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Line breaks will be recognized. The Modular Degree, Congruence Primes and Multiplicity One \ShortTitle The Modular Degree and Congruences %%%%% Running title for odd numbered pages, ONE line, please. %%%%% If none is given, \Title will be used instead. \SubTitle %%%%% A possible subtitle goes here. \Author %%%%% Put here name(s) of authors. Line breaks will be recognized. Amod Agashe\\ Kenneth A. Ribet\\ William A. Stein \ShortAuthor Agashe, Ribet, Stein %%%%%% Running title for even numbered pages, ONE line, please. %%%%%% If none is given, \Author will be used instead. \EndTitle \Abstract %%%%% Put here the abstract of your manuscript. %%%%% Avoid macros and complicated TeX expressions, as this is %%%%% automatically translated and posted as an html file. The modular degree and congruence number are two fundamental invariants of an elliptic curve over the rational field. Frey and M{\"u}ller have asked whether these invariants coincide. Although this question has a negative answer, we prove a theorem about the relation between the two invariants: one divides the other, and the ratio is divisible only by primes whose squares divide the conductor of the elliptic curve. We discuss the ratio even in the case where the square of a prime does divide the conductor, and we study analogues of the two invariants for modular abelian varieties of arbitrary dimension. \EndAbstract \MSC %%%%% 2000 Mathematics Subject Classification: \EndMSC \KEY %%%%% Keywords and Phrases: \EndKEY %%%%% All 4 \Address lines below must be present. To center the last %%%%% entry, no empty lines must be between the following \Address %%%%% and \EndAddress lines. \Address %%%%% Address of first Author here \Address Amod Agashe Insert Current Address \Address Kenneth A. Ribet Insert Current Address \Address William A. Stein Department of Mathematics Harvard University Cambridge, MA 02138 {\tt was@math.harvard.edu} \EndAddress %% %% Make sure the last tex command in your manuscript %% before the first \end{document} is the command \Addresses %% %%---------------------Here the prologue ends--------------------------------- %%--------------------Here the manuscript starts------------------------------ \section{Introduction} Let~$E$ be an elliptic curve over~$\Q$. By \cite{breuil-conrad-diamond-taylor}, we may view~$E$ as an abelian variety quotient over $\Q$ of the modular Jacobian $J_0(N)$, where $N$ is the conductor of~$E$. After possibly replacing $E$ by an isogenous curve, we may assume that the kernel of the map $J_0(N)\to E$ is connected, i.e., that~$E$ is an {\em optimal quotient} of $J_0(N)$. Let $f_E = \sum a_n q^n \in S_2(\Gamma_0(N))$ be the newform attached to $E$. The {\em congruence number}~$\re$ of~$E$ is the largest integer such that there is an element $g =\sum b_n q^n \in S_2(\Gamma_0(N))$ with integer Fourier coefficients $b_n$ that is orthogonal to~$f_E$ with respect to the Peterson innner product, and congruent to~$f_E$ modulo~$\re$ (i.e., $a_n \equiv b_n\pmod{\re}$ for all~$n$). The {\em modular degree}~$\me$ is the degree of the composite map $X_0(N)\to J_0(N)\to E$, where we map $X_0(N)$ to $J_0(N)$ by sending $P\in X_0(N)$ to $[P]-[\infty] \in J_0(N)$. Section~\ref{congintro} is about relations between~$\re$ and~$\me$. For example, $\me \mid \re$. In \cite[Q.~4.4]{frey-muller}, Frey and M{\"u}ller~ asked whether $\re = \me$. We give examples in which $\re \neq \me$, then conjecture that for any prime $p$, $\ord_p(\re/\me) \leq \frac{1}{2}\ord_p(N)$. We prove this conjecture when $\ord_p(N)\leq 1$. In Section~\ref{sec:quotients}, we consider analogues of congruence primes and the modular degree for optimal quotients that are not necessarily elliptic curves; these are quotients of~$J_0(N)$ and $J_1(N)$ of any dimension associated to ideals of the relevant Hecke algebras. In Section~\ref{sec:main} we prove the main theorem of this paper, and in Section~\ref{sec:mult1} we give some new examples of failure of multiplicity one motivated by the arguments in Section~\ref{sec:main}. % For an introduction and the motivation for studying % the objects in the title of the paper, the reader % may read Sections~\ref{sec:elliptic} % and~\ref{sec:quotients}, skipping the proofs. \smallskip {\bf \noindent Acknowledgment.} The authors are grateful to A.~Abbes, R.~Coleman, B.~Conrad, J.~Cremona, H.~Lenstra, E.~de Shalit, B.~Edixhoven, L.~Merel, and R.~Taylor for several discussions and advice regarding this paper. \section{Congruence Primes and the Modular Degree} \label{sec:elliptic} Let~$N$ be a positive integer and let $X_0(N)$ be the modular curve over~$\Q$ that classifies isomorphism classes of elliptic curves with a cyclic subgroup of order~$N$. The Hecke algebra~$\T$ of level~$N$ is the subring of the ring of endomorphisms of $J_0(N)=\Jac(X_0(N))$ generated by the Hecke operators $T_n$ for all $n\geq 1$. Let~$f$ be a newform of weight~$2$ for~$\Gamma_0(N)$ with integer Fourier coefficients, and let $I_f$ be kernel of the homomorphism $\T\to \Z[\ldots, a_n(f), \ldots]$ that sends $T_n$ to $a_n$. Then the quotient $E = J_0(N)/I_f J_0(N)$ is an elliptic curve over~$\Q$. We call~$E$ the {\em optimal quotient} associated to~$f$. Composing the embedding $X_0(N)\hra J_0(N)$ that sends $\infty$ to~$0$ with the quotient map $J_0(N) \ra E$, we obtain a surjective morphism of curves $\phie: X_0(N) \ra E$. \begin{defi} The {\em modular degree} $\me$ of~$E$ is the degree of~$\phie$. \end{defi} \label{congintro} %The congruence number $\re$ and the modular degree $\me$ %are of great interest. Congruence primes have been studied by Doi, Hida, Ribet, Mazur and others (see, e.g.,~\cite[\S1]{ribet:modp}), and played an important role in Wiles's work~\cite{wiles} on Fermat's last theorem. Frey and Mai-Murty have observed that an appropriate asymptotic bound on the modular degree is equivalent to the $abc$-conjecture (see~\cite[p.544]{frey:ternary} and~\cite[p.180]{murty:congruence}). Thus, results that relate congruence primes and the modular degree are of great interest. \begin{thm}\label{thm:ribet_au} \label{ddivsr} Let $E$ be an elliptic curve over $\Q$ of conductor~$N$, with modular degree $\me$ and congruence number $\re$. Then $\me \mid \re$ and if $\ord_p(N)\leq 1$ then $\ord_p(\re) = \ord_p(\me)$. \end{thm} We will prove a generalization of Theorem~\ref{thm:ribet_au} in Section~\ref{sec:main} below. The divisibility $\me\mid \re$ was first discussed in~\cite[Th.~3]{zagier}, where it is attributed to the second author (Ribet); however in \cite{zagier} the divisibility was mistakenly written in the opposite direction. For some other expositions of the proof, see~\cite[Lem~3.2]{abbull} and~\cite{cojo-kani}. We generalize this divisibility in Proposition~\ref{ndivsm}. The second part of Theorem~\ref{thm:ribet_au}, i.e., that if $\ord_p(N) \leq 1$ then $\ord_p(\re) = \ord_p(\me)$, follows from the more general Theorem~\ref{thm:ribet_gen} below. %\edit{I made this change. --Amod} %in more generality in in Section~\ref{sec:proof_ribet} below. Note that \cite[Prop.~3.3--3.4]{abbull} implies the weaker statement that if $p\nmid N$ then $\ord_p(\re)=\ord_p(\me)$, since \cite[Prop.~3.3]{abbull} implies $$\ord_p(\re) - \ord_p(\me) = \ord_p(\#\mathcal{C}) - \ord_p(\ce) - \ord_p(\#\mathcal{D}),$$ and by \cite[Prop.~3.4]{abbull} $\ord_p(\#\mathcal{C}) =0$. (Here $\ce$ is the Manin constant of $E$, which is an integer by results of Edixhoven and Katz-Mazur; see e.g., \cite{ars} for more details.) Frey and M{\"u}ller~\cite[Ques.~4.4]{frey-muller} asked whether $\re = \me$ in general. After implementing an algorithm to compute $\re$ in Magma \cite{magma}, we quickly found that the answer is no. The counterexamples at conductor $N\leq 144$ are given in Table~\ref{table:moddeg}, where the curve is given using the notation of \cite{cremona:alg}: \begin{table}\caption{Differing Modular Degree and Congruence Number\label{table:moddeg}} \begin{center} \begin{tabular}{|l|l|l|}\hline Curve & $\me$ & $\re$\\\hline 54B1 & 2 & 6\\\hline 64A1 & 2 & 4 \\\hline 72A1 & 4 & 8 \\\hline 80A1 & 4 & 8 \\\hline 88A1 & 8 & 16 \\\hline 92B1 & 6 & 12\\\hline 96A1 & 4 & 8 \\\hline 96B1 & 4 & 8 \\\hline \end{tabular} \begin{tabular}{|l|l|l|}\hline Curve & $\me$ & $\re$\\\hline 99A1 & 4 & 12\\\hline 108A1 & 6 & 18\\\hline 112A1 & 8 & 16\\\hline 112B1 & 4 & 8\\\hline 112C1 & 8 & 16\\\hline 120A1 & 8 & 16\\\hline 124A1 & 6 & 12\\\hline 126A1 & 8 & 24\\\hline \end{tabular} \begin{tabular}{|l|l|l|}\hline Curve & $\me$ & $\re$\\\hline 128A1 & 4 & 32\\\hline 128B1 & 8 & 32\\\hline 128C1 & 4 & 32\\\hline 128D1 & 8 & 32\\\hline 135A1 & 12 & 36\\\hline 144A1 & 4 & 8 \\\hline 144B1 & 8 & 16 \\\hline &&\\\hline \end{tabular} \end{center} \end{table} %$$54, 64, 72, 80, 88, 92, 96, 99, 108, 112, 120, 124, 126, 128, 135, %\text{ and } 144.$$ For example, the elliptic curve 54B1, given by the equation $y^2 + xy + y = x^3 - x^2 + x - 1$, has $\re=6$ and $\me=2$. To see explicitly that $3 \mid \re$, observe that the newform corresponding to~$E$ is $f=q + q^2 + q^4 - 3q^5 - q^7 + \cdots$ and the newform corresponding to $X_0(27)$ if $g=q - 2q^4 - q^7 + \cdots$, so $g(q) + g(q^2)$ appears to be congruent to~$f$ modulo~$3$. To prove this congruence, we checked it for $18$ Fourier coefficients, where the sufficiency of precision to degree $18$ was determined using \cite{sturm:cong}. % In accord with Theorem~\ref{thm:ribet_au}, %since $\ord_3(\re) \neq \ord_3(\ce)$, we have $\ord_3(54)\geq 2$. In our computations, there appears to be no absolute bound on the~$p$ that occur. For example, for the curve 242B1 of conductor $N=2\cdot 11^2$ we have\footnote{The curve 242a1 in ``modern notation.''} $$ \me = 2^4 \neq \re = 2^4\cdot 11. $$ We propose the following replacement for Question~4.4 of \cite{frey-muller}: \begin{conj}\label{conj:rm} Let~$E$ be an optimal elliptic curve of conductor~$N$ and~$p$ be any prime. Then $$ \ord_p\left(\frac{\re}{\me}\right) \leq \frac{1}{2}\ord_p(N). $$ \end{conj} We verified Conjecture~\ref{conj:rm} using Magma for every optimal elliptic curve quotient of $J_0(N)$, with $N\leq 539$. If $p\geq 5$ then $\ord_p(N)\leq 2$, so a special case of the conjecture is $$ \ord_p\left(\frac{\re}{\me}\right) \leq 1\qquad\text{ for any }p\geq 5. $$ \begin{rmk} It is often productive to parametrize elliptic curves by $X_1(N)$ instead of $X_0(N)$ (see, e.g., \cite{stevens:param} and \cite{MR2135139}). Suppose $E$ is an optimal quotient of $X_1(N)$, let $m_E'$ be the degree of the modular parametrization, and let $r_E'$ be the $\Gamma_1(N)$-congruence number, which is defined as above but with $S_2(\Gamma_0(N))$ replaced by $S_2(\Gamma_1(N))$. For the optimal quotient of $X_1(N)$ isogenous to 54B1, we find using Magma that $m_E' = 18$ and $r_E'=6$. Thus the equality $m_E'=r_E'$ fails, and the analogous divisibility $m_E'\mid r_E'$ no longer holds. Also, for a curve of conductor $38$ we have $m_E'=18$ and $r_E'=6$, so equality need not hold even if the level is square free. We hope to investigate this in a future paper. %> N := 38; D := ND(NS(CS(ModularSymbols(Gamma1(N))))); %> ModularDegree(D[1]); CongruenceModulus(D[1]); %18 %6 % %> N := 54; D := ND(NS(CS(ModularSymbols(Gamma1(N))))); %> ModularDegree(D[1]); CongruenceModulus(D[1]); %18 %18 %> ModularDegree(D[2]); CongruenceModulus(D[2]); %18 %6 \end{rmk} %\subsection{Proof of Theorem~\ref{thm:ribet_au}}\label{sec:proof_ribet} \section{Modular abelian varieties of arbitrary dimension} \label{sec:quotients} For $N\geq 4$, let~$\Gamma$ be a fixed choice of either~$\Gamma_0(N)$ or~$\Gamma_1(N)$, let~$X$ be the modular curve over~$\Q$ associated to~$\Gamma$, and let~$J$ be the Jacobian of~$X$. Let~$I$ be a {\em saturated} ideal of the corresponding Hecke algebra $\T\subset\End(J)$, so $\T/ I$ is torsion free. Then $A = A_I = J/IJ$ is an optimal quotient of~$J$ since $IJ$ is an abelian subvariety. \begin{defi} If~$f=\sum a_n(f)q^n \in S_2(\Gamma)$ and $I_f=\ker(\T\to \Z[\ldots,a_n(f),\ldots])$, then $A=A_f=J/I_f J$ is the {\em newform quotient} associated to~$f$. It is an abelian variety over~$\Q$ of dimension equal to the degree of the field $\Q(\ldots,a_n(f),\ldots)$. \end{defi} In this section, we generalize the notions of the congruence number and the modular degree to quotients~$A=A_I$, and state a theorem relating the two numbers, which we prove in Sections~\ref{sec:firstpart}--\ref{sec:secondpart}. Let $\phi_2$ denote the quotient map $J \ra A$. By Poincare reducibility over $\Q$ there is a unique abelian subvariety $A^{\vee}$ of $J$ that projects isogenously to the quotient $A$ (equivalently, which has finite intersection with $\ker(\phi_2)$), and so by Hecke equivariance of $J \to A$ it follows that $A^{\vee}$ is $\T$-stable. Let $\phi$ be the composite isogeny $$ \phi: \Adual \stackrel{\po}{\lra} J \stackrel{\pt}{\lra} A. $$ \begin{rmk} Note that $A^{\vee}$ is the dual abelian variety of $A$. More generally, if~$C$ is any abelian variety, let $C^{\vee}$ denote the dual of~$C$. There is a canonical principal polarization $J \cong \Jdual$, and dualizing $\phi_2$, we obtain a map $\phi_2^\vee: \Adual \ra \Jdual$, which we compose with $\theta^{-1}: \Jdual \cong J$ to obtain a map $\po: \Adual \ra J$. Note also that $\vphi$ is a polarization (induced by pullback of the theta divisor). \end{rmk} %\begin{prop} \label{modular:isogeny0} %The map $\phi$ is a polarization. %\end{prop} % \begin{proof} % Let $i$ be the injection $\phi_2^{\vee}:\Adual \ra \Jdual$, and let % $\Theta$ denote the theta divisor. From the definition of the % polarization attached to an ample divisor, we see that the map~$\phi$ % is induced by the pullback $i^*(\Theta)$ of the theta divisor. The % theta divisor is effective, and hence so is $i^*(\Theta)$. %By~\cite[\S6, Application~1, p. 60]{mumford:av}, $\ker \phi$ is %finite. Since the dimensions of $A$ and~$\Adual$ are the same, $\phi$ %is an isogeny. % Since $\Theta$ is ample, some power of it is % very ample. Then the pullback of this very ample power by~$i$ is again % very ample, and hence a power of $i^*(\Theta)$ is very ample, so % $i^*(\Theta)$ is ample (by~\cite[II.7.6]{hartshorne:ag}). % \end{proof} The {\em exponent} of a finite group~$G$ is the smallest positive integer~$n$ such that every element of~$G$ has order dividing~$n$. \begin{defi}\label{defi:modular} The {\em modular exponent} of~$A$ is the exponent of the kernel of the isogeny~$\phi$, and the {\em modular number} of~$A$ is the degree of~$\phi$. \end{defi} We denote the modular exponent of~$A$ by~$\nAe$ and the modular number by~$\nA$. When~$A$ is an elliptic curve, the modular exponent is equal to the modular degree of~$A$, and the modular number is the square of the modular degree (see, e.g.,~\cite[p.~278]{abbull}). %(see \cite[p.~276]{abbull}). %When~$A$ is an elliptic curve, $\na$ is just the %modular degree of~$A$. If~$R$ is a subring of~$\C$, let $S_2(R)=S_2(\Gamma;R)$ denote the subgroup of~$S_2(\Gamma)$ consisting of cups forms whose Fourier expansions at the cusp~$\infty$ have coefficients in~$R$. (Note that $\Gamma$ is fixed for this whole section.) Let $S_2(\Gamma;\Z)[I]^{\perp}$ denote the orthogonal complement of $S_2(\Gamma;\Z)[I]$ in $S_2(\Gamma;\Z)$ with respect to the Petersson inner product. The following is well known, but we had difficulty finding a good reference. \begin{prop} The group $S_2(\Gamma;\Z)$ is of finite rank as a $\Z$-module. \end{prop} \begin{proof} Using the standard pairing between $\T$ and $S_2(\Gamma,\Z)$ (see also~\cite[Theorem~2.2]{ribet:modp}) we see that $S_2(\Gamma,\Z) \isom \Hom(\T,\Z)$. Thus $S_2(\Gamma,\Z)$ is finitely generated over~$\Z$ if and only if~$\T$ is finitely generated over~$\Z$. But the action of~$\T$ on $\H_1(J,\Z)$ is a faithful representation that embeds~$\T$ into $\Mat_{2d}(\Z) \isom \Z^{(2d)^2}$. But~$\Z$ is Noetherian, so~$\T$ is finitely generated over~$\Z$. \end{proof} \begin{defi}\label{def:congexp} The exponent of the quotient group \begin{equation}\label{eqn:congexp} \frac{S_2(\Gamma; \Z)} { S_2(\Gamma; \Z)[I] + S_2(\Gamma;\Z)[I]^{\perp}} \end{equation} is the {\em congruence exponent} $\rAe$ of~$A$ and its order is the {\em congruence number} $\rA$. \end{defi} \begin{rmk} Note that $S_2(\Gamma,\Z)\tensor_\Z R = S_2(\Gamma,R)$; see, e.g., the discussion in \cite[\S12]{diamond-im}. Thus the analogue of Definition~\ref{def:congexp} with $\Z$ replaced by an algebraic integer ring (or even $\Zbar$) gives a torsion module whose annihilator ideal meets~$\Z$ in the ideal generated by the congruence exponent. \end{rmk} Our definition of~$\rA$ generalizes the definition in Section~\ref{congintro} when~$A$ is an elliptic curve (see \cite[p.~276]{abbull}), and the following generalizes Theorem~\ref{thm:ribet_au}: \begin{thm}\label{thm:ribet_gen} If $f \in S_2(\C)$ is a newform, then \begin{itemize} \item[(a)] We have $\nAfe \mid \rAfe$, and \item[(b)] If $p^2 \nmid N$, then $\ord_p(\rAfe) = \ord_p(\nAfe)$. \end{itemize} % $p \nmid \frac{\rAfe}{\nAfe}$. \end{thm} %We give the proof of this theorem in the next two sections. %The rest of the section is devoted to proving Proposition~\ref{ndivsm} %below, which asserts that if~$f$ is a newform, then $\nAfe \mid %\rAfe$. \begin{rmk}\label{rem:24} When $A_f$ is an elliptic curve, Theorem~\ref{thm:ribet_gen} implies that the modular degree divides the congruence number (since for an elliptic curve the modular degree and modular exponent are the same), i.e., $\sqrt{\nAf} \mid \rAf$. In general, the divisibility $\nAf\mid r^2_{A_f}$ need not hold. For example, there is a newform of degree $24$ in $S_2(\Gamma_0(431))$ such that $$\nAf = (2^{11}\cdot 6947)^2 \,\,\nmid\,\, r^2_{A_f} = (2^{10}\cdot 6947)^2.$$ Note that $431$ is prime and mod~$2$ multiplicity one fails for $J_0(431)$ (see \cite{kilford}). %The following Magma session illustrates how to verify the above %assertion about $\nAf$ and $\rAf$. The commands are parts of Magma %V2.11 or greater. \vspace{-1ex} %{\small %\begin{verbatim} % > A := ModularSymbols("431F"); % > Factorization(ModularDegree(A)); % [ <2, 11>, <6947, 1> ] % > Factorization(CongruenceModulus(A)); % [ <2, 10>, <6947, 1> ] %\end{verbatim} %} \end{rmk} \section{Proof of the Main Theorem}\label{sec:main} In this section we prove Theorem~\ref{thm:ribet_gen}. We continue using the notation introduced so far. \subsection{Proof of Theorem~\ref{thm:ribet_gen} (a)} \label{sec:firstpart} We begin with a remark about compatibilities. In general, the polarization of~$J$ induced by the theta divisor need not be Hecke equivariant, because if~$T$ is a Hecke operator on~$J$, then on~$\Jdual$ it acts as $W_N T W_N$, where $W_N$ is the Atkin-Lehner involution (see e.g.,~\cite[Rem.~10.2.2]{diamond-im}). However, on~$J^{\rm new}$ the action of the Hecke operators commutes with that of~$W_N$, so if the quotient map $J \ra A$ factors through~$J^{\rm new}$, then the Hecke action on~$\Adual$ induced by the embedding $\Adual \to J^{\vee}$ and the action on $\Adual$ induced by $\phi_1:\Adual\to{}J$ are the same. Hence $\Adual$ is isomorphic to $\po(\Adual)$ as a $\T$-module. Recall that $f$ is a newform, $I_f = {\rm Ann}_\T (f)$, and $J=J_0(N)$. Let $B = I_fJ$, so that $\Adual+B=J$, and $J/B\isom A$. The following lemma is well known, but we prove it here for the convenience of the reader. \begin{lem}\label{lem:homzero} $\Hom_\Q(\Adual,B)=0$. \end{lem} \begin{proof} % Suppose there were a nonzero element of $\Hom_\Q(\Adual,B)$. Since % $A$ is simple, for all~$\ell$ the Tate module % $V_{\ell}(\Adual)=\Q\tensor\varprojlim_n \Adual[\ell^n]$ would % be a factor of $V_{\ell}(B)$. % Thus the characteristic polynomial % The Eichler-Shimura relation then implies that the characteristic % polynomial of each % One could then extract almost all % prime-indexed coefficients of the corresponding eigenforms from the % Tate modules, which would violate multiplicity one for systems of % Hecke eigenvalues . Pick a prime $\ell$. Then $\Qbar_{\ell} \tensor V_{\ell} (J)^{\ss}$ as a $\Qbar_{\ell}[G_\Q]$-module is a direct sum of copies of the representations $\rho_g$ as $g$ ranges through all normalized eigenforms of weight $2$ and level $N$ with coefficients in $\Qbar$; by a well-known result of the second author, these representations are absolutely irreducible. Now since~$f$ is a newform and $A^{\vee} \to A$ is an isogeny, $\Qbar_{\ell} \tensor V_{\ell}(A^{\vee})^{\ss}$ is a direct sum of copies of $\rho_{\sigma(f)}$ as $\sigma$ ranges over all embeddings of $K_f$ into $\Qbar$. Thus, by the analytic theory of multiplicity one (see \cite[Cor.~3, pg.~300]{winnie:newforms}), the Galois modules $V_{\ell}(A^{\vee})$ and $V_{\ell}(B) = V_{\ell}(J)/V_{\ell}(A^{\vee})$ share no common Jordan-H\"older factors even when coefficients are extended to $\Qbar_{\ell}$, so $\Hom_\Q(A',B) = 0$. \end{proof} Let $\T_1$ be the image of~$\T$ in $\End(\Adual)$, and let $\T_2$ be the image of $\T$ in $\End(B)$. We have the following commutative diagram with exact rows: \begin{equation}\label{eqn:diagram} \xymatrix@=2em{ 0\ar[r] & {\T} \ar[r]\ar[d] & {\T_1\oplus \T_2} \ar[r]\ar[d] & {\displaystyle \frac{\T_1 \oplus \T_2}{\T}}\ar@{.>}[d]\ar[r] & 0\\ 0\ar[r] & {\End(J)} \ar[r] & {\End(\Adual)\oplus\End(B)} \ar[r] & {\displaystyle \frac{\End(\Adual)\oplus\End(B)}{\End(J)}}\ar[r] & 0.\\ } \end{equation} Let $$ e=(1,0)\in \T_1 \oplus \T_2, $$ and let $e_1$ and $e_2$ denote the images of~$e$ in the groups $(\T_1 \oplus \T_2)/\T$ and $(\End(\Adual) \oplus \End(B))/\End(J)$, respectively. It follows from Lemma~\ref{lem:homzero} that the two quotient groups on the right hand side of (\ref{eqn:diagram}) are finite, so~$e_1$ and~$e_2$ have finite order. Note that because $e_2$ is the image of $e_1$, the order of $e_2$ is a divisor of the order of $e_1$. %this will be used in the proof of Proposition~\ref{ndivsm} %below. The {\em denominator} of any $\vphi\in\End(J)\tensor\Q$ is the smallest positive integer~$n$ such that $n\vphi\in\End(J)$. % Explicitly, the denominator of~$\vphi$ is the least common multiples % of the denominators of the entries of any matrix that represents the % action of $\vphi$ on the lattice $\H_1(J,\Z)$. Let $\piAd, \piB \in \End(J)\tensor\Q$ be projection onto $\Adual$ and $B$, respectively. Note that the denominator of $\piAd$ equals the denominator of $\piB$, since $\piAd + \piB = 1_J$, so that $\piB = 1_J - \piAd$. \begin{lem}\label{lem:ord_e2} The element $e_2\in (\End(\Adual) \oplus \End(B))/\End(J)$ defined above has order $\nAe$. \end{lem} \begin{proof} Let $n$ be the order of $e_2$, so~$n$ is the denominator of $\piAd$, which, as mentioned above, is also the denominator of $\piB$. We want to show that $n$ is equal to~$\nAe$, the exponent of $\Adual\cap B$. Let $i_{\Adual}$ and $i_B$ be the embeddings of $\Adual$ and $B$ into $J$, respectively. Then $$\vphi = (n\piAd,n\piB)\in\Hom(J,\Adual\times B)$$ and $\vphi\circ (i_{\Adual} + i_B) = [n]_{\Adual\times B}.$ We have an exact sequence $$ 0\to \Adual\cap B\xra{x\mapsto (x,-x)}\Adual\times B \xra{i_{\Adual} + i_B} J \to 0. $$ Let $\Delta$ be the image of $\Adual\cap B$. Then by exactness, $$ [n]\Delta = (\vphi\circ (i_{\Adual} + i_B))(\Delta) = \vphi\circ ((i_{\Adual} + i_B)(\Delta)) = \vphi(\{0\}) = \{0\}, $$ so $n$ is a multiple of the exponent~$\nAe$ of $\Adual\cap B$. To show the opposite divisibility, consider the commutative diagram $$ \xymatrix@=4em{ 0 \ar[r] & {\Adual \cap B} \ar[r]^{x\mapsto (x,-x)}\ar[d]^{[\nAe]}& {\Adual \times B}\ar[d]^{([\nAe],0)} \ar[r]& J \ar[r]\ar@{.>}[d]^{\psi} & 0\\ 0 \ar[r] & {\Adual \cap B} \ar[r]^{x\mapsto (x,-x)}& {\Adual \times B} \ar[r]& J \ar[r] & 0, } $$ where the middle vertical map is $(a,b)\mapsto (\nAe a,0)$ and the map~$\psi$ exists because $[\nAe](\Adual\cap B)=0$. But $\psi = \nAe \piAd$ in $\End(J)\tensor\Q$. This shows that $\nAe \piAd \in \End(J)$, i.e., that $\nAe$ is a multiple of the denominator~$n$ of $\piAd$. \end{proof} Let $\Ext^1 = \Ext^1_{\Z}$ denote the first $\Ext$ functor in the category of $\Z$-modules. \begin{lem}\label{lem:compare_with_dual} The group $(\T_1 \oplus \T_2)/\T$ is isomorphic to the quotient (\ref{eqn:congexp}) in Definition~\ref{def:congexp}, so $\rA = \#((\T_1 \oplus \T_2)/\T)$ and $\rAe$ is the exponent of $(\T_1 \oplus \T_2)/\T$. More precisely, $\Ext^1((\T_1 \oplus \T_2)/\T,\Z)$ is isomorphic as a $\T$-module to the quotient (\ref{eqn:congexp}). \end{lem} \begin{proof} Apply the $\Hom(-,\Z)$ functor to the first row of (\ref{eqn:diagram}) to obtain a three-term exact sequence \begin{equation}\label{eqn:dualseq} 0 \to \Hom(\T_1\oplus \T_2,\Z) \to \Hom(\T,\Z) \to \Ext^1((\T_1\oplus\T_2)/\T,\Z) \to 0. \end{equation} %The term $\Ext^1(\T_1\oplus \T_2,\Z)$ is $0$ is because %$\Ext^1(M,\Z)=0$ for any finitely generated free abelian group. Also, %$\Hom((\T_1\oplus\T_2)/\T,\Z)=0$ since $(\T_1\oplus\T_2)/\T$ is %torsion. There is a $\T$-equivariant bilinear pairing $\T\times S_2(\Z)\to\Z$ given by $(t,g)\mapsto a_1(t(g))$, which is perfect by \cite[Lemma~2.1]{abbull} (see also~\cite[Theorem~2.2]{ribet:modp}). Using this pairing, we transform (\ref{eqn:dualseq}) into an exact sequence $$ 0 \to S_2(\Z)[I_f] \oplus S_2(\Gamma;\Z)[I_f]^{\perp} \to S_2(\Z) \to \Ext^1((\T_1\oplus\T_2)/\T,\Z) \to 0 $$ of $\T$-modules. Here we use that $\Hom(\T_2,\Z)$ is the unique saturated Hecke-stable complement of $S_2(\Z)[I_f]$ in $S_2(\Z)$, hence must equal $S_2(\Z)[I_f]^{\perp}$. Finally note that if~$G$ is any finite abelian group, then $\Ext^1(G,\Z)\approx G$ as groups, which gives the desired result. \end{proof} \begin{lem}\label{lem:ord_e1} The element $e_1 \in (\T_1 \oplus \T_2)/\T$ has order $\rAe$. \end{lem} \begin{proof} By Lemma~\ref{lem:compare_with_dual}, the lemma is equivalent to the assertion that the order~$r$ of~$e_1$ equals the exponent of $M=(\T_1 \oplus \T_2)/\T$. Since $e_1$ is an element of~$M$, the exponent of~$M$ is divisible by~$r$. To obtain the reverse divisibility, consider any element $x$ of~$M$. Let $(a,b)\in\T_1\oplus \T_2$ be such that its image in~$M$ is~$x$. By definition of $e_1$ and~$r$, we have $(r,0)\in\T$, and since $1=(1,1)\in\T$, we also have $(0,r)\in\T$. Thus $(\T{}r,0)$ and $(0,\T{}r)$ are both subsets of $\T$ (i.e., in the image of $\T$ under the map $\T\to\T_1\oplus \T_2$), so $r(a,b) =(ra,rb)=(ra,0)+(0,rb)\in \T$. This implies that the order of~$x$ divides~$r$. Since this is true for every $x \in M$, we conclude that the exponent of~$M$ divides~$r$. \end{proof} \begin{prop} \label{ndivsm} If $f \in S_2(\C)$ is a newform, then $\nAfe \mid \rAfe$. \end{prop} \begin{proof} Since~$e_2$ is the image of~$e_1$ under the right-most vertical homomorphism in (\ref{eqn:diagram}), the order of~$e_2$ divides that of~$e_1$. Now apply Lemmas~\ref{lem:ord_e2} and \ref{lem:ord_e1}. \end{proof} This finishes the proof of the first statement in Theorem~\ref{thm:ribet_gen}. \subsection{Proof of Theorem~\ref{thm:ribet_gen} (b)} \label{sec:secondpart} Let $\T'$ be the saturation of $\T=\Z[\ldots, T_n,\ldots]$ in $\End(J_0(N))$, i.e., the set of elements of $\End(J_0(N)) \tensor \Q$ some positive multiple of which lie in~$\T$. %so %$$ % \T' = (\T\tensor\Q) \cap \End(J_0(N)), %$$ %where the intersection is taken inside $\End(J_0(N))\tensor\Q$. The quotient $\T'/\T$ is a finitely generated abelian group because both $\T$ and $\End(J_0(N))$ are finitely generated over~$\Z$. Since $\T'/\T$ is also a torsion group, it is finite. In Section~\ref{sec:multone}, we will give some conditions under which $\T$ and~$\T'$ agree locally at maximal ideal of~$\T$. In Section~\ref{sec:degrees}, we will explain how the ratio of the congruence number to the modular degree is closely related to the order of~$\T'/\T$, and finally deduce that this ratio is $1$ (for quotients associated to newforms) locally at a prime~$p$ such that $p^2 \nmid N$. \subsubsection{Multiplicity One} \label{sec:multone} %Suppose for the moment that $M=1$, so $p=pM$. Fixt an integer $N$ and a prime $p\mid N$. Suppose for a moment that $N$ is prime, so $p=N$. In \cite{mazur:eisenstein}, Mazur proves that $\T=\T'$; he combines this result with the equality $$ \T\tensor\Q = \End(J_0(p)) \tensor\Q, $$ to deduce that $\T=\End(J_0(p))$. This result, combined with Ribet's result \cite{ribet:endo} or \cite{ribet:endalg} to the effect that $\T\tensor\Q = (\End_{\Qbar} J_0(N)) \tensor \Q$, shows that~$\T$ is the full ring of endomorphisms of $J_0(N)$ over $\Qbar$. When $N$ is no longer necessarily prime, the method of \cite{mazur:eisenstein} shows that $\T$ and $\T'$ agree locally at a maximal ideal $\m$ of $\T$ that satisfies a simple condition involving differentials form mod $\ell$, where $\ell$ is the residue characteristic of $\m$. %has dimension at most one. \comment{ $\Supp_{\T}(\T'/\T)$ contains no maximal ideal $\m$ of~$\T$ for which his space $\H^0(X_0(pM)_{\Fell},\Omega)[\m]$ has dimension $\leq 1$. (Here $\ell$ is the residue characteristic of $\m$.) In other words, multiplicity one for $\H^0(X_0(pM)_{\Fell}, \Omega)[\m]$ implies that $\T$ and $\T'$ agree at~$\m$. Mazur's argument (see \cite[pg.~95]{mazur:eisenstein}) is quite general; it relies on a multiplicity $1$ statement for spaces of differentials in positive characteristic (see \cite[Prop.~9.3, pg.~94]{mazur:eisenstein}). } For the sake of completeness, we state and prove a lemma that can be easily extracted from~\cite{mazur:eisenstein}. Let $m$ be the largest square dividing $N$ and let $R = \Z[\frac{1}{m}]$. Let $X_0(N)_{R}$ denote the minimal regular model of $X_0(N)$ over~$R$. %Let $\m$ be a maximal ideal of the Hecke algebra of %residue characteristic~$\ell$ and suppose $\ell^2 \nmid N$. Let $\Omega=\Omega_{X_0(N)/R}$ denote the sheaf of regular differentials on $X_0(N)_{R}$, as in~\cite[\S2(e)]{mazur:rational}. If~$\ell$ is a prime such that $\ell^2 \nmid N$, then $X_0(N)_{\F_\ell}$ denotes the special fiber of $X_0(N)_{R}$ at the prime~$\ell$. \comment{ His method shows in the general case (where~$M$ is no longer constrained to be~$1$) that $\Supp_{\T}(\T'/\T)$ contains no maximal ideal $\m$ of~$\T$ for which his space $\H^0(X_0(pM)_{\Fell},\Omega)[\m]$ has dimension $\leq 1$. (Here $\ell$ is the residue characteristic of $\m$.) In other words, multiplicity one for $\H^0(X_0(pM)_{\Fell}, \Omega)[\m]$ implies that $\T$ and $\T'$ agree at~$\m$. We record this fact as a lemma (see also Section~\ref{sec:dataind} for related data). } \begin{lem}[Mazur]\label{lem:m1} Let $\m$ be a maximal ideal of $\T$ of residue characteristic~$\ell$ such that $\ell^2 \nmid N$. Suppose that $$ \dim_{\T/\m} \H^0(X_0(N)_{\Fell},\Omega)[\m] \leq 1. $$ Then $\T$ and $\T'$ agree locally at~$\m$. %$\m$ is not in the support of $\T'/\T$. \end{lem} \begin{proof} Let $M$ denote the group $H^1(X_0(N)_R, \OO_{X_0(N)})$, where $\OO_{X_0(N)}$ is the structure sheaf of $X_0(N)$. As explained in~\cite[p.~95]{mazur:eisenstein}, we have an action of $\EJ$ on~$M$, and the action of $\T$ on~$M$ via the inclusion $\T \subseteq \EJ$ is faithful, so likewise for the action by $\T'$. Hence we have an injection $\phi: \T' \hookrightarrow {\rm End}_{\T} M$. % is a free module over~$\T$. Suppose~$\m$ is a maximal ideal of~$\T$ that satisfies the hypotheses of the lemma. To prove that $\T_\m=\T_\m'$ it suffices to prove the following claim:\\ \noindent {\em Claim:} The map~$\phi|_{\T}$ is surjective locally at~$\m$. \begin{proof} By Nakayama's lemma, to show that $M$ is generated as a single element over~$\T$ locally at~$\m$, %to show that $\phi$ is surjective, i.e., %to show that $M \tensor \T_\m$ is generated by a single element over %$\T\tensor \T_\m$, it suffices to check that the dimension of the ${\T/\m}\,$-vector space $M / \m M$ is at most one. % for each maximal ideal~$\m$ dividing~$\ell$ in~$\T$. Since \mbox{$\ell^2 \nmid N$}, %$H^1(X_0(N)_{\F_\ell}, \OO) / \m H^1(X_0(N)_{\F_\ell}, \OO)$ $M/ \m M$ is dual to $H^0(X_0(N)_{\F_\ell}, \Omega) [\m]$ (see, e.g.,~\cite[\S2]{mazur:rational}). Since we are assuming that ${\rm dim}_{\T/\m} H^0(X_0(N)_{\F_\ell}, \Omega) [\m] \leq 1$, we have ${\rm dim}_{\T/\m} (M/ \m M) \leq 1$, which proves the claim. \end{proof} %We shall use the subscript~$(\m)$ to denote localization at~$\m$. %Thus $\Mm$ is free of rank one over~$\Tm$. The %composite $\psi: \Tpm \ra {\rm End}_{\Tm} (\Mm) %\stackrel{{\phi}^{-1}}{\ra} \Tm$ gives a section of the inclusion %$\Tm \hookrightarrow \Tpm$. %Let $x \in \Tpm$, and let $n$ be an integer such that $nx \in \Tm$. %Let $y = \psi(x) \in \Tm$. Then $nx = \psi ( \phi (nx) ) = \psi(nx) %= n \psi(x) = ny$. Since $\Tm$ is torsion-free, this means that %$x = y \in \Tm$. Thus $\Tm = \Tpm$, as was to be shown. \comment{ Thus $M \tensor \T_\m$ is free of rank one over~$\T_\m$. The composite $\EJ \tensor \T_\m \ra {\rm End}_{\T_\m} (M \tensor \T_\m) \stackrel{{\phi}^{-1}}{\ra} \T_\m$ gives a section of the inclusion $\T_\m \hookrightarrow \EJ \tensor \T_\m$. This shows that $\T_\m$ is saturated in $\EJ \tensor \T_\m$, i.e., that $\T$ and $\T'$ agree locally at~$\m$. } \end{proof} If $\m$ is a maximal ideal of the Hecke algebra~$\T$ of residue characteristic~$\ell$, we say that $\m$ satisfies {\em multiplicity one for differentials} if $$ \dim (\H^0(X_0(N)_{\F_\ell},\Omega)[\m]) \leq 1. $$ By Lemma~\ref{lem:m1}, multiplicity one for $\H^0(X_0(N)_{\Fell}, \Omega)[\m]$ implies that $\T$ and $\T'$ agree at~$\m$. There is quite a bit of literature on the question of multiplicity~$1$ for $\H^0(X_0(N)_{\Fell},\Omega)[\m]$. The easiest case is that~$\ell$ is prime to the level $N$: \begin{lem}[Mazur]\label{lem:m_ell} If $\m$ is a maximal ideal of $\T$ of residue characteristic~$\ell$ such that $\ell \nmid N$, then $$\dim_{\T/\m} \H^0(X_0(N)_{\Fell},\Omega)[\m] \leq 1.$$ %If $\ell \nmid pM$, then~$\ell\nmid \#(\T'/\T)$. \end{lem} \begin{proof} Mazur deduces this lemma from injectivity of the $q$-expansion map. The reader may find the following alternative approach to part of the argument easier to follow than the one on p.~95 of \cite{mazur:eisenstein}. We have an $\Fell$-vector space that embeds in $\Fell[[q]]$, for example a space~$V$ of differentials that is killed by a maximal ideal $\m$. This space is a $\T/\m$-vector space, and we want to see that its dimension over $\T/\m$ is at most~$1$. Mazur invokes tensor products and eigenvectors; alternatively, we note that~$V$ embeds in $\Hom_{\Fell}(\T/\m,\Fell)$ via the standard duality that sends~$v\in V$ to the linear form whose value on a Hecke operator~$T$ is the $q$th coefficient of $v{|T}$. The group $\Hom_{\Fell}(\T/\m,\Fell)$ has the same size as $\T/\m$, which completes the argument because $\Hom_{\Fell}(\T/\m,\Fell)$ has dimension $1$ as a $\T/\m$-vector space. \end{proof} % proves that %$$\dim_{\T/\m} \H^0(X_0(pM)_{\Fell},\Omega)[\m] \leq 1$$ for %all $\m\mid \ell$. Now apply Lemma~\ref{lem:m1} %\edit{Is there a problem if $\ell=2$? How do Lloyd Kilford's examples %fit into this, where I guess $N=1$ and $\ell=2$ and multiplicity %one in $J_0(p)$ fails. Is it still OK in Mazur's %differentials? -WAS} In the context of Mazur's paper, where the level~$N$ is prime, we see from Lemma~\ref{lem:m_ell} that $\T$ and $\T'$ agree away from~$N$. Locally at $N$, Mazur proved that $\T=\T'$ by an analogue of the arguments that he used away from $N$; see Chapter II of \cite{mazur:eisenstein} (and especially Prop.~9.4 and 9.5 of that chapter) as well as \cite{mazur-ribet}, where these arguments are taken up in a context where the level is no longer necessarily prime (and where one works locally at a prime whose square does not divide the level). %At~$N$, we can still use the $q$-expansion principle %because of the arguments in \cite[Ch.II~\S4]{mazur:eisenstein}. Thus in the prime level case, $\T=\T'$, as we asserted above. Now\label{NPnotation} let $p$ be a prime such that $p \parallel N$, and let $M = N/p$. The question of multiplicity $1$ at $p$ for $\H^0(X_0(pM)_{\Fp}, \Omega)[\m]$ is discussed in \cite{mazur-ribet}, where the authors establish multiplicity~$1$ for maximal ideals $\m\mid p$ for which the associated mod~$p$ Galois representation is irreducible and {\em not} $p$-old. (A representation of level $pM$ is $p$-old if it arises from $S_2(\Gamma_0(M))$.) If~$\m$ is a maximal ideal of~$\T$ of residue characteristic~$\ell$, then we say that~$\m$ is ordinary if $T_\ell \not\in \m$ (note that $T_\ell$ is often denoted $U_\ell$ if $\ell \mid N$). For our purposes, the following lemma is convenient: \begin{lem}[Wiles]\label{lem:wiles} If $\m$ is an ordinary maximal ideal of $\T$ of characteristic~$p$, then $$ \dim_{\T/\m} \H^0(X_0(pM)_{\Fp},\Omega)[\m] \leq 1. $$ %and $\ord_{\ell}(pM)=1$, then $\m$ is not in the support of $\T'/\T$. \end{lem} This is essentially Lemma~2.2 in~\cite[pg.~485]{wiles}; \comment{, which proves, under a suitable hypothesis, that $\H^0(X_0(pM)_{\F_p},\Omega)[\m]$ is $1$-dimensional if $\m$ is a maximal ideal of~$\T$ that divides~$p$. The ``suitable hypothesis'' is that $\m$ is ordinary, in the sense that $T_p \not\in\m$. (Note that $T_p$ is often denoted $U_p$ in this context.) It follows from Wiles's lemma that $\T'=\T$ locally at~$\m$ whenever $\m$ is an ordinary prime whose residue characteristic exactly divides the level (which is $pM$ here). } we make a few comments about how it applies on our situation: \begin{enumerate} \item Wiles considers $X_1(M,p)$ instead of $X_0(pM)$, which means that he is using $\Gamma_1(M)$-structure instead of $\Gamma_0(M)$-structure. This surely has no relevance to the issue at hand. \item Wiles assumes (on page 480) that $p$ is an odd prime, but again this assumption is not relevant to our question. \item The condition that $\m$ is ordinary does not appear explicitly in the statement of Lemma~2.2 in~\cite{wiles}; instead it is a reigning assumption in the context of his discussion. \item We see by example that Wiles's ``ordinary'' assumption is less stringent than the assumption in \cite{mazur-ribet}; note that \cite{mazur-ribet} rule out cases where $\m$ is both old and new at $p$, whereas Wiles is happy to include such cases. (On the other hand, Wiles's assumption is certainly nonempty, since it rules out maximal ideals $\m$ that arise from non-ordinary (old) forms of level~$M$. Here is an example with $p=2$ and $M=11$, so $N=22$: There is a unique newform $f=\sum a_n q^n$ of level~$11$, and $\T=\Z[T_2] \subset \End(J_0(22))$, where $T_2^2-a_2 T_2 + 2 =0$. Since $a_2=-2$, we have $\T\isom \Z[\sqrt{-1}]$. We can choose the square root of $-1$ to be $T_2+1$. Then $T_2$ is a generator of the unique maximal ideal $\m$ of $\T$ with residue characteristic~$2$, and this maximal ideal is not ordinary.) \end{enumerate} %\end{proof} We now summarize the conclusions we can make from the lemmas so far: \begin{prop} \label{prop:TT'} The modules~$\T$ and $\T'$ agree locally at each maximal ideal~$\m$ that is either prime to~$N$ or that satisfies the following supplemental hypothesis: the residue characteristic of~$\m$ divides~$N$ only to the first power and $\m$ is ordinary. \end{prop} \begin{proof} This follows easily from Lemmas~\ref{lem:m1}, \ref{lem:m_ell}, and~\ref{lem:wiles}. \end{proof} \comment{ Wiles's lemma and the standard $q$-expansion argument (Lemma~\ref{lem:m_ell} and Lemma~\ref{lem:wiles}) imply that~$\T$ and~$\T'$ agree locally at each rational prime that is prime to the level $pM$, and also at each maximal ideal~$\m$ dividing~$p$ that is ordinary, in the sense that $T_p \not\in \m$. A more palatable description of the situation involves considering the Hecke algebra~$\T$ and its saturation~$\T'$ at some level $N\geq 1$. Then $\T=\T'$ locally at each maximal ideal $\m$ that is either prime to~$N$ or that satisfies the following supplemental hypothesis: the residue characteristic of~$\m$ divides~$N$ only to the first power and~$\m$ is ordinary. } In Mazur's original context, where the level~$N$ is prime, we have $T_N^2=1$ because there are no forms of level~$1$. Accordingly, each~$\m$ dividing~$N$ is ordinary, and we recover Mazur's equality $\T=\T'$ in this special case. \subsubsection{Degrees and Congruences} \label{sec:degrees} %Let $e\in \T\tensor\Q$ be an idempotent, and let $A\subset J_0(pM)$ %be the abelian variety image of $e$, i.e., the image of the homomorphism %$ne\in \T$, where the integer $n\geq 1$ is a multiple of the denominator of $e$. %Let~$B$ be the image of the complementary idempotent $1-e$. %Then $J_0(pM)=A+B$, and $A\cap B$ is a finite group whose exponent %divides the denominator of $e$. %\edit{We can just say that $e$ is as in the previous section. %Note that $A$ was~$\Adual$ in the previous section. --Amod} Let $e \in \T\tensor\Q$ be as in Section~\ref{sec:firstpart}, and let $p,N,M$ be as before Lemma~\ref{lem:wiles}. The image of $e$ in $J_0(pM)$ is the $\T$-stable abelian subvariety denoted $\Adual$ in Section~\ref{sec:firstpart}, but since we shall now exclusively work with this subvariety rather than the corresponding optimal quotient of $J_0(pM)$ (which was denoted $A$ earlier), we will now write $A$ to denote the image of $e$ (without risk of confusion). We also write $B$ to denote the unique $\T$-stable abelian subvariety of $J_0(pM)$ complementary to~$A$. For $t \in \T$, let $t_A$ be the restriction of~$t$ to $A$, and let~$t_B$ be the image of~$t$ in $\End(B)$. Let $\T_A$ be the subgroup of $\End(A)$ consisting of the various $t_A$, and define $\T_B$ similarly. As before, we obtain an injection $ j : \T \hra \T_A \times \T_B $ with finite cokernel. Because~$j$ is an injection, we refer to the maps $\pi_A:\T\to \T_A$ and $\pi_B : \T \to \T_B$, given by $t \mapsto t_A$ and $t\mapsto t_B$, respectively, as ``projections''. \begin{defi} The {\em congruence ideal} associated with the projector~$e$ is $I=\pi_A(\ker(\pi_B)) \subset \T_A.$ \end{defi} Viewing $\T_A$ as $\T_A\times \{0\}$, we may view $\T_A$ as a subgroup of $\T\tensor\Q \isom (\T_A\times \T_B)\tensor\Q$. Also, we may view $\T$ as embedded in $\T_A\times \T_B$, via the map~$j$. \begin{lem}\label{lem:i_int} We have $I=\T_A\cap \T$. \end{lem} A larger ideal of $\T_A$ is $ J = \Ann_{\T_A}(A \cap B); $ it consists of restrictions to $A$ of Hecke operators that vanish on $A\cap B$. \begin{lem} We have $I\subset J$. \end{lem} \begin{proof} The image in $\T_A$ of an operator that vanishes on $B$ also vanishes on $A\cap B$. \end{proof} \begin{lem}\label{lem:j_int} We have $J = \T_A \cap \End(J_0(pM)) = \T_A \cap \T'.$ \end{lem} \begin{proof} This is elementary; it is an analogue of Lemma~\ref{lem:i_int}. \end{proof} \begin{prop}\label{prop:ji_inc} There is a natural inclusion $ J/I \hra \T'/\T $ of $\T$-modules. \end{prop} \begin{proof} Consider the map $\T\to \T\tensor\Q$ given by $t\mapsto te$. This homomorphism factors through $\T_A$ and yields an injection $\iota_A : \T_A \hra \T\tensor\Q$. Symmetrically, we also obtain $\iota_B : \T_B \hra \T\tensor\Q$. The map $(t_A, t_B) \mapsto \iota_A(t_A) + \iota_B(t_B)$ is an injection $\T_A\times \T_B \hra \T\tensor\Q$. The composite of this map with the inclusion $j:\T\hra \T_A\times \T_B$ defined above is the natural map $\T\hra \T\tensor\Q$. We thus have a sequence of inclusions $$ \T \hra \T_A \times \T_B \hra \T\tensor \Q \subset \End(J_0(pM))\tensor\Q. $$ By Lemma~\ref{lem:i_int} and Lemma~\ref{lem:j_int}, we have $I=\T_A\cap \T$ and $J=\T_A\cap \T'$. Thus $I=J\cap \T$, where the intersection is taken inside $\T'$. Thus $$ J/I = J/(J\cap \T) \isom (J+\T)/\T \hra \T'/\T. $$ \end{proof} \begin{cor}\label{cor:ji_inc} If $\m$ is a maximal ideal not in $\Supp_{\T}(\T'/\T)$, then $\m$ is not in the support of $J/I$, i.e., if $\T$ and $\T'$ agree locally at $\m$, then $I$ and $J$ also agree locally at $\m$. \end{cor} Note that the Hecke algebra $\T$ acts on $J/I$ through its quotient $\T_A$, since the action of~$\T$ on~$I$ and on~$J$ factors through this quotient. Now we specialize to the case where $A$ is ordinary at $p$, in the sense that the image of $T_p$ in $\T_A$, which we denote $T_{p,A}$, is invertible modulo every maximal ideal of $\T_A$ that divides~$p$. (This case occurs when~$A$ is a subvariety of the $p$-new subvariety of $J_0(pM)$, since the square of $T_{p,A}$ is the identity.) If $\m\mid p$ is a maximal ideal of $\T$ that arises by pullback from a maximal ideal of $\T_A$, then~$\m$ is ordinary in the sense used above. When $A$ is ordinary at~$p$, it follows from Proposition~\ref{prop:TT'} and Corollary~\ref{cor:ji_inc} that $I=J$ locally at~$p$. The reason is simple: regarding~$I$ and~$J$ as $\T_A$-modules, we realize that we need to test that $I=J$ at maximal ideals of $\T_A$ that divide~$p$. These ideals correspond to maximal ideals $\m\mid p$ of $\T$ that are automatically ordinary, so we have $I=J$ locally at $\m$ because of Proposition~\ref{prop:TT'}. By Proposition~\ref{prop:TT'}, we have $\T=\T'$ locally at primes away from the level $pM$. Thus we conclude that $I=J$ locally at all primes $\ell\nmid pM$ and also at~$p$, a prime that divides the level $pM$ exactly once. Suppose, finally, that $A$ is the abelian variety associated to a newform~$f$ of level~$pM$. %We then have $\T_A=\Z$. The ideal $I\subset \T_A$ measures congruences between~$f$ and the space of forms in $S_2(\Gamma_0(pM))$ that are orthogonal to the space generated by~$f$. Also, $A\cap B$ is the kernel in~$A$ of the map ``multiplication by the modular element~$e$''. In this case, the inclusion $I\subset J$ corresponds to the divisibility $ \tilde{n}_A \mid \tilde{r}_A, $ and we have equality at primes at which $I=J$ locally. We conclude that the congruence exponent and the modular exponent agree both at~$p$ and at primes not dividing $pM$, which completes our proof of Theorem~\ref{thm:ribet_gen}(b). \begin{rmk} The ring $$ R = \End(J_0(pM)) \cap (\T_A \times \T_B) $$ is often of interest, where the intersection is taken in $\End(J_0(pM))\tensor \Q$. We proved above that there is a natural inclusion $J/I \hra \T'/\T$. This inclusion yields an isomorphism $ J/I \xra{\sim} R/\T. $ Indeed, if $(t_A, u_B)$ is an endomorphism of $J_0(pM)$, where $t,u \in \T$, then $(t_A, u_B) - u = (t_A, 0)$ is an element of~$J$. The ideals~$I$ and~$J$ are equal to the extent that the rings~$\T$ and $R$ coincide. Even when $\T'$ is bigger than~$\T$, its subring $R$ may be not far from~$\T$. \end{rmk} \section{Failure of Multiplicity One}\label{sec:mult1} In this section, we discuss examples of failure of multiplicity one (in two different but related senses). The notion of multiplicity one, originally due to Mazur~\cite{mazur:eisenstein}, has played an important role in several places (e.g., in Wiles's proof of Fermat's last theorem~\cite{wiles}). This notion is closely related to Gorensteinness of certain Hecke algebras (e.g., see~\cite{tilouine:hecke}). Kilford~\cite{kilford} found examples of failure of Gorensteinness (and multiplicity one) at the prime~$2$ for certain prime levels. Motivated by the arguments in Section~\ref{sec:main}, in this section we give examples of failure of multiplicity one for primes (including odd primes) whose square divides the level. \subsection{Multiplicity One for Differentials}\label{sec:dataind} In connection with the arguments in Section~\ref{sec:main}, especially Lemmas~\ref{lem:m1} and \ref{lem:wiles}, it is of interest to compute the index $[\T':\T]$ for various $N$. We can compute this index in Magma, e.g., the following commands compute the index for $N=54$: ``{\tt J := JZero(54); T := HeckeAlgebra(J); Index(Saturation(T), T);}'' We obtain Table~\ref{table:index}, where the first column contains $N$ and the second column contains $[\T':\T]$: \begin{table}\caption{The Index $[\T':\T]$\label{table:index}} \begin{center} \begin{tabular}{|l|c|}\hline 11 & 1 \\\hline 12 & 1 \\\hline 13 & 1 \\\hline 14 & 1 \\\hline 15 & 1 \\\hline 16 & 1 \\\hline 17 & 1 \\\hline 18 & 1 \\\hline 19 & 1 \\\hline 20 & 1 \\\hline 21 & 1 \\\hline 22 & 1 \\\hline 23 & 1 \\\hline 24 & 1 \\\hline 25 & 1 \\\hline 26 & 1 \\\hline 27 & 1 \\\hline 28 & 1 \\\hline 29 & 1 \\\hline 30 & 1 \\\hline 31 & 1 \\\hline 32 & 1 \\\hline 33 & 1 \\\hline 34 & 1 \\\hline 35 & 1 \\\hline 36 & 1 \\\hline 37 & 1 \\\hline 38 & 1 \\\hline 39 & 1 \\\hline 40 & 1 \\\hline 41 & 1 \\\hline 42 & 1 \\\hline 43 & 1 \\\hline 44 & 2 \\\hline 45 & 1 \\\hline 46 & 2 \\\hline 47 & 1 \\\hline 48 & 1 \\\hline 49 & 1 \\\hline 50 & 1 \\\hline \end{tabular} \,\,\,\,\,\, \begin{tabular}{|l|c|}\hline 51 & 1 \\\hline 52 & 1 \\\hline 53 & 1 \\\hline 54 & 3 \\\hline 55 & 1 \\\hline 56 & 2 \\\hline 57 & 1 \\\hline 58 & 1 \\\hline 59 & 1 \\\hline 60 & 2 \\\hline 61 & 1 \\\hline 62 & 2 \\\hline 63 & 1 \\\hline 64 & 2 \\\hline 65 & 1 \\\hline 66 & 1 \\\hline 67 & 1 \\\hline 68 & 2 \\\hline 69 & 1 \\\hline 70 & 1 \\\hline 71 & 1 \\\hline 72 & 2 \\\hline 73 & 1 \\\hline 74 & 1 \\\hline 75 & 1 \\\hline 76 & 2 \\\hline 77 & 1 \\\hline 78 & 2 \\\hline 79 & 1 \\\hline 80 & 4 \\\hline 81 & 1 \\\hline 82 & 1 \\\hline 83 & 1 \\\hline 84 & 2 \\\hline 85 & 1 \\\hline 86 & 1 \\\hline 87 & 1 \\\hline 88 & 8 \\\hline 89 & 1 \\\hline 90 & 1 \\\hline \end{tabular} \,\,\,\,\,\, \begin{tabular}{|l|c|}\hline 91 & 1 \\\hline 92 & 16 \\\hline 93 & 1 \\\hline 94 & 4 \\\hline 95 & 1 \\\hline 96 & 8 \\\hline 97 & 1 \\\hline 98 & 1 \\\hline 99 & 9 \\\hline 100 & 1 \\\hline 101 & 1 \\\hline 102 & 1 \\\hline 103 & 1 \\\hline 104 & 4 \\\hline 105 & 1 \\\hline 106 & 1 \\\hline 107 & 1 \\\hline 108 & 54 \\\hline 109 & 1 \\\hline 110 & 2 \\\hline 111 & 1 \\\hline 112 & 8 \\\hline 113 & 1 \\\hline 114 & 1 \\\hline 115 & 1 \\\hline 116 & 4 \\\hline 117 & 1 \\\hline 118 & 2 \\\hline 119 & 1 \\\hline 120 & 32 \\\hline 121 & 1 \\\hline 122 & 1 \\\hline 123 & 1 \\\hline 124 & 16 \\\hline 125 & 25 \\\hline 126 & 18 \\\hline 127 & 1 \\\hline 128 & 64 \\\hline 129 & 1 \\\hline 130 & 1 \\\hline \end{tabular} \,\,\,\,\,\, \begin{tabular}{|l|c|}\hline 131 & 1 \\\hline 132 & 8 \\\hline 133 & 1 \\\hline 134 & 1 \\\hline 135 & 27 \\\hline 136 & 16 \\\hline 137 & 1 \\\hline 138 & 4 \\\hline 139 & 1 \\\hline 140 & 8 \\\hline 141 & 1 \\\hline 142 & 8 \\\hline 143 & 1 \\\hline 144 & 32 \\\hline 145 & 1 \\\hline 146 & 1 \\\hline 147 & 7 \\\hline 148 & 4 \\\hline 149 & 1 \\\hline 150 & 5 \\\hline 151 & 1 \\\hline 152 & 32 \\\hline 153 & 9 \\\hline 154 & 1 \\\hline 155 & 1 \\\hline 156 & 32 \\\hline 157 & 1 \\\hline 158 & 4 \\\hline 159 & 1 \\\hline 160 & 256 \\\hline 161 & 1 \\\hline 162 & 81 \\\hline 163 & 1 \\\hline 164 & 8 \\\hline 165 & 1 \\\hline 166 & 2 \\\hline 167 & 1 \\\hline 168 & 128 \\\hline 169 & 13 \\\hline 170 & 1 \\\hline \end{tabular} \,\,\,\,\,\, \begin{tabular}{|l|c|}\hline 171 & 9 \\\hline 172 & 8 \\\hline 173 & 1 \\\hline 174 & 4 \\\hline 175 & 5 \\\hline 176 & 512 \\\hline 177 & 1 \\\hline 178 & 1 \\\hline 179 & 1 \\\hline 180 & 72 \\\hline 181 & 1 \\\hline 182 & 1 \\\hline 183 & 1 \\\hline 184 & 1024 \\\hline 185 & 1 \\\hline 186 & 4 \\\hline 187 & 1 \\\hline 188 & 256 \\\hline 189 & 243 \\\hline 190 & 8 \\\hline 191 & 1 \\\hline 192 & 4096 \\\hline 193 & 1 \\\hline 194 & 1 \\\hline 195 & 1 \\\hline 196 & 14 \\\hline 197 & 1 \\\hline 198 & 81 \\\hline 199 & 1 \\\hline 200 & 80 \\\hline 201 & 1 \\\hline 202 & 1 \\\hline 203 & 1 \\\hline 204 & 32 \\\hline 205 & 1 \\\hline 206 & 4 \\\hline 207 & 81 \\\hline 208 & 256 \\\hline 209 & 1 \\\hline 210 & 2 \\\hline \end{tabular} \end{center} \end{table} Let $\m$ be a maximal ideal of the Hecke algebra $\T\subset\End(J_0(N))$ of residue characteristic~$p$. Recall that we say that $\m$ satisfies {\em multiplicity one for differentials} if $\dim (\H^0(X_0(N)_{\Fp},\Omega)[\m]) \leq 1$. In each case in which $[\T':\T]\neq 1$, Lemma~\ref{lem:m1} implies that there is some maximal ideal $\m$ of $\T$ such that $\dim(\H^0(X_0(N)_{\Fp},\Omega)[\m])>1$, which is an example of failure of multiplicity one for differentials. In Table~\ref{table:index}, whenever $p\mid [\T':\T]$, then $p^2\mid 2N$. This is a consequence of Proposition~\ref{prop:TT'}, which moreover asserts that when $2$ exactly divides $N$ and $2\mid[\T':\T]$ then there is a non-ordinary (old) maximal ideal of characteristic $2$ in the support of $\T'/\T$. %The first case when $2\mid\mid N$ and $2\mid %[\T':\T]$ is $N=46$, where we find (via a Magma calculation) that %$G=\T'/\T \isom \Z/2\Z$, and the Hecke operator~$T_2$ acts as~$0$ on %$G$, so the annihilator of $G$ in $\T$ is not ordinary, which does not %ontradict Proposition~\ref{prop:TT'}. Moreover, notice that Theorem~\ref{thm:ribet_gen}(b) (whose proof is in Section~\ref{sec:secondpart}) follows formally from two key facts: that $A_f$ is new and that multiplicity one for differentials holds for ordinary maximal ideals with residue characteristic $p\mid\mid N$ and for all maximal ideals with residue characteristic $p\nmid N$. The conclusion of Theorem~\ref{thm:ribet_gen}(b) does not hold for the counterexamples in Section~\ref{sec:elliptic} (e.g., for~54B1), which are all new elliptic curves, so multiplicity one for differentials does not hold for certain maximal ideals that arise from the new quotient of the Hecke algebra. Note that in all examples we have $p\mid(r/m)$ with $p^2\mid N$, which raises the question: are there non-ordinary counterexamples with $p\mid\mid N$? %\edit{I think I have been to vague. Basically, I wanted to say %that one reason why the index $[\T':\T]$ could be nontrivial %(and hence lead to failure of multiplicity one for differentials) %is due to old-ness. But even in the new part, one may have failure %of mult one for diffs, due to the level not being square-free. --Amod} % was@modular:~/comps/ind_table$ magma % Magma V2.11-10 Sun Aug 28 2005 18:34:49 on modular [Seed = 1293693469] % Type ? for help. Type -D to quit. % > J := JZero(46); % > J; % Modular abelian variety JZero(46) of dimension 5 and level 2*23 over Q % > T := HeckeAlgebra(J); % > S := Saturation(T); % > m := S/T; % > m; % Abelian Group isomorphic to Z/2 % Defined on 1 generator % Relations: % 2*m.1 = 0 % > % > S; % Sat(HeckeAlg(JZero(46))): Group of homomorphisms from JZero(46) to JZero(46) % > Basis(S); % [ % Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix), % Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix), % Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix), % Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix), % Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix) % ] % > S.1; % Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix) % > S.1 in T; % true % > S.2 in T; % false % > t2 := HeckeOperator(J,2); % > t2 in T; % true % > t2*S.2; % Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix) % > t2*S.2 in T; % true \subsection{Multiplicity One for Jacobians} We say that a maximal ideal~$\m$ of~$\T$ satisfies {\it multiplicity one} if $J_0(N)[\m]$ is of dimension two over~$\T/\m$. We sometimes use the phrase ``multiplicity one for~$J_0(N)$'' in order to distinguish this notion from the notion of multiplicity one for differentials. %\edit{I added these lines for clarity. --Amod} \begin{prop}\label{prop:mult1J} Suppose $E$ is an optimal elliptic curve over $\Q$ of conductor~$N$ and~$p$ is a prime such that $p \mid r_E$ but $p\nmid m_E$. Let $\m$ be the annihilator in $\T$ of $E[p]$. Then multiplicity one fails for $\m$, i.e., $\dim_{\T/\m} J_0(N)[\m] > 2$. %\edit{Earlier the conclusion said $\dim_{\T/\m} J_0(N)[\m] > 1$. %--Amod} \end{prop} \begin{proof} Using the principal polarization $E \isom E^{\vee}$ we view~$E$ as an abelian subvariety of $J=J_0(N)$ and consider the complementary $\T$-stable abelian subvariety $A$ of $E$ (thus $A$ is the kernel of the modular parametrization map $J\to E$). In this setup, $J = E + A$, and the intersection of $E$ and $A$ is $E[m_E]$. Here we use that the composite map $ E \simeq E^{\vee} \to J^{\vee} \to J \to E $ is a polarization, and hence is multiplication by a positive integer $m_E$. Because $p\nmid m_E$, we have $E[p]\cap A = 0$. On the other hand, let $\m$ be the annihilator of $E[p]$ inside $\T$. Then $J[\m]$ contains $E[p]$ and also $A[\m]$, and because $p$ is a congruence prime, the submodule $A[\m]\subset J[\m]$ is nonzero. Thus the sum $E[p] + A[\m]$ is a direct sum and is larger than $E[p]$, which is of dimension $2$ over $\T/\m = \Z/p\Z$. Hence the dimension of $J[\m]$ over $\T/\m$ is bigger than $2$, as claimed. %\edit{I changed the last two lines. --Amod} \end{proof} Proposition~\ref{prop:mult1J} implies that any example in which simultaneously $p\nmid m_E$ and $\ord_p(r_E)\neq \ord_p(m_E)$ produces an example in which multiplicity one for $J_0(N)$ fails. For example, for the curve 54B1 and $p=3$, we have $\ord_3(r_E)=1$ but $\ord_3(m_E)=0$, so multiplicity one at $3$ fails for $J_0(54)$. %Also, for 242B1, we have $r_E = 11\cdot 2^4$ and $m_E = 2^4$, so %multiplicity one for $J_0(242)$ fails at $11$. For $N=242$ we also %have $[\T':\T]=121$, so multiplicity one at $11$ also fails for %differentials %\edit{Doesn't this work for 54B1 as well? Why give another %example? --Amod} %(see Section~\ref{sec:dataind} above). %\edit{I commented out the old section, which had mistakes. --Amod} \comment{ \subsection{Multiplicity one (old section)} \edit{william: I think this section should be deleted in light of the above two new sections.} %\edit{[This is still very rough, and will need to be cleaned up. --Amod]} Let $\m$ be a maximal ideal of the Hecke algebra $\T\subset\End(J_0(N))$ of residue characteristic~$p$. We say that $\m$ satisfies {\em strong multiplicity one} if $\dim (\H^0(X_0(N)_{\Fp},\Omega)[\m]) = 1$. \edit{or should be it be equal to~$1$? Also ``strong'' is just something I came up with; may want to change the name. In fact, I may have got strong and weak mixed up, since Tilouine calls above ``weak multiplicity one''. --Amod} The proof of Theorem~\ref{thm:ribet_gen}(b) follows formally from two key facts: that $A_f$ is new and that strong multiplicity one holds for ordinary maximal ideals if~$p^2 \nmid N$. The conclusion of Theorem~\ref{thm:ribet_gen}(b) does not hold for the counterexamples in Section~\ref{sec:elliptic} (at levels $54$,~$64$, etc.), which are all new elliptic curves, which shows that strong multiplicity one does not hold for certain ordinary\edit{William: I totally don't get this. When $p^2\mid N$ the ideals are {\em NOT} ordinary. We only got ordinary in the proof above because $p\mid\mid N$. For example, for the curve 54b, we have $a_3=0$, and the ideal is not ordinary.} maximal ideals for the corresponding levels (in all of them, $p^2 \mid N$). We record this observation: \begin{prop} There are ordinary maximal ideals for which strong multiplicity one fails.\edit{William: But our examples are not ordinary!} \end{prop} There is another notion of multiplicity one: suppose~$\rho_{\m}$, the representation attached to~$\m$, is absolutely irreducible. Then one says that $\m$ satisfies {\em weak multplicity one} if $J_0(N)[\m]$ is isomorphic to a single copy of~$\rho_{\m}$. By standard arguments, strong multiplicity one implies weak multiplicity one when $\overline{\rho}_{\m}$ is absolutely irreducible. \edit{William: I think this is wrong. In connection with Remark~\ref{rem:24}, we find (again via a Magma computation) that $\T'=\T$ when $N=431$. This was also already known via work of Mazur \cite{mazur:eisenstein}, and illustrates that multiplicity one for differentials need not imply multiplicity one for $J_0(N)$. So I'm confused.} %\edit{Will: I just realized that for this implication, %one also needs the representation~$\rho_{\m}$ %to be absolutely irreducible -- it might be worth checking %if this holds in our couterexamples} which in turn implies that the Hecke algebra~$\T$ is Gorenstein.\edit{or maybe weak multiplicity one is equivalent to Gorensteinness, and both follow from strong multiplicity one; a good reference might be Tilouine's article in the FLT conference.} Wiles proves that strong multiplicity one holds for ordinary maximal ideals provided $p^2 \nmid N$, which he used to show the Gorensteinness of certain Hecke algebras. This Gorensteinness property was a key step in the proof of Fermat's last theorem. Our finding above shows that the hypothesis $p^2 \nmid N$ is essential, and thus gives a limit to how far the standard argument for proving Gorensteinness works. Note that while in our examples, we know that strong multiplicity one fails, we do not know if weak multiplicity one or Gorensteinness fails. It would be interesting to do calculations for the latter, like were done by Kilford. } %\bibliographystyle{amsalpha} %\bibliography{biblio} %\end{document} \providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace} \providecommand{\MR}{\relax\ifhmode\unskip\space\fi MR } % \MRhref is called by the amsart/book/proc definition of \MR. \providecommand{\MRhref}[2]{% \href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2} } \providecommand{\href}[2]{#2} \begin{thebibliography}{BCDT01} \bibitem[ARS06]{ars} A.~Agashe, K.~Ribet and W.~Stein, \emph{The {M}anin {C}onstant}, QJPAM, Coates Volume (2006), to appear. \bibitem[AU96]{abbull} A.~Abbes and E.~Ullmo, \emph{\`{A} propos de la conjecture de {M}anin pour les courbes elliptiques modulaires}, Compositio Math. \textbf{103} (1996), no.~3, 269--286. \bibitem[BCP97]{magma} W.~Bosma, J.~Cannon, and C.~Playoust, \emph{The {M}agma algebra system. {I}. {T}he user language}, J. Symbolic Comput. \textbf{24} (1997), no.~3--4, 235--265, Computational algebra and number theory (London, 1993). %\MR{1 484 478} \bibitem[BCDT01]{breuil-conrad-diamond-taylor} C.~Breuil, B.~Conrad, F.~Diamond, and R.~Taylor, \emph{On the modularity of elliptic curves over {$\bold Q$}: wild 3-adic exercises}, J. Amer. Math. Soc. \textbf{14} (2001), no.~4, 843--939 (electronic). %\MR{2002d:11058} \bibitem[CK04]{cojo-kani} Alina~Carmen Cojocaru and Ernst Kani, \emph{The modular degree and the congruence number of a weight 2 cusp form}, Acta Arith. \textbf{114} (2004), no.~2, 159--167. \bibitem[Cre97]{cremona:alg} J.\thinspace{}E. Cremona, \emph{Algorithms for modular elliptic curves}, second ed., Cambridge University Press, Cambridge, 1997, \mbox{}\\{\tt http://www.maths.nott.ac.uk/personal/jec/book/}. \bibitem[CSS97]{boston} G.~Cornell, J.\thinspace{}H. Silverman, and G.~Stevens (eds.), \emph{Modular forms and \protect{F}ermat's last theorem (boston,ma, 1995)}, New York, Springer-Verlag, 1997, Papers from the Instructional Conference on Number Theory and Arithmetic Geometry held at Boston University, Boston, MA, August 9--18, 1995. \bibitem[DI95]{diamond-im} F.~Diamond and J.~Im, \emph{Modular forms and modular curves}, Seminar on {F}ermat's {L}ast {T}heorem, Providence, RI, 1995, pp.~39--133. \bibitem[Fre97]{frey:ternary} G.~Frey, \emph{On ternary equations of {F}ermat type and relations with elliptic curves}, Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 1997, pp.~527--548. \bibitem[FM99]{frey-muller} G.~Frey and M.~M{\"u}ller, \emph{Arithmetic of modular curves and applications}, Algorithmic algebra and number theory (Heidelberg, 1997), Springer, Berlin, 1999, pp.~11--48. \bibitem[Har77]{hartshorne:ag} R.~Hartshorne, \emph{Algebraic geometry}, Springer-Verlag, New York, 1977, Graduate Texts in Mathematics, No. 52. \bibitem[Kil02]{kilford} L.\thinspace{}J.\thinspace{}P. Kilford, \emph{Some non-{G}orenstein {H}ecke algebras attached to spaces of modular forms}, J. Number Theory \textbf{97} (2002), no.~1, 157--164. \bibitem[Lan83]{lang:av} S.~Lang, \emph{Abelian varieties}, Springer-Verlag, New York, 1983, Reprint of the 1959 original. \bibitem[Li75]{winnie:newforms} W-C. Li, \emph{Newforms and functional equations}, Math. Ann. \textbf{212} (1975), 285--315. \bibitem[Maz77]{mazur:eisenstein} B.~Mazur, \emph{Modular curves and the \protect{Eisenstein} ideal}, Inst. Hautes \'Etudes Sci. Publ. Math. (1977), no.~47, 33--186 (1978). \bibitem[Maz78]{mazur:rational} B.~Mazur, \emph{Rational isogenies of prime degree (with an appendix by {D}. {G}oldfeld)}, Invent. Math. \textbf{44} (1978), no.~2, 129--162. \bibitem[MR91]{mazur-ribet} B.~Mazur and K.~A. Ribet, \emph{Two-dimensional representations in the arithmetic of modular curves}, Ast\'erisque (1991), no.~196--197, 6, 215--255 (1992), Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). %\MR{MR1141460 (93d:11056)} \bibitem[Mum70]{mumford:av} D.~Mumford, \emph{Abelian varieties}, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970. \bibitem[Mur99]{murty:congruence} M.\thinspace{}R. Murty, \emph{Bounds for congruence primes}, Automorphic forms, automorphic representations, and arithmetic (Fort Worth, TX, 1996), Amer. Math. Soc., Providence, RI, 1999, pp.~177--192. %\MR{2000g:11038} \bibitem[Rib75]{ribet:endo} K.\thinspace{}A. Ribet, \emph{Endomorphisms of semi-stable abelian varieties over number fields}, Ann. Math. (2) \textbf{101} (1975), 555--562. %\MR{51 \#8120} \bibitem[Rib81]{ribet:endalg} K.~A. Ribet, \emph{Endomorphism algebras of abelian varieties attached to newforms of weight {$2$}}, Seminar on Number Theory, Paris 1979--80, Progr. Math., vol.~12, Birkh\"auser Boston, Mass., 1981, pp.~263--276. % \MR{82m:10044} \bibitem[Rib83]{ribet:modp} K.\thinspace{}A. Ribet, \emph{Mod $p$\ {H}ecke operators and congruences between modular forms}, Invent. Math. \textbf{71} (1983), no.~1, 193--205. \bibitem[Ste89]{stevens:param} G.~Stevens, \emph{Stickelberger elements and modular parametrizations of elliptic curves}, Invent. Math. \textbf{98} (1989), no.~1, 75--106. \bibitem[Stu87]{sturm:cong} J.~Sturm, \emph{On the congruence of modular forms}, Number theory (New York, 1984--1985), Springer, Berlin, 1987, pp.~275--280. \bibitem[Til97]{tilouine:hecke} J.~Tilouine, \emph{Hecke algebras and the {G}orenstein property}, Modular forms and Fermat's last theorem (Boston, MA, 1995), Springer, New York, 1997, pp.~327--342. \bibitem[Vat05]{MR2135139} V.~Vatsal, \emph{Multiplicative subgroups of {$J\sb 0(N)$} and applications to elliptic curves}, J. Inst. Math. Jussieu \textbf{4} (2005), no.~2, 281--316. \bibitem[Wil95]{wiles} A.\thinspace{}J. Wiles, \emph{Modular elliptic curves and \protect{F}ermat's last theorem}, Ann. of Math. (2) \textbf{141} (1995), no.~3, 443--551. \bibitem[Zag85]{zagier} D.~Zagier, \emph{Modular parametrizations of elliptic curves}, Canad. Math. Bull. \textbf{28} (1985), no.~3, 372--384. \end{thebibliography} \end{document}