\documentclass[11pt]{article}
\usepackage{url}
\newcommand{\thisdocument}{Project Description}
\include{macros}
\usepackage{natbib}
\setlength{\bibsep}{.9ex}
\bibpunct{[}{]}{,}{a}{}{;}
\begin{document}
%\tableofcontents
%\newpage
\begin{center}
\Large\bf
Explicit Approaches to the Birch and Swinnerton-Dyer Conjecture
\end{center}
\section{Introduction}
My research reflects the rewarding interplay of theory
with explicit computation in number theory, as illustrated by
Bryan Birch~\cite{birch:bsd}:
\begin{quote}
I want to describe some computations undertaken by myself and
Swin-nerton-Dyer on EDSAC by which we have calculated the
zeta-functions of certain elliptic curves. As a result of these
computations we have found an analogue for an elliptic curve of
the Tamagawa number of an algebraic group; and conjectures (due to
ourselves, due to Tate, and due to others) have proliferated.
\end{quote}
The goal of this proposal is to carry out a wide range of
computational and theoretical investigations on elliptic
curves and abelian varieties motivated by
the Birch and
Swinnerton-Dyer conjecture (BSD conjecture).
This will hopefully
improve our practical
computational capabilities, extend the data that
researchers have available for formulating
conjectures, and deepen our understanding of
theorems about the BSD conjecture.
The PI is one of the more sought after people by the worldwide
community of number
theorists, for computational confirmation of conjectures, for modular
forms algorithms, for data, and for
ways of formulating
problems so as to make them more accessible to algorithms. The
PI has also been successful at involving numerous
undergraduate and graduate students at all levels in
his research.
\subsection{Prior Support}
The PI was partly supported by NSF postdoctoral fellowship during
2000--2004 (DMS-0071576) in the amount of \$90,000.
The PI was also awarded NSF grant DMS-0555776 (and DMS-0400386)
from the ANTC program in the amount of \$177,917
for the period 2004--2007. The PI's work under
DMS-0555776 succeeded
at improving the modular forms database and
resulted in numerous papers on the arithmetic of elliptic curves,
modular forms and abelian varieties
\cite{stein:vismw,bsdalg1,mazur-stein-tate:padic, stein-watkins:ns, jetchev-stein:higher, agashe-stein:bsd, agashe-ribet-stein:manin,
agashe-ribet-stein:modular, stein:bsdmagma}, one completed book
\cite{stein:modform} on computing with modular forms,
and progress on another book on number theory
\cite{stein:ent}. It has also led to a new
software initiative (see Section~\ref{sec:sage} below).
Funds from DMS-0555776 were used to run a
successful workshop at UCSD and to purchase a
16-processor compute server with 64GB RAM.
\subsection{The BSD Conjecture}
The Birch and Swinnerton-Dyer conjecture is a
central problems in number theory, and
this proposal is based on a group of ideas
related to this conjecture.
An {\em elliptic curve} is a projective
genus one curve with a distinguished rational point.
Every such curve is the projective closure of a
nonsingular affine curve given by
$
y^2 + a_1 xy + a_3 y = x^3 + a_2 x^2 + a_4 x + a_6.
$
An {\em abelian variety} is a projective group
variety---the one dimensional abelian varieties are exactly
the elliptic curves.
\begin{conjecture}[BSD Conjecture]\label{conj:bsd}
Let $A$ be an abelian variety over~$\Q$.
(The objects and notation in the formula are discussed
below.)
\begin{enumerate}%
\item The rank $r$ of $A(\Q)$ equals $\ord_{s=1}L(A,s)$.%
\item We have
$$
\frac{L^{(r)}(A,1)}{r!} =%
\frac{\#\Sha(A) \cdot \Omega_{A} \cdot \Reg_A}%
{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}} \cdot {\displaystyle \prod_{\ell\mid N} c_\ell}.
$$
\end{enumerate}
\end{conjecture}
In the conjecture,
$L(A,s)$ is the Hasse-Weil $L$-series of $A$.
The real volume
$\Omega_{A}$ is the measure of $A(\R)$ with respect to a basis of
differentials for the N\'eron model of $A$. For each prime $\ell\mid
N$, the integer $c_\ell=\#\Phi_{A,\ell}(\F_\ell)$ is the {\em Tamagawa
number} of~$A$ at~$\ell$, where $\Phi_{A,\ell}$ denotes the component
group of the N\'eron model of~$A$ at $\ell$. The abelian variety
dual of $A$ is denoted $A^{\vee}$, and in the conjecture
$A(\Q)_{\tor}$ and
$A^{\vee}(\Q)_{\tor}$ are the torsion subgroups. The {\em
Shafarevich-Tate group} of $A$ is
\[
\Sha(A) = \Ker\left(\H^1(\Q,A) \to \bigoplus_{p\leq
\infty} \H^1(\Q_p,A)\right),
\]
which is a group that measures the failure of a local-to-global
principle. It is implicit in the statement of the
conjecture that $\Sha(A)$ is finite, though this is only
known in some cases.
%when $L(A,1)\neq 0$ and in some cases when $L(A,1)=0$
%(see \cite{kato:secret} and \cite{kolyvagin-logachev:finiteness}).
The regulator $\Reg_A$ is the absolute value of
the discriminant of the N\'eron-Tate canonical
height pairing $A(\Q)_{/\tor} \times A(\Q)_{/\tor}\to \R$.
If $A$ is an elliptic curve then $\#A(\Q)_{\tor}$,
$\# A^{\vee}(\Q)_{\tor}$, $\Omega_A$, and $c_\ell$ are relatively
easy to compute; none of the other quantities
are known to be computable in general,
even when $A$ is an elliptic curve, though many
can in practice be computed.
\begin{conjecture}[$\BSD(A,p)$]\label{conj:bsdp}
Let $A$ be an abelian variety over $\Q$ of rank $r$ and
let $p$ be a prime. Then
$$
\ord_p\left(\frac{L^{(r)}(A,1)}{r!\cdot \Reg_A \cdot \Omega_A}\right) =%
\ord_p\left(\frac{\#\Sha(A)}%
{\#A(\Q)_{\tor}\cdot \#A^{\vee}(\Q)_{\tor}} \cdot {\displaystyle \prod_{\ell\mid N} c_\ell}\right).
$$
\end{conjecture}
In Conjecture~\ref{conj:bsdp} the fraction on the left side is not known to be a rational number (except when $r\leq 1$), so
its rationality is part of the conjecture.
Tate \cite{tate:bsd} formulated the BSD conjecture for
any abelian variety over a global field~$K$, and proved
(generalizing work of Cassels)
that if $A$ and $B$ are related by an isogeny,
then $\BSD(A,p)$ is true if and only if $\BSD(B,p)$
is true.
\subsection{Modular Abelian Varieties}
Modular abelian varieties are a special class of abelian
varieties over $\Q$ that have been studied intensively.
Computation with modular abelian varieties is attractive because they
are easier to describe than arbitrary abelian varieties,
have extra hidden structure,
and their $L$-functions are reasonably well understood.
We recall Shimura's construction \cite{shimura:factors}
of modular abelian varieties.
Let~$f=\sum a_n q^n$ be a weight~$2$ newform
on $\Gamma_1(N)$. Then~$f$ corresponds to a differential on the
modular curve $X_1(N)$, which is a curve whose affine points
over~$\C$ correspond to isomorphism classes of pairs $(E,P)$,
where~$E$ is an elliptic curve and $P\in E$ is a point of
order~$N$. We view the Hecke algebra
\[\T=\Z[T_1,T_2,T_3,\ldots]\]
as a subring of the endomorphism ring of the Jacobian $J_1(N)$
of $X_1(N)$. Let $I_f$ be the kernel of the homomorphism $\T\to
\Z[a_1,a_2,a_3, \ldots]$ that sends $T_n$ to $a_n$, and attach to~$f$ the
quotient $$A_f=J_1(N)/I_f J_1(N).$$ Then $A_f$ is a simple abelian
variety over~$\Q$ of dimension equal to the degree of the field
$\Q(a_1,a_2,a_3,\ldots)$ generated by the coefficients of~$f$. We
also sometimes consider a similar construction with $J_1(N)$
replaced by the Jacobian $J_0(N)$ of the modular curve $X_0(N)$
that parametrizes isomorphism classes of pairs $(E,C)$, where $C$
is a cyclic subgroup of $E$ of order~$N$.
\begin{definition}[Modular abelian variety]
An abelian variety over a number field is a {\em
modular abelian varieties} if it is a quotient of $J_1(N)$ for
some~$N$.
\end{definition}
%Newform abelian varieties are simple (but not necessarily
%absolutely simple), and every modular
%abelian variety is isogenous to a product of copies of newform
%abelian varieties. Newform abelian varieties need not be
%absolutely simple.
Algorithms for computing many of the invariants in the BSD
conjecture for modular abelian varieties has been a major
part of the PI's research program.
The celebrated modularity theorem of C.~Breuil, B.~Conrad, F.~Diamond,
R.~Taylor, and A.~Wiles \cite{breuil-conrad-diamond-taylor}
asserts that every elliptic curve over~$\Q$ is
a modular abelian variety. Also, it is now known
(due to very recent work of Khare, Wintenberger, and Dieulefait)
that every abelian variety of $\GL_2$-type (see \cite{ribet:abvars})
and odd conductor is a modular abelian variety.
The PI recently completed a book on computing
with modular forms \cite{stein:modform} that
will be published by the AMS. He is working on a graduate
textbook with Ken Ribet on modular abelian varieties, and
an undergraduate text on number theory, both intended for
publication by Springer-Verlag. He led a 2-week
high-school student workshop on the Birch and Swinnerton-Dyer
conjecture (see \cite{simuw}).
\subsection{Software for Algebra and
Geometry Experimentation}\label{sec:sage}
The PI is the principal author of SAGE---Software for Algebra and
Geometry Experimentation (see \cite{sage}).
Substantial work on SAGE has been done jointly with students
(7 undergraduates and 5 graduate students).
The goal of SAGE is to create an optimal {\em
open source} software environment
for research in algebra, geometry, number theory,
and related areas.
The PI intends to make all data and algorithms
developed as part of the proposed
research freely available online and
from SAGE.
The PI is the author
of the modular forms and modular abelian varieties components of
Magma \cite{magma}. When possible many of the proposed computations
will be independently verified using Magma.
When we describe a result that relies on computation
below, there
is an implicit assumption that certain software
produced correct output.
Also, the ranges of computations, e.g., ``all curves
of conductor up to 1000'', are in many cases arbitrary.
Our {\em primary} motivation for
doing these computations is to
motivate the development of new conjectures and
computational and theoretical tools.
\section{The BSD Formula: Computing $\Sha$}
Much of this research proposal is about computing $\Sha$,
which is the
most difficult to compute invariant appearing in
the BSD conjecture.
\subsection{Applying Theorems of Kato and Kolyvagin}
The PI, 3 undergraduates and a graduate student
proved the following in \cite{bsdalg1}:
\begin{theorem}[Stein et al.]\label{thm:main}
Suppose that $E$ is a non-CM elliptic curve of rank $\leq 1$,
conductor $\leq 1000$ and that $p$ is a prime. If $p$ is odd,
assume further that the mod~$p$ representation $\rhobar_{E,p}$ is
irreducible and~$p$ does not divide any Tamagawa number of~$E$. Then
$\BSD(E,p)$ is true.
\end{theorem}
The proof involves an application of results of Kato and Kolyvagin, new
refinements of Kolyvagin's theorem, explicit 2-descent and 3-descent
and much explicit calculation. This is a first step toward
the following goals:
\begin{goal}\label{prob:bsd01}
Verify the BSD Conjecture for every
elliptic curve over $\Q$ of conductor $<1000$, except
for the $18$ curves of rank $2$.
\end{goal}
\begin{goal}\label{prob:bsd2}
For each curve $E$ over $\Q$ of conductor $<1000$
and rank $2$, prove that $\Sha(E)[p]=0$ for all
$p<1000$.
\end{goal}
We hope to make further progress toward
Goal~\ref{prob:bsd2} using $p$-adic methods (see Section~\ref{sec:padic} below),
since unconditional computation of $\Sel^{(p)}(E/\Q)$ directly using standard algebraic
number theory techniques for $p>5$ appears
to not be practical. Another approach is to improve on
work of Kolyvagin---Theorem~\ref{thm:main}
excludes divisors of Tamagawa numbers because Kolyvagin's theorem
is not sufficiently precise at such primes.
\begin{goal}\label{kolybsd}
Refine Kolyvagin's bound when a prime $p$ divides
a Tamagawa number.
\end{goal}
Dimitar Jetchev (who began working with the PI as an
undergraduate) has been working
on Goal~\ref{kolybsd} in consultation with the PI.
%So far he has proved the following, which will allow
%the PI to improve Theorem~\ref{thm:main}.
Let~$E$ be an elliptic curve over~$\Q$, let~$K$
be a field that satisfies the {\em Heegner hypothesis}
for~$E$, i.e., such that each prime dividing
$N$ splits in $K$. Assume that~$E$ has analytic rank~$1$ over~$K$,
and let $y_K\in E(K)$ be the Heegner point.
Suppose $p\geq 5$ and that $\rhobar_{E,p}$ is surjective.
Jetchev has made great progress toward
the following:
\begin{goal}\label{goal:jetchev}
If $\ord_p(\prod c_\ell)\geq 1$, where the $c_\ell$
are the Tamagawa numbers of $E$, prove that
$
\ord_p(\#\Sha(E/K)) \leq \ord_p([E(K): \Z y_K]) - 2.
$
\end{goal}
Kolyvagin gives a formula (see, e.g., \cite{kolyvagin:structure_of_selmer,gross:kolyvagin,mccallum:kolyvagin})
for $\#\Sha(E/K)$ which involves global divisibility of
Heegner points and Jetchev links those global divisibility
exponents to Tamagawa numbers. Work toward Goal~\ref{goal:jetchev}
uses Poitou-Tate global
duality and the Chebotarev density theorem to show that mod $p$ there are no
nontrivial Kolyvagin systems;
this is equivalent to showing that all Heegner points $P_n$ on $E$ over ring class
fields are globally divisible by $p$.
\subsection{Complex Multiplication Curves}
A {\em complex multiplication} (CM) elliptic curve $E$
over a number field is one such that $\End(E_{\C})\neq \Z$.
The PI and Aron Lum proved the following result:
\begin{theorem}[Stein, Lum]\label{thm:cm}
Suppose $E$ is a CM elliptic curve over $\Q$ with rank at most~$1$
and conductor at most $5000$. Then $\BSD(E,p)$ is
true for all primes $p\geq 5$ of good reduction for $E$.
\end{theorem}
This is an application of Rubin \cite{rubin:main-conjectures}
for rank 0 curves
and Kolyvagin \cite[Cor.~D]{kolyvagin:euler_systems} for
rank 1 curves. The PI to compute to much higher conductor.
%A number of interesting issues arise in translating
%the theoretical results of \cite{blah} into practical
%algorithms.
\begin{goal}\label{prob:bsdcmbad}
Suppose $E$ is a complex multiplication elliptic curve over $\Q$
and that $p\geq 5$ is a prime of bad reduction for $E$.
Find and implement a {\em practical} algorithm to verify $\BSD(E,p)$,
then apply it to all $E$ of conductor up to $5000$.
\end{goal}
Goal~\ref{prob:bsdcmbad} is difficult mainly because
the mod $p$ representation is
not ``as surjective as possible'', so the
methods used for Theorem~\ref{thm:cm} do not apply.
One approach to Goal~\ref{prob:bsdcmbad}
is to try do computations using
Theorem~\ref{thm:schneider} below.
\subsection{$p$-adic Methods}\label{sec:padic}
Suppose $E$ is an elliptic curve over a number field $K$
and $p\geq 5$ is a prime (of $\mathcal{O}_K$) of good ordinary reduction for $E$.
In \cite{mazur-stein-tate:padic} the PI, Mazur, and Tate give a new
approach to computing $p$-adic heights
$$h_p:E(K) \to \Q_p$$
that leverages
Kedlaya's fast algorithm \cite{kedlaya:counting_mw, kedlaya:mw2}
for explicit computation of Monsky-Washnitzer
cohomology groups. We made our algorithm explicit only
when $K=\Q$ or $K$ is a quadratic imaginary field,
though the key ideas for creating a general explicit
algorithm are given in \cite{mazur-stein-tate:padic}.
The PI, David Harvey,
Jen Balakrishnan, and Liang Xiao have implemented and
optimized this algorithm over $\Q$ and quadratic imaginary
$K$ in both SAGE and Magma (this was a major project
at an MSRI graduate student workshop that the PI ran in August 2006).
\begin{goal}
Design and implement a general algorithm for
computing $p$-adic heights (for all primes $p$) on
elliptic curves over arbitrary number fields.
\end{goal}
The natural height pairing here is not very well understood;
indeed, it still only conjectural that is nondegenerate.
It can be verified in particular cases via computation:
\begin{conjecture}[Schneider]
Suppose $E$ is an elliptic curve over $\Q$ and $p$ is a prime
of good ordinary reduction.
Then the $p$-adic cyclotomic
height pairing on $E(\Q)$ is nondegenerate.
\end{conjecture}
There are very few general results toward this conjecture (except
in the CM case). It
can be verified in particular cases:
\begin{goal}\label{prob:padicreg}
Create a table of $p$-adic regulators to precision $O(p^{5})$ for every
elliptic curve of conductor up to $120000$ and the first
five primes $p\geq 5$ of good ordinary reduction for $E$.
%In particular,
%verify for five primes $p$ the nondegeneracy of the $p$-adic height
%pairing.
\end{goal}
The data from Goal~\ref{prob:padicreg} will also
be of interest in investigations about congruences
between algebraic parts of $p$-adic $L$-functions.
%Computation of $p$-adic heights is unclear when
%$p=2,3$, so we propose:
%\begin{goal}
%Find and implement an algorithm for computing $p$-adic
%heights when $p=2,3$ is prime of good ordinary reduction.
%\end{goal}
\begin{theorem}[Schneider \cite{schneider:iwasawa} and Perrin-Riou]\label{thm:schneider}
Suppose $p$ is an odd prime of good ordinary reduction for $E$,
and let $\Reg_E^{(p)}\in \Q_p$ be the $p$-adic regulator of $E$, i.e.,
the discriminant of the $p$-adic height pairing on $E(\Q)$.
If the $p$-primary part $\Sha(E/\Q)(p)$ of $\Sha(E/\Q)$
is finite, then the leading term of the algebraic
$p$-adic $L$-function of $E$ has the same $p$-adic valuation as
$$
\frac{\#\Sha(E) \cdot \Reg_E^{(p)}}%
{\#E(\Q)_{\tor}^2} \cdot \#E(\F_p)^2 \cdot \prod_{\ell \mid N} c_\ell.
$$
\end{theorem}
%As explained in Section~\ref{sec:padic}, we
%can compute $p$-adic regulators quickly.
The following is needed in order to apply Theorem~\ref{thm:schneider}:
\begin{goal}\label{goal:padicl}
Create software and better algorithms for computing with $p$-adic $L$-functions of elliptic curves and modular abelian varieties.
\end{goal}
Much work toward Goal~\ref{goal:padicl}
in Magma has already been done by the PI,
Robert Pollack, and Christian Wuthrich.
Substantial work on the main
conjecture of Iwasawa theory
connects the algebraic and analytic (computable) $L$-functions.
\begin{remark}
There are other cases where we can apply
analogues of Theorem~\ref{thm:schneider}, e.g.,
when $p$ is a prime of good supersingular
reduction (see \cite{pr:exp}).
% See \cite{stein-wuthrich}
%for a survey of such results in the context of computing $\#\Sha$.
\end{remark}
\subsection{Constructing Nonzero Elements of $\Sha(E)$}
There are 6581 optimal curves in \cite{cremona:onlinetables} of conductor up to $120000$
for which $p$ divides
the BSD conjectural order of $\Sha(E)$ for some $p\geq 3$.
Of these, 1387 have conjectural order divisible by a prime $p\geq 5$ and
$339$ have conjectural order divisible by a prime $p\geq 7$.
Of these $1387$ curves, the mod~$p$ representation is surjective
except in $11$ cases.
%33825o1 25 [1]
%40455k1 25 [1]
%52094m1 25 [1]
%59643a1 25 CM
%61446de1 49 [1]
%62400dt1 25 [1]
%82418u1 25 [1]
%97104cp1 25 [1]
%108489m1 25 [1]
%119025ck1 25 [-1]
%119025a1 25 CM
\begin{goal}\label{prob:constructsha}
For the $1387$ curves of conductor up to $120000$
for which a prime $p\geq 5$ divides the conjectured
$\#\Sha(E)$, construct an element of order~$p$
in $\Sha(E)$.
\end{goal}
One approach to Goal~\ref{prob:constructsha} for small $p$, e.g.,
$p=5$, is to construct $\Sha(E)$ using visibility, i.e., by finding
an elliptic curve $F$ of rank $2$ such that $E[p]\isom F[p]$,
and using the visibility techniques of
\cite{cremona-mazur, agashe-stein:visibility, agashe-stein:bsd}.
A second approach, which works in many cases (because the
mod~$p$ representation is often surjective),
is to use results of \cite{grigor:phd}, which gives
an explicit criterion in terms of modular symbols to
construct nonzero elements of $\Sha(E)[p]$.
Initial computations of the PI and Grigorov (discussed in
\cite{grigor:phd}) suggest that this approach will succeed
in many cases.
A third approach is to use Theorem~\ref{thm:schneider}
and explicit calculation of $p$-adic regulators and $p$-adic
$L$-functions to at least prove that $\Sha(E)(p)$ has
the conjectured order. This approach doesn't give an
explicit construction of $\Sha(E)(p)$.
A fourth approach is to explicitly compute Kolyvagin
cohomology classes $c_{n,p} \in \H^1(K,E)[p]$ and show that
they are in fact elements of $\Sha(E)[p]$. Jetchev, Lauter,
and the author have done such a computation in a few cases
(unpublished).
Recent exciting work of Skinner and Urban also addresses
the question of providing very general explicit lower bounds on $\#\Sha(E)$.
\subsection{Tamagawa numbers}
The following is a consequence of the BSD conjecture.
\begin{conjecture}\label{conj:tamprod}
Suppose $E$ is an elliptic curve and that $p\geq 5$ is odd prime
such that $\rhobar_{E,p}$ is irreducible.
Then
$$
\ord_p\left(\prod_{\ell\mid N} c_\ell\right) \leq \ord_{p}\left(\frac{L(E,1)}{\Omega_E}\right)
$$
\end{conjecture}
Let $f = f_E$ be the newform corresponding to $E$.
Under the hypotheses of Conjecture~\ref{conj:tamprod},
\cite{ribet:lowering} implies that there
is a newform $g$ of level a proper divisor of the
conductor of $E$ whose coefficients (of index coprime to
the conductor) are congruent to the coefficients of $f$.
The PI showed in unpublished work how to use this congruence, in some
cases, to prove that
$L(E,1)/\Omega_E \con 0\pmod{p}$.
The PI proposes to write up the details of this argument
(jointly with Jetchev), then attempt
to refine the argument to yield the
congruence of Conjecture~\ref{conj:tamprod}. The PI
also hopes to find connections between the components
group of $E$ and the $\T$-module (defined using modular symbols)
whose order is the $p$-part of $L(E,1)/\Omega_E$.
This approach gives a conceptual
explanation of part of the BSD conjecture.
\section{The Rank}
Let $E$ be an elliptic curve over $\Q$.
If $r_{\an} = \ord_{s=1} L(E,s) \leq 1$,
then the rank part of the BSD conjecture is known
for $E$; moreover, in principle, and often
in practice (as explained elsewhere in this proposal) one
can verify the full BSD conjecture.
There isn't a single $E$ with $r_{\an}\geq 2$ for which
the PI is aware of even a plausible strategy
for proving that $\Sha(E)$ is finite, let alone
verifying the full BSD conjecture---it is even unknown
that $L''(E,1)/(\Omega_E\cdot \Reg_E)\in\Q$ for any~$E$.
One can verify in particular cases the rank part of the
BSD conjecture if $r_{\an} \leq 3$; the PI is aware of no
strategy
to verify the rank statement for even
a single example when $r_{\an}\geq 4$.
Morever, if $E$ has analytic
rank~$2$ or larger, then the PI is aware of no
conjectural construction of $E(\Q)$ analogous
to that of Gross-Zagier in the rank~$1$ case.
Thus new ideas are needed when $r_{\an}\geq 2$, and
the PI hopes the computations he proposes might play a role
in finding them.
\subsection{Kolyvagin's Cohomology Classes}\label{sec:kolyclass}
Let $E$ be an elliptic curve over $\Q$ with conductor $N$,
and to simplify the discussion assume that $E$ does not
have complex multiplication (there are analogues of
everything below even if $E$ has CM).
Let $K$ be a quadratic imaginary field that satisfies
the Heegner hypothesis for $E$. Fix a prime $p$ such that $E[p]$
is irreducible (we could also replace $p$ by an integer
$n$ and remove the requirement that the representation
be irreducible).
Kolyvagin defined (see, e.g., \cite{gross:kolyvagin})
classes $c_{n, p} \in \H^1(K,E[n])$
for infinitely many squarefree integers $n$ satisfying a Chebotarev condition.
Let $d_{n,p}$ be the image in $\H^1(K,E)[p]$ of $c_{n,p}$.
Kolyvagin proved that $\res_{v}(d_{n,p})=0$ for all $v\nmid n$
and computed the order of $\res_{\ell}(d_{n,p})$
in terms of local properties of the point $y_K$.
He used these classes to prove his celebrated results
toward the BSD conjecture.
Dimitar Jetchev, Kristin Lauter and the PI have solved the
following problem in a handful of cases, and intend to continue
working on refining our methods.
\begin{goal}
Find and implement a practical algorithm to compute the order of
$c_{n,p}$ and the order of its image $d_{n,p}$.
\end{goal}
For simplicity, assume that $n=\ell$ is a prime.
By the modularity theorem there is a surjective homomorphism
$\pi: X_0(N)\to E$.
The two degeneracy maps $X_0(N\ell)\to X_0(N)$ induce
maps $\delta_1$ and $\delta_\ell$ from $J_0(N)$ to $J_0(N\ell)$.
We say that $d_{\ell,p}$ is {\em visible of level $N\ell$}
if $d_{\ell, p}$ maps to $0$ under the map on cohomology
induced by
$$
E \xrightarrow{\,\,\,\,\pi^*\,\,} J_0(N)
\xrightarrow{\,\,\,\,(\delta_1 \pm \delta_\ell)^*\,\,} J_0(N\ell)
$$
for either choice of sign.
\begin{goal}\label{prob:kolyvisalg}
Find and implement a practical algorithm to
determine whether or not a Kolyvagin class $d_{\ell,p}$
is visible at level $N\ell$.
\end{goal}
\begin{goal}
Based on the data obtained from Goal~\ref{prob:kolyvisalg}
formulate a conjecture about visibility of the classes
$d_{\ell,p}$.
\end{goal}
%Carry out large-scale verification of Kolyvagin's conjecture
%(data gathering problem).
Suppose $E$ is an elliptic curve over $\Q$ with analytic rank
$\geq 2$. %, so by \cite{gross-zagier} the Heegner point $y_K$ is torsion.
Fix an odd prime~$p$ such that $E[p]$ is irreducible. Then
the Kolyvagin classes $d_{n,p}$ are elements of $\Sha(E)[p]$
and the classes $c_{n,p}$ lie in $\Sel^{(p)}(E)$.
\begin{conjecture}[Kolyvagin]
Let $E$ and $p$ be as above. Then $\Sel^{(p)}(E)$
is generated by the classes $c_{n,p}$.
\end{conjecture}
The PI intends to develop a
theory for computing with the subgroup
of $\Sel^{(p)}(E)$ generated by a given
finite collection of classes $c_{n,p}$.
Kolyvagin hints at doing this in
\cite[Pg.~120]{kolyvagin:structureofsha}.
For example, let $E$ be the rank $2$ elliptic curve
$y^2 + y = x^3 + x^2 - 2x$
of conductor $389$.
The PI, Jetchev, and Lauter computed the
class $c_{5,3}$ and showed that it defines a
nonzero element of $\Sel^{(3)}(E)$---this computation
did not require computing $E(\Q)$. This computation
thus gives an explicit construction using Heegner points
of the image in $\Sel^{(3)}(E)$ of a nonzero element of
the rank~$2$ Mordell-Weil group $E(\Q)$. Making this computation
practical has already involved interesting
techniques (e.g., using $p$-adic methods to verify global
non-divisibility of Heegner points).
Moreover, we are computationally verifying Kolyvagin's
conjecture, which has interesting consequences
(unpublished work of Cornut, Nekovar).
\subsection{Mazur and Rubin's Shadow Lines}
When~$E$ is an elliptic curve over $\Q$ of
rank~$2$ there is a construction of Mazur and Rubin, using $p$-adic heights, that attaches to appropriate quadratic imaginary fields $K$
a certain line in the $p$-adic completion of the Mordell-Weil group; these are sometimes referred to as ``shadow lines''. Conjecturally, for every quadratic imaginary field~$K$ such that the Mordell-Weil group of the twist of~$E$ by the quadratic character
of~$K$ has rank one (and such that the discriminant of~$K$ is prime to the conductor of~$E$)
we have such a ``shadow line''.
On the one hand the shadow line is the image of universal norms
in the $p$-adic completion $E(\Q)\tensor\Z_p$ of
the Mordell-Weil group of the elliptic curve over layers of
the $p$-adic anticyclotomic tower attached to $K$, and on
the other hand the shadow line is
the null space of the $p$-adic anti-cyclotomic
height pairing.
\begin{goal}\label{prob:shadow1}
Gather extensive numerical data about Mazur-Rubin shadow lines.
In particular, are they uniformly
distributed as we vary $K$?
\end{goal}
The PI has done computations in this direction using slow methods (see
\cite{mazur-rubin:pairings_arith}). More recently,
he and Liang Xiao have been applying the methods of Section~\ref{sec:padic}.
%\begin{goal}
%Study connections between shadow lines and
%the Kolyvagin cohomology classes $d_{n,p}$, which we can
%explicitly compute as in Section~\ref{sec:kolyclass}.
%\end{goal}
\section{Databases}
The modular forms database (see \cite{mfd})
is a freely-available collection of data about objects attached to
modular forms. It is analogous to Neil Sloane's tables of integer
sequences, and generalizes John Cremona's tables of elliptic curves \cite{cremona:onlinetables}
to
dimension bigger than one and weight bigger than two. The database is
used by many prominent number theorists.
The PI proposes to greatly expand the databases with more information
about modular forms, elliptic curves, and modular abelian
varieties.
In each enumeration problem below, we intend to store the result of
the computation in two forms:
\begin{enumerate}
\item A plain text file that can be easily parsed, similar to
\cite{cremona:onlinetables}.
\item SAGE provides robust support
for saving nearly arbitrary individual objects.
Since SAGE is free and every version of SAGE is archived
in multiple locations, this data will not be lost
because of changes to SAGE.
\end{enumerate}
\subsection{Elliptic Curves}
The elliptic curve over $\Q$ with rank $4$ and smallest known
conductor is the curve
$
y^2 + xy = x^3 - x^2 - 79x + 289,
$
with conductor $234446 = 2 \cdot 117223$. Nobody knows
if there is a curve with smaller conductor and rank $4$,
though the PI showed that any such curve has composite
conductor by finding all curves of prime conductor
up to $234446$ (see \cite{rank4}).
\begin{goal}\label{prob:enum234446}
Enumerate every elliptic curve up to isogeny of conductor up to 234446.
\end{goal}
Cremona \cite{cremona:onlinetables} has spent years methodically
enumerating all curves of conductor up to $130000$ and
computing the invariants in the BSD conjecture
(except $\#\Sha$). The PI believes an independent
computation whose goal is just to find all curves would be extremely
valuable, as the following illustrates:
\begin{quote}
Bad news: Bill Allombert found an elliptic curve not in my database,
conductor $97200$ (it was $[0,0,0,0,15]$ !!!!) and I just found that there
are $40$ conductors in the range 90k--100k where there are curves in
Stein-Watkins and not in Cremona. Misery! \\
\mbox{}\hfill -- email to me from John Cremona, 2006-09-07
\end{quote}
The Stein-Watkins table \cite{stein-watkins:ants5, bmsw:bulletins} mentioned above is a massive
table by the PI and Mark Watkins of over 100 million elliptic
curves, which was made by systematically enumerating
Weierstrass equations. Though it does not contain every curve
of a given conductor, a substantial fraction are there.
Cremona enumerates elliptic curves by computing the matrix
of the Hecke operator $T_2$ on the space of weight $2$
modular symbols for $\Gamma_0(N)$,
then finding the kernels of $T_2-2$, $T_2-1$, $T_2$, $T_2+1$, $T_2+2$.
Next, he computes the Hecke operator $T_3$ restricted to each of the kernels,
and decomposes those kernels under $T_3$, and likewise for $T_5$,
etc. The crucial point is that for the purposes of finding elliptic
curves it is not necessary to
compute the minimal polynomial of $T_2$,
which is extremely difficult when $N$ is large, e.g.,
if $N=200003$ then this dimension is $16667$.
Another key fact is that Cremona computes Hecke operators
using modular symbols, which yield dense matrices, so
the linear algebra is difficult.
%goal would be to find the curves up to isogeny,
%and not necessary the equations, finding even just the Hecke eigenvalues
%(using modular symbols, the method of graphs, or quaternion algebras)
%would be enough.
\begin{goal}\label{prob:enum_mestre}
Use the Mestre method of graphs \cite{mestre:graphs}
to find all elliptic curves of conductor $N\leq 234446$,
for all integers $N$ that are either prime or of the form
$pM$ with $p$ prime and $M\leq 10$ or $M=12,13,16,18$.
\end{goal}
The method of graphs, when applicable, very quickly
produces {\em extremely sparse matrices} of the Hecke operators
$T_p$ on $S_2(\Gamma_0(N))_{p-\new}$ for small $p$ (e.g., $p=2,3,5,7,11$, say).
Sparse linear algebra then yields an
upper bound on the number of isogeny classes of elliptic
curves of conductor $N$. Consulting the tables of Stein-Watkins
and Cremona yields a lower bound. If
these disagree, it is likely that we have found a new curve; we then
prove this by sparse linear algebra computations over $\Q$,
which we do via a multimodular algorithm (see \cite[Ch.~7]{stein:modform}).
Ifti Burhanuddin did
partial work toward Goal~\ref{prob:enum_mestre}
at the MSRI summer graduate student
workshop that the PI organized.
David Kohel, Lassina Dembele, and the PI have been working on a strategy
to carry out the following computation:
\begin{goal}\label{prob:enum_quat}
Use quaternion algebra arithmetic over~$\Q$ (much as in
\cite{dembele:hilbert5}) to compute sparse matrices
for several Hecke operators, then proceed as
above to find all curves of conductor $\le 234446$ for which
some prime exactly divides~$N$.
\end{goal}
Computing the quaternion algebra presentation and the Hecke operators
in the first place can be time consuming---however, for large levels
linear algebra with matrices of Hecke operators dominates the runtime,
so obtaining sparse matrices using quaternion algebras or the method
of graphs is extremely helpful.
There are only $960$ integers up to $234446$ that
do not satisfy the conditions of Goal~\ref{prob:enum_quat}
(which is about 0.4\%). For these remaining conductors
we might use modular symbols to compute all rational eigenforms:
\begin{goal}\label{enum:modsym}
Compute using modular symbols all elliptic curves of the $960$ conductors up
to $234446$ not covered by Goal~\ref{prob:enum_quat}.
\end{goal}
%We emphasize that in Goals~\ref{prob:enum_mestre}--\ref{enum:modsym}
%we are concerned with finding
%data that determines all curves of given conductor.
These calculations
are easy to parallelize, and the linear algebra
involved in some cases requires a huge amount of memory (which is one
reason the PI is requesting a memory upgrade to 128GB for his main server).
\subsection{Modular Abelian Varieties}
In each case the elliptic curves enumeration in
Goals~\ref{prob:enum_mestre}--\ref{enum:modsym}
starts by computing matrices of Hecke operators.
The PI will store these matrices and also use them
to enumerate modular abelian varieties (and, equivalently,
newforms in $S_2(\Gamma_0(N))$).
The linear algebra involved in searching for higher-degree factors
is much more prohibitive than in searching for elliptic curves.
\begin{goal}\label{prob:j0n}
Compute every quotient $A_f$ of $J_0(N)$
for all $N \leq 10000$.
\end{goal}
Note, for example, that $\dim S_2(\Gamma_0(10000)) = 1411$, so computing
and factoring the relevant minimal polynomials should be possible.
\begin{goal}\label{prob:j1n}
Compute every quotient $A_f$ of $J_1(N)$
for all $N\leq 1000$.
\end{goal}
For Goal~\ref{prob:j1n}, we directly compute the spaces
$S_2(\Gamma_1(N), \eps)$ for each Dirichlet character $\eps$ of
modulus $N$. The main difficulty is linear algebra
over large cyclotomic fields. Recent work in progress of the PI on
algorithms for multimodular linear algebra over cyclotomic fields
will be very helpful (see \cite[Ch.~7]{stein:modform}).
\newpage
\bibliographystyle{amsalpha}
\addtolength{\itemsep}{-3pt}
\bibliography{biblio}
\end{document}