmr9 \font\eightrm=cmr8 \font\sixrm=cmr6 \font\ninei=cmmi9 \font\eighti=cmmi8 \font\sixi=cmmi6 \font\ninesy=cmsy9 \font\eightsy=cmsy8 \font\sixsy=cmsy6 \font\ninebf=cmbx9 \font\eightbf=cmbx8 \font\sixbf=cmbx6 \font\nineit=cmti9 \font\eightit=cmti8 \font\ninett=cmtt9 \font\eighttt=cmtt8 \font\ninesl=cmsl9 \font\eightsl=cmsl8 \newskip\ttglue % switch to 8-point type \def\eightpoint{\def\rm{\fam0\eightrm} \textfont0=\eightrm \scriptfont0=\sixrm \scriptscriptfont0\fiverm \textfont1=\eighti \scriptfont1=\sixi \scriptscriptfont1\fivei \textfont2=\eightsy \scriptfont2=\sixsy \scriptscriptfont2\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3\tenex \textfont\itfam=\eightit\def\it{\fam\itfam\eightit}% \textfont\slfam=\eightsl\def\sl{\fam\slfam\eightsl}% \textfont\ttfam=\eighttt\def\tt{\fam\ttfam\eighttt}% \textfont\gothfam=\eightgoth\scriptfont\gothfam=\sixgoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\tengoth} \textfont\bffam=\eightbf\scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\eightbf}% \tt\ttglue=.5em plus.25em minus.15em \normalbaselineskip=9pt \setbox\strutbox\hbox{\vrule height7pt depth2pt width0pt}% \let\big=\eightbig\normalbaselines\rm} % switch to 9-point type \def\ninepoint{\def\rm{\fam0\ninerm} \textfont0=\ninerm \scriptfont0=\sixrm \scriptscriptfont0\fiverm \textfont1=\ninei \scriptfont1=\sixi \scriptscriptfont1\fivei \textfont2=\ninesy \scriptfont2=\sixsy \scriptscriptfont2\fivesy \textfont3=\tenex \scriptfont3=\tenex \scriptscriptfont3\tenex \textfont\itfam=\nineit\def\it{\fam\itfam\nineit}% \textfont\slfam=\ninesl\def\sl{\fam\slfam\ninesl}% \textfont\ttfam=\ninett\def\tt{\fam\ttfam\ninett}% \textfont\gothfam=\ninegoth\scriptfont\gothfam=\sixgoth \scriptscriptfont\gothfam=\fivegoth \def\goth{\fam\gothfam\tengoth} \textfont\bffam=\ninebf\scriptfont\bffam=\sixbf \scriptscriptfont\bffam=\fivebf \def\bf{\fam\bffam\ninebf}% \tt\ttglue=.5em plus.25em minus.15em \normalbaselineskip=11pt \setbox\strutbox\hbox{\vrule height8pt depth3pt width0pt}% \let\big=\ninebig\normalbaselines\rm} \def\bibliography#1\par{\vskip0pt plus.3\vsize\penalty-250\vskip0pt plus-.3\vsize\bigskip\vskip\parskip \message{Bibliography}\leftline{\bf Bibliography}\nobreak\smallskip\noindent \ninepoint\frenchspacing#1} \def\quod{$\!\quad$} \def\qued{$\!\!\!\quad$} \def\cd{\!\cdot\!} \def\ZZ{{\bf Z}} \def\QQ{{\bf Q}} \def\TT{{\bf T}} \def\FF{{\bf F}} \def\mm{{\goth m}} \def\zmod#1{\,\,({\rm mod}\,\,#1)} \font\twelvebf=cmbx12 at 15pt {\eightpoint \noindent Supernew preliminary version \hfill Roma, January 31, 2001} \hrule \bigskip\bigskip \centerline{\twelvebf Some computations with Hecke rings } \bigskip \centerline{\twelvebf and deformation rings} \bigskip\bigskip \centerline{\bf Joan-C. Lario \ and \ Ren\'e Schoof} \bigskip \centerline{\bf (with an appendix by William Stein)}\bigskip\bigskip\bigskip \def\AT{{1}} % Antwerp IV \def\CR{{2}} \def\HD{{3}} %Darmon \def\DS{{4}} %DeShalit CSS \def\SSR{{5}} %SmitSchoofRubin \def\DI{{6}} %Diamond \def\DR{{7}} %Diamond-Ribet CSS \def\ED{{8}} %Edixhoven \def\GE{{9}} %Gelbart \def\MA{{10}} %Mazur CSS \def\RI{{11}} %Rio \def\SE{{12}} %Serre \def\SEINV{{13}} %Serre, Invent. 15 \def\WS{{14}} %was \def\TW{{15}} %Taylor-Wiles \def\TI{{16}} %Tilouine CSS \def\WI{{17}} %Wiles {\eightpoint \noindent {\bf Abstract.} In the proof by Wiles, completed by Taylor-Wiles, of the fact that all semi-stable elliptic curves over~$\QQ$ are modular, certain deformation rings play an important role. In this note we explicitly compute these rings for the elliptic curve $Y^2+XY=X^3-X^2-X-3$ of conductor~142}. \beginsection 1. Introduction. In order to prove that semi-stable elliptic curves $E$ over~${\bf Q}$ are modular, A.~Wiles~[\WI] considers the Galois representation $\overline{\rho}:{\rm Gal}(\overline{\QQ}/\QQ)\longrightarrow{\rm GL}_2(\FF_3)$ provided by the 3-torsion points~$E[3]$ of a semi-stable elliptic curve~$E$. He proves that certain rather restricted deformations of $\overline{\rho}$ are modular. It follows then that, in particular, the deformation provided by the Galois representation ${\rho}_E^{}:{\rm Gal}(\overline{\QQ}/\QQ)\longrightarrow{\rm GL}_2(\ZZ_3)$ associated to the 3-adic Tate module of~$E$ is modular. This implies that $E$ is modular. Wiles proceeds in two steps. First he considers certain {\it minimal} deformations of $\overline{\rho}$. Using the Langlands-Tunnell Theorem~[\GE,~Thm.1.3], and some so-called `level lowering theorems'~[\ED] he constructs a normalized eigenform of weight~2 and minimal level $N$ whose associated Galois representation $$ \rho_{\rm min}:{\rm Gal}(\overline{\QQ}/\QQ)\longrightarrow{\rm GL}_2(R) $$ is a {\it minimal} deformation of~$\overline{\rho}$. Here the ring $R$ is a finite extension of~$\ZZ_3$. R.~Taylor and Wiles~[\TW] then show that the {\it universal} minimal deformation ring is isomorphic to a ring~$\TT$ of Hecke operators of level~$N$. It follows that the minimal deformations are all modular. It may happen that the representation $\rho_E$ associated to the 3-adic Tate module of~$E$ is {\it not} minimal. Therefore Wiles's second step is to consider deformations of $\overline{\rho}$ that are not necessarily minimal, but satisfy somewhat more relaxed conditions at the primes that divide the conductor of~$E$. From the fact that the minimal deformation ring is a Hecke ring he then deduces that the corresponding universal deformation rings are also isomorphic to certain Hecke rings and it follows that the representation $\rho^{}_E$ is modular. \smallskip In this short note we explicitly compute the relevant universal deformation rings for a specific elliptic curve~$E$. The main result is Theorem~3.2. Our example is the elliptic curve $E$ of conductor 142 which is denoted by ``142A" in the Antwerp Tables~[\AT] (it is the curve 142C1 in J.~Cremona's Tables~[\CR]). A Weierstrass equation for~$E$ is given by $$ Y^2+XY=X^3-X^2-X-3. $$ As we explain below, the representation $$ \overline{\rho}:{\rm Gal}(\overline{\QQ}/\QQ)\longrightarrow{\rm GL}_2(\FF_3) $$ provided by the 3-torsion points~$E[3]$ is unramified outside 3~and~71. Since $\overline{\rho}$ is not ramified at the prime~2, the representation $\rho^{}_E$ associated to the 3-adic Tate module of $E$ is {\it not} a minimal deformation of~$\overline{\rho}$. In this case the minimal universal deformation ring is isomorphic to a ring $\TT$ of Hecke operators of level~71. We compute it in section~2. It has two $\ZZ_3$-valued points, one of which corresponds to $\rho_{\rm min}$ and we take as the ``origin" of the deformation space~${\rm Spec}(\TT)$. In section~3 we study deformations $\rho$ of $\overline{\rho}$ that satisfy somewhat more relaxed conditions at the prime~2: they need not be unramified at~2, but should be of ``type~(A)". This means that the image ${\rho}(I_{2})$ of the inertia group~$I_{2}$ at any prime over~2 is contained in a subgroup conjugate to~$\pmatrix{1&*\cr0&1\cr}$. The deformation $\rho^{}_E$ is of type~(A). In this case the universal deformation ring is a ring of Hecke operators of level $284=4\cdot71$. We show that it is a complete intersection algebra of rank~8 over~$\ZZ_3$. Since the elliptic curve $E$ has conductor~142, one might have expected this universal deformation ring to be isomorphic to a Hecke ring of level~$2\cdot71$ rather than~$4\cdot 71$. Therefore we also determine the Hecke ring of level~$142$ in section~5. It has rank~5 over~$\ZZ_3$ and it turns out not to be a complete intersection or even a Gorenstein ring. For the three Hecke rings $\TT$ of levels 71, 142 and~284, we also compute two invariants associated to the $\ZZ_3$-algebras~$\TT$ and the morphisms $\pi:\TT\longrightarrow\ZZ_3$ provided by $\rho_{\rm min}$. Writing $I={\rm ker}(\pi)$, we determine the {\it congruence ideal} $\eta=\pi({\rm Ann}(I))$ and the {\it cotangent space} $I/I^2$ of~$\TT$. We have that $\#(I/I^2)=[\ZZ_3:\eta]$ if and only if $\TT$ is a complete intersection~[\SSR,~Crit.I]. \smallskip We conclude this introduction by giving some relevant information concerning the curve~142A. Counting points on~$E$ modulo the primes $p$ with $3\le p\le67$ one finds that $\#E(\FF_p)=p+1-a^{}_p$ with $a^{}_p$ as in Table~1.1. \bigskip \vbox{\noindent {\bf Table 1.1} {\sl Fourier coefficients $a_p$.} \medskip \centerline {\vbox {\offinterlineskip \hrule\halign{&\vrule#&\strut\qued\hfil#\qued\cr height3pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit& &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &$p$&&3&&5&&7&&11&&13&&17&&19&&23&&29&&31&&37&&41&&43&&47&&53&&59&&61&&67&\cr height3pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit& &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr \noalign{\hrule} height2pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit& &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &$a_{p}^{}$&&0&&2&&0&&6&&4&&6&&$-8$&&$-4$&&$-2$&&$-8$&&10&&$-2$&&$-8$&&$-4$&&0&&10&&$-8$&&2&\cr height3pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit& &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr }\hrule}}} \bigskip\noindent For $p\not=3,71$, the characteristic polynomial of a Frobenius automorphism $\varphi_p^{}$ acting on~$E[3]$ is given by $X^2-a^{}_pX+p$. As in~[\SEINV,~sect.5.5] one deduces from the table that $\overline{\rho}$ is surjective and hence irreducible. Alternatively, the $X$-coordinates of the 3-torsion points are the zeroes of the polynomial $3X^4-3X^3-6X^2-36X+8$. It is not difficult to show directly that the Galois group of this polynomial is isomorphic to~$S_4$, and deduce that the representation $\overline{\rho}$ is surjective. We briefly discuss the behavior of the critical primes 2, 3 and~71 with respect to the representation~$\overline{\rho}$. The discriminant of the curve 142A is equal to~$-2^671$. \smallskip \item{--} Since $E$ has good reduction modulo~3, the representation $\overline{\rho}$ is ``flat at~3", i.e. the Zariski closure of the 3-torsion points in the N\'eron model of~$E$ over~$\ZZ_3$ is a finite flat group scheme. Since $a_3=0$, the elliptic curve $E$ is supersingular modulo~3 and the representation $\overline{\rho}$ is non-ordinary. \smallskip \item{--} The curve $E$ has multiplicative reduction modulo~71. Since the 71-adic valuation of the discriminant is not divisible by~3, the theory of the Tate curve implies that the representation $\overline{\rho}$ is ramified at 71. Moreover, $\overline{\rho}$ is of type (A) at~71. This means that the image $\overline{\rho}(I_{71})$ of the inertia group~$I_{71}$ at any prime over~71 is contained in a subgroup conjugate to~$\pmatrix{1&*\cr0&1\cr}$. \smallskip \item{--} The curve $E$ has multiplicative reduction at~2 as well, but since the 2-adic valuation of $\Delta$ is divisible by~3, the representation $\overline{\rho}$ is unramified and hence flat at~2. Alternatively, one can show directly that the number field generated by a zero of the quartic polynomial above is only ramified at~3 and~71. Because of this, the Galois representation $\rho^{}_E$ associated to the 3-adic Tate module of~$E$ is not a minimal deformation of $\overline{\rho}$ in the sense of Wiles. However, it follows from the theory of the Tate curve that $\rho^{}_E$ is a deformation of type~(A) at~2. See~[\RI] for similar octahedral calculations. \beginsection 2. The Hecke ring of level~71. In this section we consider {\it minimal} deformations $\rho:{\rm Gal}(\overline{\QQ}/\QQ)\longrightarrow{\rm GL}_2(R)$ of the representation $\overline{\rho}:{\rm Gal}(\overline{\QQ}/\QQ)\longrightarrow{\rm GL}_2(\FF_3)$ associated to the 3-torsion points of the elliptic curve 142A mentioned in the introduction. Here $R$ is a local Noetherian $\ZZ_3$-algebra with residue field~$\FF_3$ and the diagram $$ \def\normalbaselines{\baselineskip20pt\lineskip3pt \lineskiplimit3pt} \matrix{&&{\rm GL}_2(R)\cr&\!\!\!\!\!\!\!\!\!\!\hbox{$\scriptstyle \rho$}\nearrow&\Big\downarrow\cr {\rm Gal}(\overline{\QQ}/\QQ)& \mathop{\longrightarrow}\limits^{\overline{\rho}}&{\rm GL}_2(\FF_3)\cr} $$ is commutative. We consider deformations $\rho$ of the following type: \smallskip \item{$\bullet$} $\rho$ is unramified outside $\{3,71\}$; \item{$\bullet$} $\rho$ is flat at~3; \item{$\bullet$} $\rho$ is of type (A) at 71; \item{$\bullet$} the determinant of $\rho$ is the cyclotomic character. \smallskip\noindent The first condition ensures that $\rho$ is minimal. See~[\MA, section~29]. The representation $\overline{\rho}$ is irreducible. Therefore, by [\MA, sections 26, 29 and~31] there exists a universal deformation of this type: $$ \rho_{\rm univ}:{\rm Gal}(\overline{\QQ}/\QQ)\longrightarrow{\rm GL}_2(R_{\rm univ}). $$ Wiles shows~[\DS,~Thm 8] that the universal deformation ring $R_{\rm univ}$ is isomorphic to a certain local 3-adic ring~$\TT$ of Hecke operators~$T_n$. By~[\DR,~section 3], the ring $\TT$ is the $\ZZ_3$-subalgebra $$ \TT\quad\subset\quad\tilde{\TT}\,=\,\prod_{f}O_{f} $$ generated by the vectors $T_p=(a_p(f))_f$ with $p\not=3$ prime. Here $f$ runs over the normalized 3-adic eigenforms of level~71 whose Fourier coefficients $a_p(f)$, $p\not=3$ are congruent to the coefficients $a^{}_p$ associated to the curve~142A. The rings $O_f$ denote the rings of integers of the extension of $\QQ_3$ generated by the Fourier coefficients $a_p(f)$ of~$f$. It follows from~[\DR,~Lemma~3.2] that the Hecke operator~$T_3$ is in~$\TT$ as well. This means that the ring $\TT$ is generated by {\it all} $T_p$. By Theorem~1 of the appendix, $\TT$ is then generated as a $\ZZ_3$-module by the operators $T_n$ with $n\le 2\cdot 72/12=12$. The space $S_2(\Gamma_0(71))$ of weight~2 cusp forms of level~71 has dimension~6 and a basis can be found in~[\HD] or can be computed with W.A.~Stein's package~[\WS]. The Hecke action decomposes $S_2(\Gamma_0(71))$ over $\QQ$ into a product of two subspaces of dimension~3 corresponding to a pair of newforms $f^{}_{71}$ and $f'_{71}$. The Fourier coefficients $a_n(f^{}_{71})$ of~$f^{}_{71}$ or $a_n(f'_{71})$ of~$f'_{71}$ generate the same totally real cubic field of discriminant~257, generated by $u$ where $u^3-5\,u+3=0$. For $n\le12$ they are listed in Table~2.1. \bigskip \vbox{\noindent {\bf Table 2.1} {\sl Hecke operators $T_n$.} \medskip \centerline {\vbox {\offinterlineskip \hrule\halign{&\vrule#&\strut\quad\hfil#\quad\cr height3pt &\omit&&\omit&&\omit&&\omit&\cr &$n$&&$a_n(f^{}_{71})$&&$a_n(f'_{71})$&&mod 81&\cr height3pt &\omit&&\omit&&\omit&&\omit&\cr \noalign{\hrule} height2pt &\omit&&\omit&&\omit&&\omit&\cr &1&&1&&1&&(1,1)&\cr &2&&$u$&&$3-u-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&(60,69)&\cr &3&&$3-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$& &$-3+u+u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&(48,12)&\cr &4&&$-2+u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$1+u$&&$(34,61)$&\cr &5&&$-1-u$&&$5-2u-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$(20,11)$&\cr &6&&$3-2u$&&$-3-u$&&$(45,18)$&\cr &7&&$-6+2u+2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$& &$-6+2u+2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&(24,24)&\cr &8&&$-3+u$&&$-u$&&$(57,21)$&\cr &9&&$6-3u-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$& &$u$&&(33,60)&\cr &10&&$-u-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$6+u- u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$(66,30)$&\cr &11&&$6-2u-2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$2u$&&(57,39)&\cr &12&&$-6+3u$&&$-6+3u+2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$ (12,3)$&\cr height3pt &\omit&&\omit&&\omit&&\omit&\cr }\hrule}}} \bigskip \noindent The polynomial $X^3-5X+3\in\ZZ_3[X]$ factors as a product of a linear and an irreducible quadratic factor. Mapping $u$ to the unique root $u_0$ of $X^3-5X+3$ in $\ZZ_3$, we obtain a 3-adic eigenform which we also denote by $f_{71}^{}$ and whose Fourier coefficients $a_p(f_{71}^{})$ are congruent to the coefficients $a_{p}^{}$ associated to the curve~142A. See Table~1.1. On the other hand, mapping $u$ to a root of the irreducible quadratic divisor of $X^3-5X+3$, gives rise to a 3-adic eigenform whose Fourier coefficients are {\it not} congruent to the coefficients $a_{p}^{}$. The same happens with the other eigenform~$f'_{71}$. Using the approximation $u_0\equiv 60\zmod{3^4}$, we list the vectors $T_n=(a_n(f_{71}),(a_n(f'_{71})\rlap)$ in the rightmost column of Table~2.1 with a precision of~$O(3^4)$. The Hecke ring $\TT$ is the $\ZZ_3$-submodule of $\tilde{\TT}=\ZZ_3\times\ZZ_3$ generated by the vectors $T_n$ for $n\le12$. The following Lemma enables us to do the 3-adic computations with finite precision. \proclaim Lemma 2.2. Let $R$ be a local ring with maximal ideal~$\mm$ and let $M$ be a finitely generated $R$-module. Let $M_1$ and $M_2$ be two submodules of~$M$ and suppose there is an integer $k>0$ for which \smallskip \item{--} $M_1+\mm^kM = M_2+\mm^kM$, \item{--} $\mm^{k-1}M\subset M_2$; \smallskip\noindent then $M_1=M_2$. \smallskip\noindent{\bf Proof.} Since $\mm^{k-1}M\subset M_2\subset M_1+\mm^kM$, Nakayama's Lemma implies that $\mm^{k-1}M\subset M_1$. This implies that $\mm^kM$ is contained in $M_1$ as well as $M_2$ so that $M_1=M_2$ as required. \proclaim Theorem 2.3. There is an isomorphism between complete intersections $$ \TT\cong\ZZ_3[[X]]/(X^2-9X). $$ \smallskip\noindent{\bf Proof.} It is easy to see that the row vectors in the rightmost column of Table~2.1 generate the submodule $T'$ of $\tilde\TT=\ZZ_3\times\ZZ_3$ with basis $(1,1)$ and $(0,9)$. Therefore $3^2\tilde\TT\subset T'$. Now we apply Lemma~2.2 to the $\ZZ_3$-module $M=\tilde\TT$ and to the submodules $M_1=\TT$ and $M_2=T'$. Let $k=3$. Then $3^{k-1}\tilde\TT\subset T'$. Since the computations were done with an accuracy of $O(3^4)$ we have that $M_1=M_2$ modulo~${3^k}\tilde{\TT}$. It follows that $\TT$ is the $\ZZ_3$-module generated by $1=(1,1)$ and $x=(0,9)$. As a $\ZZ_3$-algebra $\TT$ is therefore generated by~$x$. Since $x^2=9x$, the homomorphism $\ZZ_3[X]/(X^2-9X)\longrightarrow\TT$ given by $X\mapsto x$ is an isomorphism. Since $x^n\rightarrow0$ in $\TT$ as $n\rightarrow\infty$, this isomorphism extends to the ring $\ZZ_3[[X]]/(X^2-9X)$. This proves the Theorem. \medskip The two $\ZZ_3$-valued points $\pi,\pi':\ZZ_3[[X]]/(X^2-9X)\longrightarrow\ZZ_3$ given by $\pi(X)=0$ and $\pi'(X)=9$, are precisely the algebra-homomorphisms given by, say, $T_n\mapsto a_n(f^{}_{71})$ and $T_n\mapsto a_n(f'_{71})$ respectively. The $\eta$-invariant with respect to the first point is the $\ZZ_3$-ideal $\eta=\pi({\rm Ann}_{\bf T}(I))$ where $I={\rm ker}(\pi)=(X)$. Since $\TT$ is a complete intersection, $\eta$ is also the ideal for which $\#\ZZ_3/\eta=\#(X)/(X^2)$. Using either description one easily checks that $\eta$ is the $\ZZ_3$-ideal generated by $3^2$. See~[\SSR,~Crit.I]. \beginsection 3. The Hecke ring of level~284. In this section we consider deformations $\rho:{\rm Gal}(\overline{\QQ}/\QQ)\longrightarrow{\rm GL}_2(R)$ of the representation $\overline{\rho}$ that are not necessarily minimal at the prime~2. They are supposed to satisfy the following more relaxed conditions: \smallskip \item{$\bullet$} $\rho$ is unramified outside $\{2,\,3,\,71\}$; \item{$\bullet$} $\rho$ is flat at~3; \item{$\bullet$} $\rho$ is of type (A) at 2 and 71; \item{$\bullet$} the determinant of $\rho$ is the cyclotomic character. \smallskip\noindent It follows from the theory of the Tate curve that the deformation given by the 3-adic Tate module of the curve 142A is of this type. According to Wiles, the universal deformation ring is isomorphic to a certain Hecke ring~$\TT$ of level $284=4\cdot71$ rather than $142=2\cdot 71$. The reason for this is the fact that 2 divides the order of ${\rm GL}_2({\FF_3})$ and that therefore, deformations $\rho$ of type $(A)$ at 2 could have Artin conductor divisible by~4. By~[\DR,~section 3], the Hecke operators $T_p$ in the algebra $\TT$ of level~284 can be represented as row vectors $(a^{}_p(f))_f$ with $f$ running through the normalized 3-adic newforms of level dividing~284 whose Fourier coefficients $a_p(f)$ are, for prime $p\not=2,3$, congruent to the coefficients $a^{}_p$ associated to the elliptic curve 142A. The levels of these forms are equal to~71, 142 or~284. The forms of level~71 were already described in the previous section. The new part of the space of weight~2 cusp forms $S_2(\Gamma_0(142))$ is a product of 1-dimensional eigenspaces for the action of the Hecke algebra. The Fourier coefficients of the eigenforms can be found in the Antwerp Tables~[\AT]. The Fourier coefficients of three of these, viz. 142A, 142F and 142G, are congruent to the coefficients $a^{}_p$ of the curve 142A. Finally, there is up to conjugacy only one newform of level~284 whose Fourier coefficients $a_p(f_{284})$ are congruent to the coefficients $a^{}_p$ of the curve 142A. Its coefficients are contained in the cubic field $\QQ_3(t)$ where $t^3+3\,t^2-3=0$. This totally ramified extension of~$\QQ_3$ has discriminant~$81$. Its ring of integers is~$\ZZ_3[t]$. In this way the Hecke ring $\TT$ becomes the $\ZZ_3$-subalgebra of~$\tilde{\TT}$ $$ \TT\quad\subset\quad\tilde\TT=\ZZ_3\times\ZZ_3\times\ZZ_3\times\ZZ_3\times\ZZ_3\times\ZZ_3[t] $$ generated by the Hecke operators $T_p$ with $p\not=2,3$ prime. Here $T_p$ is identified with the row vector consisting of the $p$-th Fourier coefficients of the eigenforms $f^{}_{71}$, $f_{71}'$, $142A$, $142F$, $142G$ and $f_{284}$ as displayed in the $p$-th row of Table~3.1. The $q$-expansion of the newform of level~284 was computed using modular symbols as in~[\WS]. It follows from [\DR,~Lemma~3.2] that $\TT$ is generated as a $\ZZ_3$-algebra by the $T_p$ with $p\not=2$ prime. By Theorem~1 of the appendix, $\TT$ is then generated as a $\ZZ_3$-module by the operators $T_n$ with $n$ odd and $n\le 2\cdot6\cdot72/12=72$. The additive group of $\tilde\TT$ is a free $\ZZ_3$-module of rank~8. Choosing the $\ZZ_3$-basis $\{1,t,t^2\}$ for $\ZZ_3[t]$, we view the Hecke operators $T_p$ as vectors in $\ZZ_3^8$. For example, the Hecke operator $T_{25}$ in Table~3.1 becomes the vector $$ (-4+2u+u^2,\,8-3u-u^2\,,-1\,,11\,,-1\,,5\,,9\,,2). $$ We do the computations modulo~$3^7$. As in section~2, let $u$ denote the unique root in $\ZZ_3$ of the polynomial $X^3-5X+3$. We have that $$ u=2\cd3 + 2\cd3^3 + 3^4 + 2\cd3^5 + 2\cd3^6 + O(3^{7}). $$ \bigskip \vbox{\noindent {\bf Table 3.1} {\sl Hecke operators $T_n$.} \medskip \centerline {\vbox {\offinterlineskip \hrule\halign{&\vrule#&\strut\quod\hfil#\quod\cr height3pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &$n$&&$a_n(f^{}_{71})$&&$a_n(f'_{71})$&&142A&&142F&&142G&&$a^{}_n(f_{284})$&\cr height3pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr \noalign{\hrule} height2pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &1&&1&&1&&1&&1&&1&&1&\cr &3&&$3-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$-3+u+u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$0$&&$-3$&&$3$&&$t$&\cr &5&&$-1-u$&&$5-2u-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$2$&&$-4$&&$2$&&$-1-3t-t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &7&&$-6+2u+2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$-6+2u+2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$0$&&$-3$&&$-3$&&$-6+2t+2t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &9&&$6-3u-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$u$&&$-3$&&6&&6&&$-3+t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &11&&$6-2u-2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$2u$&&$6$&&$0$&&$-6$&&$2t$&\cr &13&&$4$&&$-2-2u$&&$4$&&$1$&&$-5$&&$4-6t-4t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &15&&$-6+2u+u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$-6-u+u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&0&&12&&6&&$-3-t$&\cr\ &17&&$-6+2u+2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$6-2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$6$&&$0$&&$6$&&$-6+6t+4t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &19&&$7-u-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$7-u-2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$-8$&&$-5$&&$1$&&$-2-t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &21&&$-12+2u+2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$6+2u$&&0&&9&&$-9$&&$3-6t-t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &23&&$-4+2u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$-4$&&$-4$&&$-7$&&$5$&&$8-2t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &25&&$-4+2u+u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$8-3u-u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&$-1$&&$11$&&$-1$&&$5+9t+2t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr height2pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &\vdots\ &&\vdots\quad&&\vdots\quad&&\vdots\ &&\vdots\ &&\vdots\ &&\vdots\quad&\cr height2pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr &67&&$-4+4u$&&$-4-2u$&&$2$&&$-4$&&$2$&&$-4 + 2 t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &69&&$-12+6u$&&$12-4u-4u^{\rlap{\hbox{$ \scriptstyle 2$}}}$&&0&&21&&15&&$-6+8t+6 t^{\rlap{\hbox{$ \scriptstyle 2$}}}$&\cr &71&&$1$&&$1$&&$1$&&$1$&&$1$&&$1$&\cr height3pt &\omit&&\omit&&\omit&&\omit&&\omit&&\omit&&\omit&\cr }\hrule}}} \bigskip \noindent Applying Hermite reduction to the row vectors, we find the matrix below. We let $T'$ denote the $\ZZ_3$-submodule of $\tilde\TT$ spanned by its rows. Incidentally, we find the same matrix using only the rows $n=1$, $3,\ldots,17$. This means that the module~$T'$ is already generated by the $T_n$ with $n$ odd and $n\le 17$. It is easy to see that $3^5\tilde\TT\subset T'$. Now we apply Lemma~2.2 with $M=\tilde\TT$, $M_1=\TT$ and $M_2=T'$. We take $k=6$. Then $3^{k-1}\tilde\TT\subset T'$ and, since we did the computations with an accuracy of $O(3^7)$, we have $M_1+3^k\tilde{\TT}=M_2+3^k\tilde{\TT}$. It follows that $\TT=T'$. Thus, as a $\ZZ_3$-module, $\TT$ is generated by the rows of the matrix $$ \pmatrix{1&1&1&1&1&1&0&0\cr0&9&0&6&18&0&0&1\cr 0&0&3&0&54&0&-1&-1\cr 0&0&0&9&0&0&0&1\cr 0&0&0&0&81&0&0&1\cr 0&0&0&0&0&3&0&0\cr 0&0&0&0&0&0&3&0\cr 0&0&0&0&0&0&0&3\cr}. $$ For later use we observe that the determinant of the matrix and hence the index $[\tilde\TT:\TT]$ are equal to~$3^{12}$. The first row of the matrix is the unit element $1\in\TT$. We let $x=(0,\ 9,\ 0,\ 6,\ 18\ ,t^2)$ be the element corresponding to the second row and $y=(0,\ 0,\ 3,\ 54,\ -t-t^2)$ be the one corresponding to the third row. The morphism $\varphi:\ZZ_3[X,Y]\longrightarrow\TT$ given by $f(X,Y)\mapsto f(x,y)$ extends to a $\ZZ_3$-algebra morphism $$ \widehat\varphi:\ZZ_3[[X,Y]]\longrightarrow \TT. $$ This follows from the fact that $x^n,y^n\rightarrow 0$ in $\tilde\TT$ as $n\rightarrow\infty$. \proclaim Theorem 3.2. The ring homomorphism $\widehat\varphi$ induces an isomorphism of $\ZZ_3$-algebras $$ \ZZ_3[[X,Y]]/(F_1,F_2)\mathop{\longrightarrow}\limits^{\cong}\TT, $$ where $$ \eqalign{F_1&=29412Y -9804Y^2 -91158XY -1641XY^2+ 11618X^2Y-787(X^3-15X^2+54X),\cr F_2&=8514Y-477Y^2-8204XY+1741XY^2+ 2369X^2Y-787Y^3.\cr} $$ \smallskip\noindent{\bf Proof.\ } We compute the first few monomials in $x$ and~$y$ in the subring $\TT$ of $\tilde{\TT}$: $$\vbox{\baselineskip=15pt\halign{$\hfil#$ &&\ \hfil#\hfil\cr {\bf 1}=(\,1,&1,&1,&1,&1,&1&),\cr x=(\,0,&9,&0,&6,&18,&$t^2$&),\cr y=(\,0,&0,&3,&0,&54,&$-t-t^2$&),\cr x^2=(\,0,&81,&0,&36,&324,&$-9+3t+9t^2$&),\cr xy=(\,0,&0,&0,&0,&972,&$6-3t-6t^2$&),\cr y^2=(\,0,&0,&9,&0,&2916,&$-3+3t+4t^2$&),\cr x^2y=(\,0,&0,&0,&0,&17496,&$45-18t-39t^2$&),\cr xy^2=(\,0,&0,&0,&0,&52488,&$-27+12t+24t^2$&).\cr}} $$ Using the $\ZZ_3$-basis $\{1,t,t^2\}$ for the ring $\ZZ_3[t]$, we write these eight vectors as row vectors in $\ZZ_3^8$. The determinant of the resulting $8\times 8$ matrix is equal to $-2^23^{12}787$. This implies that the Hecke ring $\TT$ is equal to the $\ZZ_3$-span of the eight monomials above. It follows that $\varphi$ and hence $\widehat\varphi$ are surjective. In order to determine the kernels of~$\varphi$ and~$\widehat\varphi$, we express $x^3$, $y^3$ and $x^2y^2$ as $\ZZ_3$-linear combinations of the eight monomials above. Solving the corresponding linear systems of eight equations in eight unknowns one finds that $$ \eqalign{787x^3&=29412y -9804y^2 -91158xy -1641xy^2+ 11618x^2y-787(54x-15x^2),\cr 787y^3&=8514y-477y^2-8204xy+1741xy^2+ 2369x^2y,\cr 787x^2y^2&=15444y-5148y^2 -15870xy+12657xy^2+ 6219x^2y.\cr} $$ In other words, $F_1(x,y)=F_2(x,y)=F_3(x,y)=0$ where $F_1$ and~$F_2$ are as above and $$ F_3=15444Y-5148Y^2 -15870XY+12657XY^2+ 6219X^2Y-787X^2Y^2. $$ Therefore $J=(F_1,F_2,F_3)\subset{\rm ker}(\varphi)$. Since $\ZZ_3[X,Y]/J$ has $\ZZ_3$-rank at most~8, it follows that ${\ker}(\varphi)=J$. Writing $J'$ for the $\ZZ_3[[X,Y]]$-ideal $(F_1,F_2,F_3)$, there is a commutative diagram $$ \def\normalbaselines{\baselineskip20pt\lineskip3pt \lineskiplimit3pt} \matrix{\ZZ_3[X,Y]/J&\mathop{\longrightarrow}\limits^{\cong}&\TT.\cr \Big\downarrow&\nearrow\cr \ZZ_3[[X,Y]]/J'\cr} $$ Here the diagonal arrow is the morphism induced by~$\widehat\varphi$. It is surjective. Reducing everything modulo~3, we obtain the diagram $$ \def\normalbaselines{\baselineskip20pt\lineskip3pt \lineskiplimit3pt} \matrix{\FF_3[X,Y]/\overline{J}&\mathop{\longrightarrow}\limits^{\cong}&\TT/3\TT,\cr \Big\downarrow&\nearrow\hbox{$\scriptstyle \overline\varphi$}\cr \FF_3[[X,Y]]/\overline{J}'\cr} $$ where $\overline{J}$ and $\overline{J}'$ denote the ideals generated by the reduced polynomials $\overline{F}_1$, $\overline{F}_2$ and~$\overline{F}_3$ in the rings $\FF_3[X,Y]$ and $\FF_3[[X, Y]]$ respectively and $\overline\varphi:\FF_3[[X,Y]]/\overline{J}'\longrightarrow\TT/3\TT$ denotes the surjective morphism induced by $\widehat\varphi$. Since $\FF_3[X,Y]/\overline{J}$ is an Artin ring, the natural map to the product of the completions at its maximal ideals is an isomorphism. Since the vertical map is the natural map from $\FF_3[X,Y]/\overline{J}$ to its completion at the maximal ideal $(X,Y)$, it is surjective. We conclude that $\overline\varphi$ is an isomorphism $$ \FF_3[[X,Y]]/\overline{J}'\ \ \mathop{\longrightarrow}\limits^{\cong}\ \ \TT/3\TT. $$ From $$\eqalign{\overline{F}_1&= -X^3-X^2Y,\cr \overline{F}_2&= -Y^3+X\, Y+X\,Y^2-X^2Y,\cr \overline{F}_3&= -X^2Y^2,} $$ we obtain the relation $$ -Y(X-1)\overline{F}_1+X^2\overline{F}_2=(1+X+Y)\overline{F}_3 $$ in $\FF_3[[X,Y]]$. Since $1+X+Y$ is a unit in $\FF_3[[X,Y]]$, the monomial $\overline{F}_3$ belongs to the ideal $(\overline{F}_1,\overline{F}_2)$ so that $\overline{J}'=(\overline{F}_1,\overline{F}_2)$. We claim that the ideal $\widehat{J}={\rm ker}(\widehat{\varphi})\subset\ZZ_3[[X,Y]]$ is generated by~$F_1$ and~$F_2$. Indeed, since the Hecke algebra $\TT$ is free over $\ZZ_3$, the exact sequence $$ 0\longrightarrow \widehat{J}\longrightarrow\ZZ_3[[X,Y]]\longrightarrow\TT\longrightarrow 0. $$ remains exact after tensoring with ${\FF}_3$. It follows that $\widehat{J}/3\widehat{J}=\overline{J}'=(\overline{F}_1,\overline{F}_2)$. Therefore $$ \widehat{J}\subset (F_1,F_2)+ 3\,\widehat{J} \subset (F_1,F_2) + \mm\, \widehat{J} $$ where $\mm=(3,X,Y)$ denotes the maximal ideal of the local ring $\ZZ_3[[X,Y]]$. Nakayama's Lemma implies then that $\widehat{J}=(F_1,F_2)$. This proves the Theorem. \medskip The ``points" of the Hecke algebra $\TT$ are $(X,Y)=(0,0)$, $(9,0)$, $(0,3)$, $(6,0)$, $(18,54)$ and $(t^2,-t-t^2)$ and its conjugates. The first two correspond to the newforms of level~71, the next three to eigenforms of level~142 and the three conjugates of $(t^2,-t-t^2)$ to an eigenform of level~284. The point $(0,3)$ corresponds to the elliptic curve 142A of the introduction. The point $(0,0)$ is our origin. The canonical morphism from $\TT$ to the minimal universal deformation ring of section~2 is given by $X\mapsto X$ and $Y\mapsto 0$. Since $\TT$ is a complete intersection, the $\eta$-invariant associated to it is equal to the $\ZZ_3$-ideal generated by $\#I/I^2$ where $I$ is the ideal~$(X,Y)$. The order of $I/I^2$ is given by the determinant of the matrix of the coefficients of the linear terms of $F_1$ and~$F_2$. Since $$\eqalign{F_1&=-787\cd 54X+29412Y\ \ \hbox{$+$ terms of deg $\ge 2$},\cr F_2&=\qquad\qquad\qquad\ 8514Y\ \ \hbox{$+$ terms of deg $\ge 2$},\cr } $$ it is the $\ZZ_3$-ideal generated by $$ \eta={\rm det}\pmatrix{-787\cd 54&29412\cr 0&8514\cr}=3^5\cdot(\hbox{unit}). $$ This is $3^3$ times the $\eta$-invariant of the minimal deformation ring computed in section~2. This agrees with [\DR, Lemma~4.4] because the contribution of the Euler factors at~2 is equal to $(1-2)((1+2)^2-a_2(f_{71}^{})^2) = -3^2+u^2$ and has 3-adic valuation equal to~3. \beginsection 4. The Hecke ring of level~142. In this section we study the 3-adic Hecke ring $\TT$ of level~142. In contrast to the Hecke rings of levels 71 and~284, we show that it is {\it not} a Gorenstein ring. The $\ZZ_3$-algebra $\TT$ is generated by the vectors $T_p=(a_p(f))_f$ for $p\not=2$. Here $f$ runs over the 3-adic normalized eigenforms of level~142 whose Fourier coefficients are congruent to the coefficients $a_p^{}$ of the curve~142A. The algebra is obtained by simply omitting the last component of the row vectors that generate the Hecke ring of level~284. Therefore the results of the previous section imply that modulo~$3^7\tilde{\TT}$, the ring $\TT$ is equal to the $\ZZ_3$-submodule $T'$ of $\tilde\TT=\ZZ_3\times\ZZ_3\times\ZZ_3\times\ZZ_3\times\ZZ_3$ generated by the rows of the matrix $$ \pmatrix{1&1&1&1&1\cr0&9&0&6&18\cr 0&0&3&0&54\cr 0&0&0&9&0\cr 0&0&0&0&81\cr }. $$ We see that $3^4\tilde\TT\subset T'$ so by Lemma~2.2 we have that $\TT=T'$. The determinant of the matrix and hence the index $[\tilde\TT:\TT]$ are equal to~$3^{9}$. We let $x\in\TT$ be the second and $y\in\TT$ be the third row of the matrix. The morphism $\varphi:\ZZ_3[X,Y]\longrightarrow\TT$ given by $f(X,Y)\mapsto f(x,y)$ extends to a $\ZZ_3$-algebra morphism $\widehat\varphi:\ZZ_3[[X,Y]]\longrightarrow \TT$. \proclaim Theorem~4.1. The morphism $\widehat\varphi$ induces an isomorphism $$ \ZZ_3[[X,Y]]/(F_1,F_2,F_3)\cong\TT, $$ where $$\eqalign{F_1&=17XY-6Y^2+18Y,\cr F_2&=17(X^3-15X^2+54X)-12Y^2+36Y,\cr F_3&=Y^3-57Y^2+162Y.\cr} $$ \smallskip\noindent{\bf Proof.}\ We have that $$\vbox{\baselineskip=15pt\halign{$\hfil#$ &&\ \hfil#\hfil\cr {\bf 1}=(\,1,&1,&1,&1,&1&),\cr x=(\,0,&9,&0,&6,&18&),\cr y=(\,0,&0,&3,&0,&54&),\cr x^2=(\,0,&81,&0,&36,&324&),\cr y^2=(\,0,&0,&9,&0,&2916&).\cr}} $$ Since the determinant of the matrix whose rows are the above five row vectors is equal to~$3^9$ times a unit, the Hecke ring~$\TT$ is generated by $1,x,x^2,y,y^2$ as a module over~$\ZZ_3$. Therefore the morphism $\psi:\ZZ_3[X,Y]\longrightarrow\TT$ given by $\psi(X)=x$ and $\psi(Y)=y$, is surjective. We have the following relations in~$\TT$: $$\eqalign{ 17xy&=6y^2-18y,\cr 17(x^3-15x^2+54x)&=12y^2-36y,\cr y^3&=57y^2-162y.\cr} $$ Using the same arguments as in proof of Theorem~3.2, the result follows. \medskip It follows that $\TT/3\TT\cong\FF_3[X,Y]/(X^3,XY,Y^3)$. This is a finite local $\FF_3$-algebra with maximal ideal ${\goth m}=(X,Y)$. The annihilator of $\mm$ is $(X^2,Y^2)$. Since the $\FF_3$-dimension of this ideal is~2 rather than~1, the ring $\TT$ is not Gorenstein by~[\TI, Prop.1.4]. For the sake of completeness we compute the two invariants. Let $\pi:\TT\longrightarrow\ZZ_3$ be the point given by $\pi(X)=\pi(Y)=0$ and let $I=(X,Y)={\rm ker}(\pi)$. It is easy to see that the group $I/I^2$ is isomorphic to $(\ZZ/9\ZZ)\times(\ZZ/27\ZZ)$. On the other hand, a somewhat more cumbersome linear algebra computation shows that ${\rm Ann}_{\TT}(I)$ is zero so that~$\eta=0$. \bigskip\bigskip\bigskip\noindent {\bibliography \item{[CSS]} G.~Cornell, J.H.~Silverman and G.~Stevens, Eds., {\sl Modular forms and Fermat's Last Theorem}, Springer-Verlag, New York 1997. \item{[\AT]} Birch, B.J. and Kuyk, W., {\sl Modular Functions of One Variable IV}, Lecture Notes in Mathematics {\bf 476}, Springer-Verlag, 1975. \item{[\CR]} Cremona, J., {\sl Algorithms for modular elliptic curves}, Cambridge University Press, Cambridge~1992. \item{[\HD]} Darmon, H., {\sl Serre's conjectures}, in Seminar on Fermat's Last Theorem 1993-1994, Kumar Murty ed. Canadian Mathematical Society, CMS Conference Proceedings {\bf 17}, 1995, pages 135--153. \item{[\DS]} De Shalit, E., {\sl Hecke rings and universal deformation rings}, Chapter~XIV in~[CSS]. \item{[\SSR]} De Smit, B., Rubin, K., Schoof, R. , {\sl Criteria for Complete Intersections}, Chapter XI in~[CSS]. \item{[\DI]} Diamond, F., {\sl The refined conjecture of Serre}, in Elliptic Curves, Modular Forms and Fermat's Last Theorem, J. Coates, S.~Yau, eds. International Press, Cambridge, 1995, pages 22--37. \item{[\DR]} Diamond, F., Ribet, K., {\sl $\ell$-adic Modular Deformations and Wiles's ``Main Conjecture"}, Chapter XII in~[CSS]. \item{[\ED]} Edixhoven, S.J., {\sl Serre's conjecture}, Chapter VII in~[CSS]. \item{[\GE]} Gelbart, S., {\sl Three lectures on the modularity of $\overline{\rho}^{}_{E,3}$ and the Langlands reciprocity conjecture}, Chapter VI in~[CSS]. \item{[\MA]} Mazur, B., {\sl Deformation theory of Galois representations}, in Galois groups over $\QQ$, in~[CSS]. \item{[\RI]} Rio, A., {\sl Representacions de Galois octa\`edriques}, Tesi Doctoral, Universitat de \hyphenation{Bar-ce-lo-na} Barcelona (1995). \item{[\SE]} Serre, J.-P., {\sl Sur les repr\'esentations de degr\'e 2 de ${\rm Gal}(\overline{\QQ}/\QQ)$}, Duke Math. Journal {\bf 54} (1987), 179--230. \item{[\SEINV]} Serre, J.-P., {\sl Propri\'et\'es galoisiennes des points d'ordre fini des courbes elliptiques}, Invent. Math. {\bf 15} (1972), 259--331. \item{[\WS]} Stein, W.A., HECKE package, Magma V2.7 or higher, 2000. \item{[\TW]} Taylor, R. and Wiles, A., {\sl Ring-theoretic properties of certain Hecke algebras}, Ann. Math. {\bf 141} (1995) 553--572. \item{[\TI]} Tilouine, J., {\sl Hecke algebras and the Gorenstein property}, Chapter X in~[CSS]. \item{[\WI]} Wiles, A., {\sl Modular elliptic curves and Fermat's Last Theorem}, Annals of Math. {\bf 141} (1995), 443--551. \item{} } \bigskip\bigskip \beginsection Appendix by W.~Stein. \proclaim Theorem. The ring ${\bf T}$ of Hecke operators acting on the space of cusp forms of weight~$k$ and level~$N$ is generated as an abelian group by the Hecke operators $T_n$ with $$ n\le {{kN}\over{12}}\prod_{p|N}(1+{1\over p}) $$ \smallskip\noindent{\bf Proof.} \bye \item{[\BF]} Bayer, P., Frey, G., {\sl Galois representations of octahedral type and $2$-coverings of elliptic curves}, Math. Zeitschrift. {\bf 207} (1991), no.3, 395--408. \item{[\DE]} Deligne, P., Serre, J.-P., {\sl Formes modulaires de poids 1}, Ann. Scient. ENS {\bf 7} (1974), 507--530. From makaiok@zdl.net Sun Feb 4 11:30:56 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f14GjkW03309 for ; Sun, 4 Feb 2001 11:45:46 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f14Gj4m05898 for ; Sun, 4 Feb 2001 11:45:05 -0500 (EST) Received: from mail.bb.ah163.net ([202.102.200.76]) by math.berkeley.edu (8.9.3/8.9.3) with SMTP id IAA24771 for ; Sun, 4 Feb 2001 08:45:03 -0800 (PST) Received: from exe([61.133.146.211]) by aimc.com(JetMail 2.5.3.0) with SMTP id jm2d3a7da90d; Sun, 4 Feb 2001 16:45:02 -0000 From: "虚拟主机最优价" Reply-To: makaiok@zdl.net Message-ID: {F7BD4EFD-6208-491D-9790-20A340F834BE}@exe Subject: 虚拟主机最优价！ To: X-Mailer: V3,1,6,1 (W95/NT) (Build: Oct 18 1999) Mime-Version: 1.0 Date: Mon, 05 Feb 2001 00:30:56 +0800 Status: R X-Status: N this message is only introduced to chinese, if you arn't,i'm sorry for you. 亲爱的朋友您好，此信您若不感兴趣，请立即删除它，并在此向您表示由衷地歉意！ 中国民网（http://www.cndem.net) 推出域名注册优惠价！ 240元=国际域名一个+40MB+10个email 390元=(国际域名一个+40MB+10个email)/2年 480元＝（国际域名一个＋150mb+10个email)/年，支持asp、access、cgi、　php4等 880元=(国际域名一个+150MB+20个email)/2年，支持asp、access、cgi　php4等。 1200元=(国际域名一个+200MB+20个email)/2年，支持asp、access、cgi　php4等 　并现已正式推出中文域名注册！ 网络时代，飞速发展，中国民网，让您轻松拥有自己的网站 （政府上网工程协办网站，中国民主建国会蚌埠市委会承办） From whitney@math.berkeley.edu Sun Feb 4 14:33:27 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f14JOpW03354 for ; Sun, 4 Feb 2001 14:24:51 -0500 Received: from shimura.math.berkeley.edu (shimura.Math.Berkeley.EDU [169.229.58.53]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f14JOBm09334 for ; Sun, 4 Feb 2001 14:24:11 -0500 (EST) Received: from localhost (whitney@localhost) by shimura.math.berkeley.edu (8.11.0/8.11.0) with ESMTP id f14JXRf29404 for ; Sun, 4 Feb 2001 11:33:28 -0800 Date: Sun, 4 Feb 2001 11:33:27 -0800 (PST) From: Wayne Whitney Sender: Reply-To: To: "William A. Stein" Subject: mfnet reboot Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N Hi William, Unless you have a strong opposition, I plan to reboot mfnet pretty soon. The proximate cause is that I would like to set up ntpd so that the clocks on all the machines are in agreement, and ntpd likes to have multicast support in the kernel. But as you and I know the real reason is that I haven't rebooted any machines in three weeks! :-) Seriously though, I imagine this will be OK, you'll just have to restart your scripts, eh? If I don't hear back from you, I'll wait, because you might not be around to restart the scripts. I'm also waiting on hearing back from Kevin. When do you return to the States? Cheers, Wyane From marlene@infomagic.com Sun Feb 4 14:42:10 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f14JQ4W03358 for ; Sun, 4 Feb 2001 14:26:04 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f14JPNm09364 for ; Sun, 4 Feb 2001 14:25:24 -0500 (EST) Received: from infomagic.com (MAX6-Port45.Downtown.InfoMagic.NET [63.229.95.145]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f14JPLB24249; Sun, 4 Feb 2001 12:25:21 -0700 Message-ID: <3A7DB091.F76B2E6C@infomagic.com> Date: Sun, 04 Feb 2001 12:42:10 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: "William A. Stein" CC: Stein Dennis , sharon thormahlen Subject: website for FFOTM Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: R X-Status: N Here's the website for our Traditional music club...just put on the web....you may or may not have time to check it out.. http://www.infomagic.net/~lloyd/ffotm.htm From hwl@math.berkeley.edu Sun Feb 4 16:30:46 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f14LVRW03394 for ; Sun, 4 Feb 2001 16:31:27 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f14LUlm12895 for ; Sun, 4 Feb 2001 16:30:47 -0500 (EST) Received: from faema.math.Berkeley.EDU (faema.Math.Berkeley.EDU [169.229.58.25]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id NAA03479 for ; Sun, 4 Feb 2001 13:30:46 -0800 (PST) From: Hendrik Lenstra Received: (from hwl@localhost) by faema.math.Berkeley.EDU (8.9.3/8.9.3) id NAA18883 for was@math.harvard.edu; Sun, 4 Feb 2001 13:30:46 -0800 (PST) Date: Sun, 4 Feb 2001 13:30:46 -0800 (PST) Message-Id: <200102042130.NAA18883@faema.math.Berkeley.EDU> To: was@math.harvard.edu Subject: Re: BP Status: R X-Status: N Dear William, Thanks for your message. Regarding the question >"Is there an abelian variety A such that OddPart(#Sha(A)) is not a >square?" I was just curious to know whether there was any reason for this to be true. To summarize my conversation with Michael Stoll: he said `no', but didn't know an example, and said that if ANYBODY knew an example then it would be you. Now, when he said `no', I would believe he did this on the basis of the type of argument in his paper with Bjorn, and that the visibility that you bring into the discussion played no role in his mind; you seem to bring up more cases in which the answer is `yes' than he may have been aware of. All the best, Hendrik From 0t367@21cn.com Fri Feb 2 13:30:21 2001 Return-Path: <0t367@21cn.com> Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f150IWW03469 for ; Sun, 4 Feb 2001 19:18:33 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f150Hqm16914 for ; Sun, 4 Feb 2001 19:17:52 -0500 (EST) Received: from 21cn.com ([202.104.32.248]) by math.berkeley.edu (8.9.3/8.9.3) with SMTP id QAA09782 for ; Sun, 4 Feb 2001 16:17:51 -0800 (PST) Message-Id: <200102050017.QAA09782@math.berkeley.edu> Received: from game1([202.96.144.131]) by 21cn.com(JetMail 2.5.3.0) with SMTP id jmf3a7df441; Mon, 5 Feb 2001 00:17:17 -0000 From: "0t367" <0t367@21cn.com> Subject: Re:Hi~ To: 0t367@21cn.com X-Mailer: DiffondiCool V3,1,6,0 (W95/NT) (Build: Oct 18 1999) Mime-Version: 1.0 Date: Sat, 03 Feb 2001 02:30:21 +0800 Content-Type: multipart/mixed; boundary="----=_NextPart_000_007F_01BDF6C7.FABAC1B0" Content-Transfer-Encoding: 7bit Status: R X-Status: N This is a MIME Message ------=_NextPart_000_007F_01BDF6C7.FABAC1B0 Content-Type: multipart/alternative; boundary="----=_NextPart_001_0080_01BDF6C7.FABAC1B0" ------=_NextPart_001_0080_01BDF6C7.FABAC1B0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable ***** This is an HTML Message ! ***** ------=_NextPart_001_0080_01BDF6C7.FABAC1B0 Content-Type: text/html; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable ------=_NextPart_001_0080_01BDF6C7.FABAC1B0-- ------=_NextPart_000_007F_01BDF6C7.FABAC1B0-- From verrill@math.ku.dk Sun Feb 4 19:52:39 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f151u4W03497 for ; Sun, 4 Feb 2001 20:56:05 -0500 Received: from localhost.localdomain (IDENT:root@ip65.opnxr1.ras.tele.dk [195.215.236.65]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f151tNm19799 for ; Sun, 4 Feb 2001 20:55:23 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id BAA28047 for ; Mon, 5 Feb 2001 01:52:40 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Mon, 5 Feb 2001 01:52:39 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: snow In-Reply-To: <01020116101103.21564@form> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N Hi William! Hey, how is the weather in Belgium? Here there were about 3 inches of snow by the time I finally looked out the window. I went for a walk and had a look at the frozen lake covered in snow, with a patch of dark water in the middle for the ducks and swans. I took photos of duck foot prints in the snow at the edges of the lake. I suppose someone has been feeding them. I've been spending the weekend implementing "kulkarni's method" for finding generators for congruence subgroups. It's advantage is that it always gets the minimal number. But it's even slower than my previous generators function, so far slower than what you have.. been trying to work out what makes it so good. May thing things like P1Classes are just really good functions to use. I have to decide what dates to go to Australia for and let David Kohel know. Still not sure yet, though I resheduled my Postdoc in Hannover to start 1st of June. Well, I hope all is well whereever you are, is it still Belgium, or somewhere else by now? Helena From weber@exp-math.uni-essen.de Mon Feb 5 04:26:50 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f159RaW03826 for ; Mon, 5 Feb 2001 04:27:36 -0500 Received: from irmgard.exp-math.uni-essen.de (irmgard.exp-math.uni-essen.de [132.252.150.18]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f159Qrm28936 for ; Mon, 5 Feb 2001 04:26:53 -0500 (EST) Received: from werner.exp-math.uni-essen.de (werner.exp-math.uni-essen.de [132.252.150.17]) by irmgard.exp-math.uni-essen.de (8.9.3/8.9.3) with SMTP id KAA19572 for ; Mon, 5 Feb 2001 10:26:51 +0100 Received: from localhost by werner.exp-math.uni-essen.de (AIX 4.1/UCB 5.64/4.03) id AA44890; Mon, 5 Feb 2001 10:26:50 +0100 Date: Mon, 5 Feb 2001 10:26:50 +0100 (MEZ) From: Georg Weber To: William A Stein Subject: a natural self-pairing Message-Id: Mime-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N Hi William! It's Monday, so it's time for interesting talk ... Now that I no longer am being on that horrible trip "search for a generalized intersection pairing", I realize that I had had all the time what was really needed: a nice, natural and easy-to-compute self pairing <.,.> on spaces of modular symbols, which is, well, not really Hecke-invariant, but really as good as being that, more precisely, we have: = Here T_n is the n-th Hecke-operator for some n and T_n^\iota is the image of T_n under the so-called main involution on Z(GL_2(Q)): \iota(a,b;c,d):=(d,-b;-c,a). If the space of modular symbols considered belongs to some \Gamma_1(N) or \Gamma_0(N) with Nebentype, then, for all p prime to N, T_p^\iota is just

^{-1} T_p, where

is the diamond bracket operator corresponding to a matrix with lower right hand entry congruent to p mod N. That is, for modular symbols spaces with some nebentype and p prime to the level, any eigenvector for a T_p just *is* also an eigenvector for the corresponding T_p^\iota, and that's all the "Hecke-invariance" we need in practice. So how does <.,.> look like? Essentially, for two *manin* symbols x = \sum_g (\sum a_i(g) X^{k-2-i}Y^i) {g0, g\infty} and y = \sum_g (\sum b_i(g) X^{k-2-i}Y^i) {g0, g\infty}, it looks like = \sum_g \sum (-1)^{i} a_i(g)*b_{k-2-i}(g) That's easy! But not really true. Well, everything is well- defined over an arbitrary ring, but for characteristics (a multiple of) 2 or 3 we get wrong results (too big dimensions) if we use the wrong definitions for our spaces, so for simplicity let's work over Q and, again for simplicity, even weights k. Due our assumptions, the space M^f of "free" modular symbols (you'll remember) is just a direct product of two spaces: one, V, is generated by the 2- and 3- relations (the kernel modded out of M^f to get the "true" modular symbols) and the other space, W, is the intersection of W_1=im(1-S)=ker(1+S) and W_2:=im(1-R)=ker(1+R+R^2) (with S, R say, (0,1;-1,0) and (-1,1,;-1,0) --- they need to be generators of PSL_2 of order 2 and 3 respectively). (In characteristic 2, e.g. it does not need to be true that ker(1-S) and im(1+S) are equal, and so some computations go awry, using the "wrong" subspaces. All goes well for V if the congruence subgroup considered is torsion free even if the characteristic is 2 or 3, but in these cases V and W have non-trivial intersection, which spoils everything of what follows). The pairing <.,.> now is defined *** on M^f ***, just as above (but only under the assumption of the weight k being even) and is such that (easily verified) < V, W > = 0. So errrm, to be able to really calculate it, we'll have to be able to lift modular symbols to the space of free modular symbols in such a way that (at least) one of the two arguments lies in the subspace W of M^f. Remark that we need to have only *one* argument in W! Now one could have calculated in W right from the beginning. In this case, the pairing is calculated with virtually no effort. This space W is canonically isomorphic to the space of modular symbols, after all, if we're working over Q resp. iff both 2 and 3 are divisible in the base ring considered. Especially, the finite fields GF(p) with p>=5 are fine. Or, we do all our calculations over Z (i.e. Q) and go over to (any!) finite characteristic only afterwards. (Over Z, the image of W in M(Z) is a lattice of full rank but with an index --- whose prime divisors can only be 2 and 3, of course). The isomorphism alluded to is, explicitly, being the restriction to W of modding out the usual kernel V out of the free space generated by the Manin symbols, i.e. M^f; so it is trivially computable in forward direction. I definitely would like to be able to do calculations in such W's using your magma package! Not many alterations would be needed, essentially the only thing to do would be a change in the basic notions of "ModSym" resp. "ambient space" to be able to encompass spaces like M^f and W --- whose elements now are expressible in "free Manin" symbols, but not as "true modular" symbols --- all that Hecke-operator stuff still works, if one uses Merel's coverings of these (sic! they let W invariant and act on it like on the "true" space of modular symbols -- at least if we're away from the level). On second thought, we only need W as "father" of all of the *dual* spaces --- remember that only *one* of the arguments of the pairing needs to be in W itself, and we trivially have "some" lifting of a modular symbol to the space of the "free" modular symbols. So much of the core, the user interface, and the basic categories need no change at all! Internally, we have the choice, in which space to do our calculations in; as before, and then the pairing gives the "other side". "Only" the many algorithms using the dual somewhere were to be changed (urrggs, sorry, this should be a mess however ...). Of course, the DualHeckeOp and so on now should act on W resp.. subspaces of W, I mean, really on their respective internal representations! Of course, from a mathematical point of view, there's a canonical isomorphism between W and the "true" modular symbols. As we need it, the StarInvolution behaves well under <.,.>, so duals for plus- and minus-spaces are not problematic. As a matter of fact, the Boundary operator is defined on all of the "free" modular symbols M^f; the intersection of its kernel and of W is a lifting of the cuspidal subspace of the "true" modular symbols. So, being interested mostly in cusp forms, one is intrigued to *define* the magma dual of the cuspidal subspace of the modular symbols as this *subspace* (call it C(W)) of W - rather than as a quotient space of it. This makes calculations pretty neat, I guess. And (now mathematically correct) the plus and minus duals of the plus and minus cuspidal subspaces would then be subspaces of SW --- all along we always have that nice little pairing between the corresponding spaces, easy to compute. In fact, in down to earth terms, it is just the one written up explicitly above! I don't know what the picture is for Atkin-Lehner; I just didn't think about it. But all in all, a function named just "DualSpace", going from ModSym to DualModSym, and vice versa, seems to be a feasible one (hmm, if only I had it right now available...the backward direction, going from W into M, is easy: just apply S to all the polynomial coefficients of the manin symbols). In the same vein: a function "Complement" between ModSym and DualModSym. The EisensteinSubspace e.g. then would just be the "Complement" of the "DualSpace" C(W) of the CuspidalSubspace, and also the "DualSpace" of "Complement" E(W) inside W of the CuspidalSubspace. (OK, OK, starting with a subspace X of M, one could take its complement inside W, the dual of that one inside M, and then the complement again to get the dual in W of the original X. Taking complements is computing kernels --- how costly is it?) Hmmmm, the pairing being Hecke-invariant, one only has to compute the old-/new-spaces to get the new-/old-spaces also, n'est-ce pas? I wonder what exactly is going wrong on the EisensteinSubspace here. By the way, here in Essen there was a thesis done on alternatives to the "Heibronn", etc. matrices, with both theoretical and computational results indicating that certain choices of sets of matrices were way better than others. I can look it up for you, if you wish. A way to compute the pairing in the magma package as it is now requires nothing but a way to lift a modular symbol in such a way that it lies not only in M^f, but already in W --- to achieve this, one can do the "graph algorithm" backwards (at least for cuspidal symbols, or if the level is square-free --- ask me about it). This should be very, very fast once the graph is there as an internal object ... ;-) Of course, since W is easily computed directly as the intersection of W_1 and W_2, and the iso from W to the space of modular symbols is trivial to write down, one could just compute it and take the inverse ... but taking inverses might be something to be avoided ... OK, I think some words are due for the proof of the above claim that the pairing has the properties I described. Well, I always tried to view it as coming from some high brow hidden fact (as it does, of course, one way or the other), but its properties actually are quickly verified by hand, using the interpretation of the space of homogeneous polynomials in two variables in degree k-2 and the matrix action on it as being the k-1-fold symmetric tensor product of Q^2 together with the natural action of GL_2(Q) on it (from the right, altered to an action on the left by the main involution \iota); the fact that S=(0,1;-1,0) acts like sending \sum_i a_i X^{k-2-i}Y^i to \sum_i (-1)^{i} a_{k-2-i} X^{k-2-i}Y^i ; the fact that the main ivolution \iota may be described as mapping a matrix M to S^{-1} M^t S, the conjugate by S of its transpose M^t, and that's essentially all, at least if we restrict ourselves to the case of even weights k. And I had done all this already in November! For the Heckeoperators, this is just writing them out. Perhaps working here in a space of modular symbols that combines all the different levels might by useful for a clean proof, since there GL_2(Q) itself can be shown to act there, but this is easily achieved. Wow, that was a long email, sorry for that, if you have any questions, comments, etc., just mail me! -- Georg P.S. I'm in direct contact with - you know whom - Robert Pollack. From sharifi@math.arizona.edu Mon Feb 5 04:22:08 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f159SSW03830 for ; Mon, 5 Feb 2001 04:28:28 -0500 Received: from hedgehog.math.arizona.edu (hedgehog.math.arizona.edu [128.196.197.57]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f159Rmm28946 for ; Mon, 5 Feb 2001 04:27:48 -0500 (EST) Received: from [10.0.2.15] (hedgehog.math.arizona.edu [128.196.197.57]) by hedgehog.math.arizona.edu (8.9.3/8.9.3) with ESMTP id CAA20298 for ; Mon, 5 Feb 2001 02:27:46 -0700 User-Agent: Microsoft-Outlook-Express-Macintosh-Edition/5.02.2022 Date: Mon, 05 Feb 2001 02:22:08 -0700 Subject: NSF postdoc From: Romyar Sharifi To: Message-ID: Mime-version: 1.0 Content-type: text/plain; charset="US-ASCII" Content-transfer-encoding: 7bit Status: R X-Status: N William, How's it going? I got the NSF postdoc, so I guess I'll be seeing you next year. Heard about the BP. Congratulations! Hey, I want to apologize for not paying for your dinner that night. Felt real bad about that later, but I hadn't actually ever taken a speaker to dinner before and was confused about etiquette. Did you get the plane ticket all worked out? Anyway, what can you tell me about the NSF postdoc and Harvard? I haven't gotten the official offer from Harvard yet. Please tell me whatever might be useful to know, but here are some leading questions. What's your office like? Did they provide you with a computer? Does Harvard provide (or supplement) health insurance? How do I look for housing? I heard something about cheap Harvard housing. How difficult is it to have a car there? Can I use some of the $7500 allowance for moving there? Are there lots of seminars? Are people accessible? Anyway, I'm definitely looking forward to it. Going to be a shame to leave this weather, though. Best, Romyar From fcale@math.berkeley.edu Mon Feb 5 04:37:29 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f159c9W03836 for ; Mon, 5 Feb 2001 04:38:09 -0500 Received: from math.berkeley.edu (gold.Math.Berkeley.EDU [169.229.58.61]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f159bUm29114 for ; Mon, 5 Feb 2001 04:37:30 -0500 (EST) Received: from pub-708c-15.math.Berkeley.EDU (pub-708c-15.Math.Berkeley.EDU [169.229.58.110]) by math.berkeley.edu (8.9.3/8.9.3) with ESMTP id BAA29566 for ; Mon, 5 Feb 2001 01:37:29 -0800 (PST) Received: from localhost (fcale@localhost) by pub-708c-15.math.Berkeley.EDU (8.9.3/8.9.3) with ESMTP id BAA12057 for ; Mon, 5 Feb 2001 01:37:29 -0800 (PST) X-Authentication-Warning: pub-708c-15.math.berkeley.edu: fcale owned process doing -bs Date: Mon, 5 Feb 2001 01:37:29 -0800 (PST) From: Frank Calegari X-X-Sender: To: William Arthur Stein Subject: AWS Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N Hi William! On your web page is says that you're going to the AWS---if so, are you staying at the Marriott? I'm going, but the Marriott has booked all its rooms by now so I need to find somewhere else to stay... Yours, Frank. From marlene@infomagic.com Mon Feb 5 08:30:56 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f15DEoW03899 for ; Mon, 5 Feb 2001 08:14:50 -0500 Received: from Mail.InfoMagic.NET (root@MAIL.InfoMagic.NET [63.229.89.2]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f15DEBm03310 for ; Mon, 5 Feb 2001 08:14:11 -0500 (EST) Received: from infomagic.com (MAX6-Port25.Downtown.InfoMagic.NET [63.229.95.125]) by Mail.InfoMagic.NET (8.11.1/8.11.1) with ESMTP id f15DEBB26141; Mon, 5 Feb 2001 06:14:11 -0700 Message-ID: <3A7EAB0F.8DF31501@infomagic.com> Date: Mon, 05 Feb 2001 06:30:56 -0700 From: Marlene Stein X-Mailer: Mozilla 4.7 [en] (Win95; I) X-Accept-Language: en MIME-Version: 1.0 To: Stein Dennis CC: "William A. Stein" Subject: Internet plane tickets Content-Type: text/plain; charset=us-ascii Content-Transfer-Encoding: 7bit Status: R X-Status: N What sites do you look at to buy cheap plane tickets? I was already told about travelocity.com and southwest.com any others? Mom From verrill@math.ku.dk Mon Feb 5 07:38:51 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f15DgEW03910 for ; Mon, 5 Feb 2001 08:42:14 -0500 Received: from localhost.localdomain (IDENT:root@verrill.math.ku.dk [130.225.103.24]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f15DfYm03981 for ; Mon, 5 Feb 2001 08:41:34 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id NAA10051 for ; Mon, 5 Feb 2001 13:38:51 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Mon, 5 Feb 2001 13:38:51 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: Re: snow In-Reply-To: <01020511571907.13137@tx-irmar-48> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N > I'm in France. Picture a thick wet grey cloud. For the last > four days that's exactly what I see when I look outside. I've > attached a tiny picture of the view outside of my window right now. Well, I guess it's a bit gloomy sky, but you do have lots of green and trees. That's nice. > You are so lucky! I miss snow! I'll send you a picture when I get then on my computer later. Snow is far nicer than wet and grey. If I get the position in Canada I will see lots snow! I've still not heard anything about that yet though. > I don't know what to say about the above. I'm not sure exactly what > you are saying. Never mind, I'll just experiment on my own. It's some of what I've mentioned before about cosets and generators etc, different ways of getting them, which is faster, most effficient etc; your method is the fastest, by far, but produces far more generators than necessary. But I'm not sure why it's quite so much faster than the other methods. If I could get my method even just half as fast as yours that would make it worth using as it gives about 10 times fewer generators. > Are you taking the postdoc in Hannover? Yep, because I can terminate it early if I get a tenure track position. So I'll probably be there at least for June to September anyway, probably. They've booked a room at the Libnitz guest house for me for a year though. Actually, if I don't get a tenure track at a reseach university, even if I get offers from other places I might turn them down as another year with no teaching maybe I can get some good papers written and get a tenure track position at a better place. > Rennes, France, which is my final European destination, aside from a > quick visit to Paris sometime later in the week. Is that the place with Eidixhoven, or is it the place where you are a superstar with special visitor status or something, or are they the same? Helena From mbaker@math.harvard.edu Mon Feb 5 09:13:01 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f15EDeW03922 for ; Mon, 5 Feb 2001 09:13:40 -0500 Received: from dirichlet.math.harvard.edu (dirichlet.math.harvard.edu [140.247.28.18]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f15ED1m05185 for ; Mon, 5 Feb 2001 09:13:01 -0500 (EST) Received: from localhost (mbaker@localhost) by dirichlet.math.harvard.edu (8.11.1/8.11.1) with ESMTP id f15ED1g09788 for ; Mon, 5 Feb 2001 09:13:01 -0500 (EST) X-Authentication-Warning: dirichlet.math.harvard.edu: mbaker owned process doing -bs Date: Mon, 5 Feb 2001 09:13:01 -0500 (EST) From: Matt Baker To: "William A. Stein" Subject: Re: Weierstrass points: I'm still not satisfied. In-Reply-To: <01020219073500.03064@tx-irmar-48> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N On Fri, 2 Feb 2001, William A. Stein wrote: > > should think about trying to solve this problem some time. How far do you > > think you could check this experimentally? Maybe infinity *is* a > > Certainly looking at some new data is a good idea, since it will be > routine for me to do this. Perhaps this is like the "does p divide > the discriminant of the Hecke algebra attached to Gamma_0(p)" problem, > in that there is a counterexample. With some effort, I think I can > check up to level 3000. I should compute q-expansion basis of > S_2(Gamma_0(N);Q) for my modular forms database, anyways, and this is > just a good excuse. Excellent. > For prime n, oo is not a Weierstrass point. I know -- I've been thinking about how one might try to generalize Ogg's proof. The same proof that works when n is prime, by the way, also works for X_0(pm), where p is prime and X_0(m) has genus 0. -- Matt From Bea.Peeters@wis.kuleuven.ac.be Mon Feb 5 09:30:19 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f15EmSW03962 for ; Mon, 5 Feb 2001 09:48:28 -0500 Received: from krimson.cc.kuleuven.ac.be (root@krimson.cc.kuleuven.ac.be [134.58.10.5]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f15Elmm06916 for ; Mon, 5 Feb 2001 09:47:49 -0500 (EST) Received: from vinip.cc.kuleuven.ac.be (vinip.cc.kuleuven.ac.be [134.58.15.35]) by krimson.cc.kuleuven.ac.be (8.9.1a/8.9.0) with SMTP id PAA22229 for ; Mon, 5 Feb 2001 15:47:44 +0100 Received: by vinip.cc.kuleuven.ac.be with VINES-ISMTP; Mon, 5 Feb 2001 15:50:41 +0100 Date: Mon, 5 Feb 2001 15:30:19 +0100 Message-ID: X-Priority: 3 (Normal) To: From: "Bea Peeters" Reply-To: Errors-to: Subject: routingnumber X-Incognito-SN: 909 X-Incognito-Version: 5.1.0.78 MIME-Version: 1.0 Content-type: text/plain; charset=us-ascii Status: R X-Status: N Dear Prof. Stein, Last week you gave a talk in the Department Mathematics. You gave your adress and bankaccountnumber to Prof. Denef to receive your travelexpenses and fee. Today I got a phonecall from the Finance Department, who asked me the ABA- of routingnumber of your bank. That number is necessary for transactions to foreign countries. I hope you can give me that information, otherwise you can get the money with a cheque from your bank. Sincerely yours, Bea PEETERS Secretary From verrill@math.ku.dk Mon Feb 5 09:25:04 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f15FSLW04028 for ; Mon, 5 Feb 2001 10:28:21 -0500 Received: from localhost.localdomain (IDENT:root@verrill.math.ku.dk [130.225.103.24]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f15FRgm09644 for ; Mon, 5 Feb 2001 10:27:42 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id PAA22790 for ; Mon, 5 Feb 2001 15:25:04 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Mon, 5 Feb 2001 15:25:04 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: Re: drawing hyperbolic triangles In-Reply-To: <01020511154500.13137@tx-irmar-48> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N > That's awesome! Way to go! Well it still needs some work, but if you want to play with drawing pictures, you can get the magma files from http://www.math.ku.dk/~verrill/fundomain_v2.m and http://www.math.ku.dk/~verrill/fundomain-drawer.m then you can type things like: load "fundomain-drawer.m"; load "fundomain_v2.m" a,b,c:=CosetRepsData([[9,0,1]]); transtriangles(a,[1,0.5,0],"/tmp/gamma09.ps"); and then a blue and orange fundamental domain for Gamma_0(9) will pop up! Actually, I really only want it to pop up if it's not already being viewed, but somehow my system command to try and see if it's already viewing the ps file does not seem to work properly. Also the domains are currently not drawn for prettiness, so might look a bit funny. You'll probably be shocked if you look in the files at how baddly the code is written... I need to learn to structure things better. Helena From verrill@math.ku.dk Mon Feb 5 09:32:48 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f15Fa6W04034 for ; Mon, 5 Feb 2001 10:36:06 -0500 Received: from localhost.localdomain (IDENT:root@verrill.math.ku.dk [130.225.103.24]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f15FZQm10169 for ; Mon, 5 Feb 2001 10:35:26 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id PAA23708 for ; Mon, 5 Feb 2001 15:32:48 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Mon, 5 Feb 2001 15:32:48 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: Re: Fwd: a natural self-pairing In-Reply-To: <01020511351103.13137@tx-irmar-48> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N Hi William, Thanks for sending this. Hum, so does this mean Georg has beaten me to it in defining the intersection pairing for higher weight modular symbols? I better read through the email carefully to work out what he is doing. Helena From verrill@math.ku.dk Mon Feb 5 10:07:39 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f15GAwW04103 for ; Mon, 5 Feb 2001 11:10:58 -0500 Received: from localhost.localdomain (IDENT:root@verrill.math.ku.dk [130.225.103.24]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f15GAIm16039 for ; Mon, 5 Feb 2001 11:10:18 -0500 (EST) Received: from localhost (verrill@localhost) by localhost.localdomain (8.9.3/8.9.3) with ESMTP id QAA27819 for ; Mon, 5 Feb 2001 16:07:40 +0100 X-Authentication-Warning: localhost.localdomain: verrill owned process doing -bs Date: Mon, 5 Feb 2001 16:07:39 +0100 (CET) From: "Helena A. Verrill" X-Sender: verrill@localhost.localdomain To: "William A. Stein" Subject: Re: Fwd: a natural self-pairing In-Reply-To: <01020517025109.13137@tx-irmar-48> Message-ID: MIME-Version: 1.0 Content-Type: TEXT/PLAIN; charset=US-ASCII Status: R X-Status: N > NO, as he doesn't do that. He claims to give *some* pairing, > but it is definitely *not* the intersection pairing. Yep, that's true, but he says it's "good enough". Anyway I will have to read the email properly, and work out how what he is defining should compare to the "real" intersection pairing. I'd guess they were pretty close. Anyway, I better get to work on it, and take a break from drawing fundamental domains. Helena From merel@math.jussieu.fr Mon Feb 5 11:15:10 2001 Return-Path: Received: from abel.math.harvard.edu (abel [140.247.28.153]) by modular.fas.harvard.edu (8.11.0/8.11.0) with ESMTP id f15GHHW04108 for ; Mon, 5 Feb 2001 11:17:17 -0500 Received: from shiva.jussieu.fr (shiva.jussieu.fr [134.157.0.129]) by abel.math.harvard.edu (8.11.0/8.11.0) with ESMTP id f15GGcm16496 for ; Mon, 5 Feb 2001 11:16:38 -0500 (EST) Received: from cantor.math.jussieu.fr (cantor.math.jussieu.fr [134.157.13.106]) by shiva.jussieu.fr (8.10.0/jtpda-5.3.3) with ESMTP id f15GGbq53646 for ; Mon, 5 Feb 2001 17:16:37 +0100 (CET) Received: from [134.157.61.16] (macmerel.institut.math.jussieu.fr [134.157.61.16]) by cantor.math.jussieu.fr (8.9.3/jtpda-5.3.3) with ESMTP id RAA00396 for ; Mon, 5 Feb 2001 17:16:36 +0100 (CET) Mime-Version: 1.0 X-Sender: merel@pophost.math.jussieu.fr Message-Id: In-Reply-To: <01020217340205.01014@tx-irmar-48> References: <01020118212300.22693@form> <01020217340205.01014@tx-irmar-48> X-Mailer: Eudora Pro F4.2.2 Date: Mon, 5 Feb 2001 17:15:10 +0100 To: was@math.harvard.edu From: =?iso-8859-1?Q?Lo=EFc?= Merel Subject: Re: spurious torsion Content-Type: text/plain; charset="iso-8859-1" ; format="flowed" Content-Transfer-Encoding: 8bit X-MIME-Autoconverted: from quoted-printable to 8bit by modular.fas.harvard.edu id f15GHHW04108 Status: R X-Status: N William, I'll be available on the following days prior to February 15: Tuesday 6, Thursday 8, Friday 9, Monday 12, Tuesday 13, Wednedsday 14, Thursday 15. I hope you'll be able to pay a visit to Paris. Lo颿