%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % TITLE: The Manin Constant, Congruence Primes, % % and the Modular Degree % % % % William A. 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Line breaks will be recognized. The Manin Constant, Congruence Primes, and the Modular Degree \ShortTitle The Manin Constant, Congruences, and Modular Degrees %%%%% 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 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. We obtain relations between the modular degree and congruence modulus of elliptic curves, and answer a question raised in a paper of Frey and M{\"u}ller about whether or not the congruence number and modular degree of elliptic curves are equal; they are not, but we prove a theorem relating them and make a conjecture. We also prove results and make conjectures about Manin constants of quotients of~$J_1(N)$ of arbitrary dimension. For optimal elliptic curves~$E$, we give a new condition under which the Manin constant of~$E$ is odd. \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 a quotient 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 $J_0(N)$. The pullback of a minimal differential on~$E$ is a multiple~$c$ of some normalized new cuspidal eigenform $f_E\in S_2(\Gamma_0(N))$. The absolute value of $c$ is the Manin constant $c_E$ of~$E$. Manin conjectured that $c_E=1$. In Section~\ref{maninintro}, we summarize results about $c_E$, then extend techniques of Abbes and Ullmo~\cite{abbull} to show that $2\nmid c_E$ under certain hypothesis. The congruence number~$\re$ of~$E$ is the largest integer such that there is a nonzero element of $S_2(\Gamma_0(N))$ that is orthogonal to~$f_E$ and congruent to~$f_E$ modulo $\re$. The modular degree~$\me$ is the degree of the composite map $X_0(N)\to J_0(N)\to E$. 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 $\ord_p(N)\leq 1$. We generalize the Manin constant, congruence primes, and modular degree to optimal quotients of~$J_0(N)$ and $J_1(N)$ of any dimension associated to ideals of the Hecke algebra. Section~\ref{sec:quotients} is concerned with the congruence number and the modular degree and Section~\ref{sec:genman} with the Manin constant. We also conjecture that the Manin constant is~$1$ for newform quotients of~$J_0(N)$ and $J_1(N)$. % 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, E.~de Shalit, B.~Edixhoven, L.~Merel, and R.~Taylor for several discussions and advice regarding this paper. They would also like to thank J.~Cremona for explaining his computations involving the Manin constant. \section{Optimal Elliptic Curve Quotients} \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}[Modular Degree] The {\em modular degree} $\me$ of~$E$ is the degree of~$\phie$. \end{defi} \subsection{The Manin Constant} \label{maninintro} Let $E_{\Z}$ denote the N\'{e}ron model of~$E$ over~$\Z$ (see, e.g., \cite[App.~C,~\S15]{silverman:aec}, \cite{silverman:aec2} and \cite{neronmodels}). Let~$\omega$ be a generator for the rank one ${\Z}$-module of invariant differential one forms on~$E_{\Z}$. The pullback of~$\omega$ to~$X_0(N)$ is a differential $\phie^*\omega$ on~$X_0(N)$. The newform~$f$ defines another differential~$2 \pi i f(z) dz = f(q)dq/q$ on~$X_0(N)$. Because the action of Hecke operators is compatible with the map $X_0(N)\to E$, \cite{atkin-lehner} implies that $\phie^*\omega = c \cdot 2 \pi i f(z) dz$ for some $c \in \Q^*$ (see also \cite[\S5]{manin:parabolic}). \begin{defi}[Manin Constant]\label{def:ce} The {\em Manin constant} $\ce$ of~$E$ is the absolute value of~$c$, where $c$ is as above. \end{defi} The Manin constant plays a role in the Birch and Swinnerton-Dyer conjecture (see Section~\ref{maninmotiv}), and its integrality is important to Cremona's computations of elliptic curves (see \cite[pg.~45]{cremona:alg}). %\subsubsection{Manin's Conjecture} The following conjecture is implicit in \cite[\S5]{manin:parabolic}. \begin{conj}[Manin]\label{conjman} $\ce = 1$. \end{conj} Significant progress has been made towards this conjecture. In the following list of theorems,~$p$ denotes a prime and~$N$ denotes the conductor of~$E$. \begin{thm}[Edixhoven~\mbox{\cite[Prop.~2]{edix:manin}}] \label{edixman} $\ce$ is an integer. \end{thm} Edixhoven proved this using an integral $q$-expansion map, whose existence and properties follow from results in \cite{katz-mazur}. We generalize his argument to quotients of arbitrary dimension in Section~\ref{maninmotiv}. \begin{thm}[Mazur,~\mbox{\cite[Cor.~4.1]{maziso}}] \label{mazman} If~$p\mid \ce$, then $p^2 \mid 4N$. \end{thm} Mazur proved this by applying theorems of Raynaud about exactness of sequences of differentials, then using the ``$q$-expansion principle'' in characteristic~$p$ and a property of the Atkin-Lehner involution. We generalize Mazur's argument in Section~\ref{maninmotiv}. The following two results refine the above results at $p=2$. \begin{thm}[Raynaud \mbox{\cite[Prop.~3.1]{abbull}}] \label{thmofraynaud} If $4\mid \ce$, then $4\mid N$. \end{thm} \begin{thm}[Abbes-Ullmo \mbox{\cite[Thm.~A]{abbull}}] \label{abulman} If $p \mid \ce$, then $p \mid N$. \end{thm} We generalize Theorem~\ref{thmofraynaud} in Section~\ref{maninmotiv}. However, it is not clear if one can generalize Theorem~\ref{abulman} to dimension greater than~$1$ (see Remark~\ref{rem:diff_to_generalize}). It would be fantastic if the theorem could be generalized, since it would imply that for newform quotients $A_f$ of $J_0(N)$, with~$N$ odd and square free, that the Manin constant is~$1$, which would be useful for computations regarding the Birch and Swinnerton-Dyer conjecture. B.~Edixhoven also has unpublished results (see~\cite{edix:thesis}) which assert that the only primes that can divide~$\ce$ are $2$, $3$, $5$, and $7$; he also gives bounds that are independent of~$E$ on the valuations of~$\ce$ at $2$, $3$, $5$, and $7$. His arguments rely on construction of certain stable integral models for $X_0(p^2)$. Cremona verified computationally that the Manin constant is~$1$ for every elliptic curve of conductor up to at least $10000$. % REF: email from Cremona to Stein on 2005-01-27 09:35 am Cremona computes lattice invariants~$c_4$ and~$c_6$ from a rational newform~$f$, and verifies in each case that~$c_4$ and~$c_6$ are the invariants of a minimal Weierstrass equation, to conclude that the Manin constant for the corresponding elliptic curve is~$1$. \begin{defi}[Congruence Number] The {\em congruence number}~$\re$ of~$E$ is the largest integer~$r$ such that there exists a cusp form~$g\in S_2(\Gamma_0(N))$ that has integer Fourier coefficients, is orthogonal to~$f$ with respect to the Petersson inner product, and satisfies $g \equiv f \pmod{r}$. The {\em congruence primes} of~$E$ are the primes that divide~$\re$. \end{defi} To the above list we add the following theorem. Our proof builds on the techniques of~\cite{abbull}. \begin{thm} \label{agell} If $p\mid \ce$ then $p^2\mid N$ or $p\mid \me$. %Suppose that $\re$ is odd. If $2\mid \ce$, then $4\mid N$. \end{thm} This theorem is a special case of Theorem~\ref{agn} below, which we prove in Section~\ref{proofsofman}. In fact, Theorem~\ref{agn} asserts that if $p\mid \ce$ then $p^2\mid N$ or $p\mid \re$. However, Theorem~\ref{thm:ribet_au} implies that when $\ord_p(N)=1$ then $\ord_p(\re) = \ord_p(\me)$. In view of Theorem~\ref{mazman}, our new contribution is that if $\me$ is odd and $\ord_2(N)=1$, then $\ce$ is odd. This hypothesis is {\em very stringent}---of the $125357$ optimal elliptic curve quotients of conductor $\leq 30000$, only~$31$ of them satisfy the hypothesis. In the notation of \cite{cremona:alg}, they are \vspace{-2ex}\\ \noindent{}14A, 46A, 142C, 206A, 302B, 398A, 974C, 1006B, 1454A, 1646A, 1934A, 2606A, 2638B, 3118B, 3214B, 3758D, 4078A, 7054A, 7246C, 11182B, 12398B, 12686C, 13646B, 13934B, 14702C, 16334B, 18254A, 21134A, 21326A, 22318A, 26126A. \vspace{-2ex} It is unknown if there are infinitely many elliptic curves that satisfy our hypothesis. The third author conjectured in \cite[Conj.~4.2]{stein-watkins:ns} that there are infinitely many elliptic curves (of prime conductor) with odd modular degree. % \begin{table} % \caption{Theorem~\ref{agell} Applies to These Curves\label{tab:curves}} % $$ % \begin{array}{|l|c|l|}\hline % \, N & \text{isogeny class} & r_E\\\hline % 2\cdot 7 & A & 1 \\ % 2\cdot 23 & A & 5 \\ % 2\cdot 71 & C & 3^2\\ % really it is B % 2\cdot 103 & A & 3\cdot 5 \\ % 2\cdot 151 & C & 3^3 \\ % really it is A % 2\cdot 199 & A & 5\cdot 11 \\ % 2\cdot 487 & C & 3^3\cdot 7 \\ % 2\cdot 503 & B & 3^2 \cdot 7 \\ % 2\cdot 727 & A & 3^6 \\\hline % \end{array} % \quad\quad % \begin{array}{|l|c|l|}\hline % \, N & \text{isogeny class} & r_E\\\hline % 2\cdot 823 & A & 5\cdot 109 \\ % 2\cdot 967 & A & 7\cdot 139 \\ % 2\cdot 1303 & A & 7\cdot 113 \\ % 2\cdot 1319 & B & 3^2\cdot 5\cdot 7 \\ % 2\cdot 1559 & B & 3^2\cdot 5^2\cdot 13 \\ % 2\cdot 1607 & B & 3\cdot 5^3 \\ % 2\cdot 1879 & D & 5\cdot 7\cdot 23 \\ % 2\cdot 2039 & A & 3\cdot 5\cdot 47 \\ % &&\\\hline % \end{array}$$ % \end{table} % %SQL query: % %select level, iso_class, modular_degree from mod, modular_degree, dimension % %where id=modular_degree.mod_id and id=dimension.mod_id and dimension=1 and % %level % 2 = 0 and not (level % 4 = 0) and modular_degree % 2 = 1 order by level \subsection{Congruence Primes and the Modular Degree} \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 (e.g., see~\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 modulus $\re$. Then $\me \mid \re$ and if $\ord_p(N)\leq 1$ then $\ord_p(\re) = \ord_p(\me)$. \end{thm} The divisibility $\me\mid \re$ was first discussed in~\cite[Th.~3]{zagier}, where it is attributed to 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 $\ord_p(N) = 1$ 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 Prop.~3.3 implies $$\ord_p(\re) - \ord_p(\me) = \ord_p(\#\mathcal{C}) - \ord_p(\ce) - \ord_p(\#\mathcal{D}),$$ and by Prop.~3.4 $\ord_p(\#\mathcal{C}) =0$. 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, we quickly found that the answer is no. The first $16$ countexamples occur at levels $$54, 64, 72, 80, 88, 92, 96, 99, 108, 112, 120, 124, 126, 128, 135, 144.$$ For example, the elliptic curve 54B1 of~\cite{cremona:alg}, with 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 $g(q) + g(q^2)$ is congruent to~$f$ modulo~$3$. To prove this congruence, we checked it for $18$ Fourier coefficients, where the precision $18$ 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 242B of conductor $N=2\cdot 11^2$ we have $$ \me = 2^4 \neq \re = 2^4\cdot 11. $$ 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} In particular, for $p\geq 5$, the conjecture simply asserts that $$ \ord_p\left(\frac{\re}{\me}\right) \leq 1, $$ because $\ord_p(N)\leq 2$ for any $p\geq 5$. As evidence, we verified Conjecture~\ref{conj:rm} for every optimal elliptic curve quotient of $J_0(N)$, with $N\leq 539$. %\subsection{Proof of Theorem~\ref{thm:ribet_au}}\label{sec:proof_ribet} \section{Quotients of arbitrary dimension: generalization of the congruence number and the modular degree} \label{sec:quotients} Let $\Gamma$ be either~$\Gamma_0(N)$ or~$\Gamma_1(N)$, for $N\geq 4$, 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~$\T$, 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}[Newform quotient] If~$f \in S_2(\Gamma)$ 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 to the degree of the field $\Q(\ldots,a_n(f),\ldots)$. \end{defi} In Section~\ref{gen-cong-mod}, we generalize the notions of the congruence number and the modular degree to quotients~$A=A_I$, and state a theorem relating the two, which we prove in Sections~\ref{sec:firstpart}--\ref{sec:secondpart}. \subsection{The congruence number and the modular degree} \label{gen-cong-mod} If~$C$ is an abelian variety, let $C^{\vee}$ denote the dual of~$C$. Let $\phi_2$ denote the quotient map $J \ra A$. There is a canonical principal polarization $\theta: J \cong \Jdual$ arising from the theta divisor % (e.g., see %\cite[Thm.~6.6]{milne:jac}). 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 $\po: \Adual \ra J$. Since $\phi_2$ is a surjection, by~\cite[\S{}VI.3, Prop 3]{lang:av}, $\ker(\phi_2^\vee)$ is finite. Since $\ker(\phi_2)$ is connected, $\ker(\phi_2^\vee)$ is trivial, so $\phi_2^\vee$ and $\po$ are injections. Let $\phi$ be the composition $$ \phi: \Adual \stackrel{\po}{\lra} J \stackrel{\pt}{\lra} A. $$ \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. Moreover, 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 $S_2(\Gamma;R)$ denote the subgroup of~$S_2(\Gamma)$ consisting of cups forms whose Fourier expansions at the cusp~$\infty$ have coefficients in~$R$. Let $W(I) =S_2(\Gamma;\Z)[I]^{\perp}$ denote the orthogonal complement of $S_2(\Gamma;\Z)[I]$ in $S_2(\Gamma;\Z)$ with respect to the Petersson inner product. \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] + W(I)} \end{equation} is the {\em congruence exponent} $\rAe$ of~$A$ and its order is the {\em congruence number} $\rA$. \end{defi} 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, 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 $$\nAf = (2^{11}\cdot 6947)^2 \,\,\nmid\,\, \rAf = (2^{10}\cdot 6947)^2.$$ Note that $431$ is prime and mod~$2$ multiplicity one fails for $J_0(431)$ (see \cite{kilford}). The following {\sc Magma} session illustrates how to verify the above assertion about $\nAf$ and $\rAf$. The commands were implemented by the second author, and are parts of {\sc 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} \subsection{Proof of Theorem~\ref{thm:ribet_gen} (a)} \label{sec:firstpart} 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[Remark 10.2.2]{diamond-im}). However, on~$J^{\rm new}$, the action of the Hecke operators commutes with that of~$W_N$. 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 for such quotients we may identify $\Adual$ with~$\po(\Adual)$ as modules over~$\T$. Recall that $f$ is a newform, $I_f = {\rm Ann}_\T (f)$, $J=J_0(N)$, %and $A=J/I_f J$. 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} $\Hom(\Adual,B)=0$. \end{lem} \begin{proof} If there were a nonzero element of $\Hom(\Adual,B)$, then for all~$\ell$, the Tate module $\Tate_{\ell}(\Adual)=\Q\tensor\varprojlim_n \Adual[\ell^n]$ would be a factor of $\Tate_{\ell}(B)$. One could then extract almost all prime-indexed coefficients of the corresponding eigenforms from the Tate modules, which would violate multiplicity one (see \cite[Cor.~3, pg.~300]{winnie:newforms}). \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 the order of $e_2$ is a divisor of the order of $e_1$, which is the crucial ingredient 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} \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 $$ 0 \to S_2(\Z)[I_f] \oplus W(I_f) \to S_2(\Z) \to \Ext^1((\T_1\oplus\T_2)/\T,\Z) \to 0 $$ 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 $S_2(\Z)[I_f]^{\perp} = W(I_f)$. Finally note that if~$G$ is any finite abelian group, then $\Ext^1(G,\Z)\approx G$ as groups, to get 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 an 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}. %In Section~\ref{sec:genman}, we generalize the notion of the %Manin constant %to quotients of~$J_0(N)$ of arbitrary dimension associated to %ideals of the Hecke algebra, and indicate which %of the above results apply in the general situation. %Finally, in Sections~\ref{proofofstein} and~\ref{proofsofman}, we prove %all the new results mentioned in Sections~\ref{intro} %and~\ref{sec:genman}. % %{\bf Acknowledgment.} The authors are grateful for conversations with %Abbes, de Shalit, Edixhoven, Merel, Ribet, and Taylor. \subsection{Proof of the Theorem~\ref{thm:ribet_gen} (b)} \label{sec:secondpart} Write $N= pM$ with~$p$ prime and $p\nmid M$. (Note: The argument below also works if $p=1$, which addresses the case when no prime exactly divides $N$.) Let $\T=\Z[\ldots, T_n,\ldots]$ be the subring of $\End(J_0(N))$ generated by the Hecke operators $T_n$ for all $n\geq 1$. Let $\T'$ be the saturation of $\T$ in $\End(J_0(N))$, 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~$\Z$. Suppose for the moment that $N=1$, so $p=pM$. In \cite{mazur:eisenstein}, Mazur proves that $\T=\T'$. He combines this result with the equality $$\T\tensor\Q = \End(J_0(p)) \tensor\Q$$ of \cite{ribet:endo} or \cite{ribet:endalg}, to deduce that $\T=\End(J_0(p))$. \subsubsection{Multiplicity One} 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}). 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. \begin{lem}\label{lem:m1} Suppose $\m$ is a maximal ideal of $\T$ of residue characteristic~$\ell$ and that $$ \dim_{\T/\m} \H^0(X_0(pM)_{\Fell},\Omega)[\m] \leq 1. $$ Then $\m$ is not in the support of $\T/\T'$. \end{lem} There is quite a bit of literature on the question of multiplicity~$1$ for $\H^0(X_0(pM)_{\Fell},\Omega)[\m]$. The easiest case is that~$\ell$ is prime to the level $pM$. \begin{lem}\label{lem:m_ell} If $\ell \nmid pM$, then~$\ell\nmid \#(\T/\T')$. \end{lem} \begin{proof} The standard $q$-expansion argument of \cite{mazur:eisenstein} 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} \end{proof} In the context of Mazur's paper, where $p=pM$, we see from Lemma~\ref{lem:m_ell} that $\T$ and $\T'$ agree away from~$p$. At~$p$, we can still use the $q$-expansion principle because of the arguments in \cite[Ch.II~\S4]{mazur:eisenstein}. Thus in this case $\T=\T'$, as we asserted above. The question of multiplicity $1$ at $p$ for $\H^0(X_0(pM)_{\Fell}, \Omega)[\m]$ 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 is $p$-old if it arises from $S_2(\Gamma_0(M))$.) \begin{lem}[Wiles]\label{lem:wiles} If $\m$ is an ordinary prime of $\T$ of characteristic $\ell$ and $\ord_{\ell}(pM)=1$, then $\m$ is not in the support of $\T'/\T$. \end{lem} \begin{proof} This follows from \cite[Lem.~2.2, pg.~485]{wiles}, 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 further comments about the proof of this lemma. \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 the lemma; 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 forms of level~$N$.) Here is an example with $p=2$ and $N=11$: 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~$2$. \end{enumerate} \end{proof} We now summarize the conclusions we can make from the lemmas so far. 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, the level~$N$ is prime. Moreover, 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} %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}. Let $A\subset J_0(pM)$ be the image of~$e$ (note that we denoted this image by $\Adual$ in Section~\ref{sec:firstpart}). 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}[Congruence Ideal] 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 $\T\tensor\Q$. 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} 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 Lemma~\ref{lem:wiles} and Proposition~\ref{prop: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 Lemma~\ref{lem:wiles}. By Lemma~\ref{lem:m_ell}, 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 degree''. In this case, the inclusion $I\subset J$ corresponds to the divisibility $ \nAfe \mid \rAfe, $ 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}. \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{Quotients of arbitrary dimension: generalization of the Manin Constant} \label{sec:genman} %Let $N$ denote a positive integer greater than~$4$, and let %$\Gamma$ denote 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~$X$. %Thus, if $\Gamma = \Gamma_0(N)$, %then $X = X_0(N)$ and $J = J_0(N)$, and if $\Gamma = \Gamma_1(N)$, %then $X = X_1(N)$ and $J = J_1(N)$. %Let~$I$ be a saturated ideal of the corresponding Hecke algebra~$\T$ %(i.e., such that $\T/ I$ is torsion free). Let %$A = J/IJ$ be the optimal quotient of~$J$ associated to~$I$ (this %quotient is optimal, because $IJ$ is an abelian subvariety, hence %connected). %In particular, if $f$ is a newform on~$\Gamma$, %and $I_f$ is the annihilator %of~$f$ in~$\T$, then the quotient abelian %variety~$J/I_f J$ is called the {\em newform (optimal) quotient} %associated to~$f$, and we denote it by~$A_f$. Let the notation be as in the beginning of Section~\ref{sec:quotients}. Let $J_{\rm old}$ denote the abelian subvariety of~$J$ generated by the images of the degeneracy maps from levels that properly divide~$N$ (e.g., see~\cite[\S2(b)]{maziso}) and let $J^{\rm new}$ denote the quotient of~$J$ by~$J_{\rm old}$. In Section~\ref{maninmotiv}, we generalize the notion of the Manin constant to quotients~$A$ as above, and conjecture that this constant is~$1$ for newform quotients of $J_0(N)$ and $J_1(N)$. In Section~\ref{section:integrality}, we show that the (generalized) Manin constant is an integer. In the next two sections, we give generalizations of some of the results from Section~\ref{sec:elliptic} to quotients~$A$ that factor through~$J_0(N)^{\rm new}$. In Section~\ref{proofofstein}, we show that if the level~$N$ is squarefree and $A$ is a factor of~$J_0(N)^{\rm new}$, then the Manin constant is a power of~$2$, whose exponent is bounded above by the dimension of~$A$ if $A$ is a newform quotient. In Section~\ref{proofsofman}, we prove that if $A$ is a newform quotient of~$J_0(N)$ and the level is squarefree, then the Manin constant is coprime to the congruence number. %Note that Sections~\ref{section:integrality},~\ref{proofofstein}, %and~\ref{proofsofman} can be read independently of each other, %after reading Section~\ref{maninmotiv}. \subsection{The definition of the generalized Manin constant and a conjecture} \label{maninmotiv} As in Section~\ref{gen-cong-mod}, if~$R$ is a subring of~$\C$, let $S_2(R)=S_2(\Gamma;R)$ denote the $\T$-submodule of~$S_2(\Gamma; {\C})$ consisting of modular cuspforms whose Fourier expansions at ~$\infty$ have coefficients in~$R$. Note that $S_2(R)\isom S_2(\Z)\tensor R$. If~$A$ is an abelian variety over~$\Q$ and~$n$ is a positive integer, let $A_{\Z[1/n]}$ denote the N\'{e}ron model of~$A$ over~$\Z[1/n]$. On a N\'eron model, the global differentials are the same as the group of invariant differentials, so the group $H^0(A_{\Z},\Omega^1_{A_\Z/\Z})$ is free of rank~$d$, where $d=\dim(A)$ and $\Omega^1_{A_\Z/{\Z}}$ is the sheaf of differentials on the N\'{e}ron model~$A_{\Z}$ of~$A$. Let $D$ be a generator of $\bigwedge^d H^0(A_{\Z},\Omega^1_{A_\Z/\Z})$. The {\em real volume}~$\Omega_A$ of~$A$ is the volume of $A(\R)$ with respect to the measure given by~$D$. This quantity is of interest because it appears in the Birch and Swinnerton-Dyer conjecture, which expresses the ratio $L(A,1)/\Omega_A$ in terms of certain arithmetic invariants of~$A$ (see \cite[Chap.~III, \S5]{lang} and \cite{agst-bsd}). Let $g_1, \ldots , g_d$ be a $\Z$-basis of $S_2(\Z)[I]$, and for $j = 1, \ldots, d$, let $$\omega'_j = 2 \pi i g_j(z) dz \in H^0(X,\Omega_{X/{\Q}}) = H^0(J,\Omega_{J/{\Q}})$$ (where we use the standard map $X \ra J$ that sends the cusp~$\infty$ to~$0$). As before, let $\phi_2$ denote the quotient map $J \ra A$. Then $\phi_2^*$ induces an isomorphism $H^0(A,\Omega_{A/{\Q}}) \ra \bigoplus_j \Q w_j'$. For $j = 1, \ldots, d$, let $\omega_j = (\phi_2^*)^{-1} \omega_j'$. % = 2 \pi i g_j(z) dz \in H^0(A,\Omega_{A/{\Q}})$ for $j = 1, \ldots, d$. %Choose a~$\Q$-basis $\omega_1, \ldots , \omega_d$ %for $H^0(A,\Omega_{A/{\Q}})$ that corresponds to a In calculations (see \cite{agst-bsd}), or while proving formulas regarding the ratio mentioned above (see \cite[\S2]{aginv}), instead of working with~$\Omega_A$, it is easier to work with the volume~$\Omega_A'$ of~$A(\R)$ with respect to the measure given by $\wedge_j \omega_j$. There exists $c \in \Q^*$ such that \mbox{$D = c \cdot \wedge_j \omega_j$}. The absolute value of~$c$ depends only on~$I$, and is independent of other choices made above. \begin{defi} Let $A$~be an optimal quotient of~$J$ attached to an ideal~$I$ of the Hecke algebra, as above. The {\em Manin constant}~$\ca$ of the optimal quotient~$A$ is the absolute value of the constant~$c$ defined above. \end{defi} \comment{ We have the following generalization of Thm.~\ref{edixman} of Edixhoven; we give the proof in Section~\ref{section:integrality}. \begin{thm}\label{maninint} The Manin constant $\ca$ is an integer. \end{thm} } If~$A$ has dimension one, then $\ca$ is as in Definition~\ref{def:ce}. The constant $c$ as defined above was considered by Gross~\cite[(2.5) on p.~222]{gross} and Lang~\cite[III.5, p.95]{lang}, although they did not explicitly state its relation to the usual Manin constant (for elliptic curves). The constant~$\ca$ was defined for a particular quotient~$A$ in~\cite{aginv}, where it was called the generalized Manin constant. In \cite{ces} it is called the Manin index. If one works with the easier-to-compute volume~$\Omega_A'$ instead of~$\Omega_A$, it is necessary to obtain information about~$\ca$ in order to make conclusions regarding the Birch and Swinnerton-Dyer conjecture, since $\Omega_A = \ca \cdot \Omega_{A'}$. This is our motivation for studying the Manin constant. Cremona's method for proving that $\ca=1$ for a specific elliptic curve, i.e., computing $c_4$ and $c_6$ and checking that they are invariants of a minimal Weierstrass model, is of little use when~$A$ has dimension greater than one, since there is no simple analogue of the minimal Weierstrass model for general~$A$. %Let $J_{\rm old}$ denote the abelian subvariety of~$J$ generated %by the degeneracy maps from levels dividing~$N$ (e.g., %see~\cite[\S2(b)]{maziso}) and let $J^{\rm new}$ denote %the quotient of~$J$ by~$J_{\rm old}$. We call $J^{\rm new}$ %the new quotient of~$J$. Note that the Manin constants $\ca$ might not equal~$1$, especially if $A$ is not a quotient of~$J^{\rm new}$ (see Remark~\ref{rmk:maninconstnotone}). At the same time, if $A$ is a newform quotient and the level~$N$ is squarefree, then Theorems~\ref{thm:stein},~\ref{thm:stein-raynaud}, and~\ref{agn} suggest that the Manin constant is~$1$ for such quotients. \comment{ the following results for quotients of the new part of~$J_0(N)$. We postpone the proofs until Sections~\ref{proofofstein} and~\ref{proofsofman}. In the following, $p$ always denotes a prime, $A$ is an optimal quotient of~$J_0(N)$ attached to a saturated ideal~$I\subset \T$, and if we write $A=A_f$, then we mean that~$I$ is the annihilator of a newform of level~$N$. The following theorem generalizes Theorem~\ref{mazman} of Mazur: \begin{thm} \label{thm:stein} Suppose the quotient map $J_0(N) \ra A$ factors through~$J_0(N)^{\rm new}$. If $p\mid \ca$, then $p^2 \mid N$ or $p=2$. \end{thm} The following theorem generalizes Theorem~\ref{thmofraynaud} of Raynaud: \begin{thm} \label{thm:stein-raynaud} If $4 \nmid N$, then ${\rm ord}_2(\caf) \leq \dim A_f$. \end{thm} We also have the following theorem (which is a generalization of Theorem~\ref{agell} whose proof builds on techniques of~\cite{abbull}): \begin{thm} \label{agn} If $p\mid \caf$, then $p^2 \mid N$ or $p \mid \rAfe$. \end{thm} Note that the techniques of the proof of this theorem can be used to prove that if the quotient map $J_0(N) \ra A$ factors through~$J_0(N)^{\rm new}$, and if $p\mid \ca$, then $p^2 \mid N$ or $p=2$ or $p \mid \rAe$ (see Remark~\ref{finalrmk}). However, this does not add anything new, in light of Theorem~\ref{thm:stein}. } %These theorems provide evidence %that the Manin constant of newform optimal %quotients equals~$1$ when the level is square free. In the case when the level is not square free, computations of \cite{FLSSSW} involving Jacobians of genus~$2$ curves that are quotients of~$J_0(N)^{\rm new}$ show that $\ca=1$ in~$28$ case of $2$-dimensional newform quotients. These include quotients having the following non-square-free levels: $$3^2\cdot 7,\quad 3^2\cdot 13, \quad 5^3,\quad 3^3\cdot 5,\quad 3\cdot 7^2, \quad 5^2\cdot 7, \quad 2^2\cdot 47, \quad 3^3\cdot 7.$$ %These $28$ cases were done %without assuming any conjectures. The above observations are evidence for the following conjecture, which generalizes Conjecture~\ref{conjman} of Manin: \begin{conj} \label{conjmannew} If $f$ is a newform on $\Gamma_0(N)$ or $\Gamma_1(N)$, then $\caf=1$. \end{conj} \begin{rmk}\label{rem:joyce} The above conjecture does not hold if all we know is that $J_0(N) \ra A$ factors through~$J_0(N)^{\rm new}$. Adam Joyce~\cite{joyce} found an optimal quotient of~$J_0(431)^{\rm new}$ whose Manin constant is~$2$ (this example is motivated by~\cite{kilford}); note that this optimal quotient is not attached to a single newform. \end{rmk} \subsection{Integrality of the Manin constant} \label{section:integrality} We continue to use the notation introduced so far in Section~\ref{sec:genman}. In this section we prove Theorem~\ref{maninint}, which asserts that the (generalized) Manin constant is an integer, which generalizes Thm.~\ref{edixman} of Edixhoven (see also \cite[\S6.1.2]{ces} for a similar argument). The proof is itself a generalization of that of~\cite[Prop.~2]{edix:manin}. The main idea is to construct an injective map on $H^0(A,\Omega^1_{A/\Q})$ using ``$q$-expansions'', then show that the image of $H^0(A_{\Z},\Omega^1_{A_{\Z}/{\Z}})$ under this map is contained in the image of $\oplus_j \Z \omega_j$. We assume that $N > 4$, which is harmless, since $J_1(N)$ has dimension $0$ for $N\leq 10$. Using the standard immersion $X \hra J$ sending $\infty$ to $0$, we have maps \begin{eqnarray} \label{xja} X \hra J \ra A. \end{eqnarray} If $X = X_1(N)$, then we obtain a map $X_1(N) \ra A$. If $X = X_0(N)$, then composing with the standard map $X_1(N) \ra X_0(N)$ we get a map $X_1(N) \ra A$. In either case, denote the resulting map $X_1(N) \ra A$ by~$\phia$. Consider the model $\Xmu(N)$ over~$\Z$ for~$X_1(N)$ whose affine points parametrize isomorphism classes of pairs $(E,i)$, where $E$ is an elliptic curve and $i:{\bf{\mu}}_N \hookrightarrow E^{\reg}$ is an immersion, as in~\cite{katz:Eis} (see also~\cite[\S9.3.6, p.~80]{diamond-im}). Since $\Xmu(N)$ is smooth over~$\Z$ (by~\cite[\S{}II.2.5]{katz:Eis}), the N\'{e}ron mapping property implies that there is a map $$\Xmu(N) \ra A_{\Z},$$ which we again denote by~$\phia$. The Tate curve $E_q$ over $\Z[[q]]$ with the canonical immersion of ${\bf \mu}_N$ gives a map (see, e.g., \cite[p.~112]{diamond-im}) \begin{equation}\label{eqn:tau} \tau: \Spec \Z[[q]] \ra \Xmu(N). \end{equation} Pulling back differentials, gives a map $$ H^0\left(\Xmu(N), \Omega^1_{\Xmu(N)/{\Z}}\right) \lra H^0\left(\Spec \Z[[q]], \Omega^1_{\Z[[q]]/\Z}\right). $$ Now $H^0(\Spec \Z[[q]], \Omega^1_{\Z[[q]]/\Z})$ is free of rank one over~$\Z[[q]]$ with generator~$dq$, so we get a map $$H^0\left(\Xmu(N), \Omega^1_{\Xmu(N)/{\Z}}\right) \lra {\Z}[[q]].$$ Let~$\qe$ denote the composite $$H^0\left(\Xmu(N), \Omega^1_{\Xmu(N)/{\Z}}\right) \lra {\Z}[[q]] \stackrel{q \cdot}{\lra} \Z[[q]], $$ where the second map is multiplication by~$q$. Next, we relate $\qe$ to the usual Fourier-expansion over~$\C$. Since $\Xmu(N) \tensor \C \isom X_1(N)_{\C}$, the Tate curve over~$\C$ (see~\cite[VII.4.2]{dera}) gives a map $$\tauc: \Spec \C[[q]] \ra X_1(N)_\C,$$ which is the base extension of (\ref{eqn:tau}) and which identifies~$q$ with the local parameter $e^{2\pi i z}$ on~$X_1(N)_\C$ at the cusp~$\infty$. As above, pulling back differentials, gives a map $$H^0\left(X_1(N)_\C, \Omega^1_{X_1(N)/{\C}}\right) \lra \C[[q]].$$ Let~$\Fe$ denote the composite $$H^0\left(X_1(N)_\C, \Omega^1_{X_1(N)/{\C}}\right) \lra {\C}[[q]] \stackrel{q \cdot}{\lra} \C[[q]], $$ where the second map is multiplication by~$q$. We obtain a commutative diagram \begin{equation}\label{comdiag} \xymatrix@=3em{ {H^0(A_{\Z}, \Omega^1_{A_{\Z}/{\Z}})}\ar[r]^{\phia^*\qquad}\ar[d]& {H^0(\Xmu(N), \Omega^1_{\Xmu(N)/\Z})}\ar[r]^{\qquad\quad\sqe}\ar[d] & {\Z[[q]]}\ar[d]\\ {H^0(A_{\C}, \Omega^1_{A_{\C}/\C})}\ar[r]^{\phia^* \tensor \C\qquad} & {H^0(X_1(N)_{\C}, \Omega^1_{X_1(N)/\C})}\ar[r]^{\qquad\qquad\sFe} & {\C[[q]]} \\ } \end{equation} in which the first and last vertical maps are injections. The relation of $\Fe$ to the Fourier expansion of cusp forms is given by the following lemma. Let $\psi$ be the isomorphism $$ \psi:S_2(\Gamma_1(N),\C) \stackrel{\isom}{\ra} H^0(X_1(N)_{\C}, \Omega^1_{X_1(N)/\C}) $$ given by $f(z) \mapsto 2\pi i f(z) dz$. \begin{lem} \label{Fourexp} Let $f \in S_2(\Gamma_1(N),\C)$, and let $\{a_n\}$ be the coefficients of the Fourier expansion of~$f$. Then $\Fe(\psi(f)) = \sum_n a_n q^n$. \end{lem} \begin{proof} If $f \in S_2(\Gamma_1(N),\C)$, and its Fourier series is $\sum_n a_n e^{2\pi i z n}$, then $\psi(f) = 2 \pi i \sum_n a_n e^{2\pi i z n} dz$. Since $\tauc$ identifies~$q$ with the local parameter $e^{2\pi i z}$, we see that the pullback of~$\psi(f)$ via~$\tauc$ to $H^0(\Spec \C[[q]], \Omega^1_{\C[[q]]/\C})$ is $\sum_n a_n q^{n-1} dq$. So $\Fe(\psi(f)) = \sum_n a_n q^n$. \end{proof} \begin{lem} \label{subgrouplemma} The group $\qe\left(\phia^*\left(H^0(A_{\Z}, \Omega^1_{A_{\Z}/{\Z}})\right)\right)$ is a subgroup of $\Fe (\psi (S_2(\Gamma_1(N),\Z)[I]))$. %and the quotient group is finite. \end{lem} \begin{proof} If $x \in H^0(\Xmu(N), \Omega^1_{\Xmu(N)/\Z})$ maps to~$y\in{}H^0(X_1(N)_{\C}, \Omega^1_{X_1(N)/\C})$, then by the commutativity of the right half of the commutative diagram above and by Lemma~\ref{Fourexp}, the Fourier expansion of $\psi^{-1}(y) \in S_2(\Gamma_1(N),\C)$ is the same as $\qe(x)$, i.e., has integral Fourier coefficients; hence $\psi^{-1}(y) \in S_2(\Gamma_1(N),\Z)$. This gives an injection $$\qe\left(H^0(\Xmu(N), \Omega^1_{\Xmu(N)/\Z})\right) \hookrightarrow \Fe( \psi( S_2(\Gamma_1(N),\Z) )).$$ Now the lemma follows from the fact that $\phia^*(H^0(A_{\Z}, \Omega^1_{A_{\Z}/{\Z}}))[I]=0$. \end{proof} \begin{prop}\label{subgroupprop} We have $H^0(A_{\Z},\Omega^1_{A_\Z}) \subseteq \oplus_j \Z \omega_j$, considered as subgroups of $H^0(A,\Omega^1_{A/\Q})$. %The Manin constant $\ca$ is the order of the group~$C_A$. \end{prop} \begin{proof} Let $\phi$ denote the composite $$H^0(A_{\C}, \Omega^1_{A_{\C}/\C}) \stackrel{\phia^* \tensor \C}{\lra} H^0(X_1(N)_{\C}, \Omega^1_{X_1(N)/\C}) \stackrel{\sFe }{\lra} \C[[q]].$$ Now $\phia^*$ is injective: if $X = X_1(N)$, this follows by considering pullbacks along the sequence of maps in~(\ref{xja}); if $X = X_0(N)$, then a similar argument works, noting that the pullback of differentials along $X_1(N) \ra X_0(N)$ is injective. Also, $\Fe$ is injective since the Fourier expansion map is injective. Thus $\phi$ is injective. By Lemma~\ref{subgrouplemma} and diagram (\ref{comdiag}), we have $\phi(H^0(A_{\Z},\Omega^1_{A_{\Z}/\Z})) \subseteq \phi(\oplus_j \Z \omega_j)$. As $\phi$ is injective, $H^0(A_{\Z},\Omega^1_{A_\Z}) \subseteq \bigoplus_j \Z \omega_j$. \end{proof} We obtain the following theorem as a corollary of Proposition~\ref{subgroupprop}: \begin{thm}\label{maninint} The Manin constant $\ca$ is an integer. \end{thm} We finish this section with a few remarks. \begin{rmk} The quotient $$\frac{\oplus_j \Z \omega_j}{H^0(A_{\Z},\Omega^1_{A_\Z})} \isom \frac{\psi(S_2(\Gamma_1(N),\Z))}{\phia^*(H^0(A_{\Z},\Omega^1_{A_\Z}))} \isom \frac{\Fe (\psi (S_2(\Gamma_1(N),\Z)[I]))}{\qe\left(\phia^*\left(H^0(A_{\Z}, \Omega^1_{A_{\Z}})\right)\right)} $$ is in fact a module over~$\T$, and hence one may in general be interested in its module structure, as opposed to just the Manin constant, which is its order. %However, we shall not go into such questions in this paper. \end{rmk} \begin{rmk} \label{x0nremark} The reason we used the model $\Xmu(N)$ was that we needed a smooth model over~$\Z$ so that we can use the N\'eron mapping property to define a $q$-expansion map over~$\Z$ that agreed with the usual one over~$\C$. When $A$ is a quotient of~$J_0(N)$, (i.e., when $J = J_0(N)$), we could use a model for~$X_0(N)$ in the proof above, as we describe now. %\edit{William: I think the %rest of this 2 page remark should be cut. My feeling is that the only %reason it is here is because it was a lot of work to write. That's %not sufficient justification for 2 pages.} By~\cite[6.6.1]{katz-mazur}, the moduli problem $[\Gamma_0(N)]$ (\cite[3.4]{katz-mazur}) is relatively representable and finite. The moduli problem $[\Gamma_0(N)]$ is also regular (by~\cite[6.6.1]{katz-mazur} again), and hence normal, and so the associated coarse moduli scheme $M([\Gamma_0(N)])$ is normal (by~\cite[8.1.2]{katz-mazur}). So one can use \cite[\S8.6]{katz-mazur} to compactify it; call the resulting compactification~$M_0(N)$. Let $M_0(N)^0$ be the open part of~$M_0(N)$ where the projection to~$\Spec \Z$ is smooth. For the case where $J = J_0(N)$, we could have used $M_0(N)^0$ instead of~$\Xmu(N)$ for proving integrality of the Manin constant. This is what was done in the proof of Prop.~2 in~\cite{edix:manin}, but some of the details were skipped, which we mention two paragraphs below. Note that $q$-expansion maps over~$\Z$ or~$\Z[1/m]$ (where $m$ is the largest square that divides~$N$) on differentials on certain models of~$X_0(N)$ have been constructed in several places in the literature (e.g., \cite[p.141]{maziso}, \cite[p.271]{abbull}), and the usual reference given is~\cite{dera}. However, this seems inadequate, since in~\cite{dera}, one has to invert~$N$ to get a moduli-theoretic interpretation at the cusps. And in~\cite{katz-mazur}, while the models are over~$\Z$, they do not give a moduli interpretation at the cusps. We now indicate how the construction of a $q$-expansion map over~$\Z$ for differentials on $M_0(N)^0$ can be justified (this is probably well-known to experts). One method, communicated to us by B.~Edixhoven, is as follows: Consider the Tate curve ${\rm Tate}(q)$ over~$\Z((q))$ as in~\cite[p.258]{katz-mazur} along with its canonical subgroup ${\bf{\mu}}_{\scriptscriptstyle N}$. This gives us an element of~$M_0(N)(\Z((q)))$ as in~\cite[\S8.11]{katz-mazur}. One then verifies that this element extends uniquely to an element of~$M_0(N)(\Z[[q]])$. Thus we get a map $\tau: \Spec \Z[[q]] \ra M_0(N)$ and composing with the map $\Spec \Z \ra \Spec \Z[[q]]$ (given by $q \mapsto 0)$), we get a point in $M_0(N)(\Z)$, called the cusp~$\infty$. The structure along $\infty$ of $M_0(N)$ is described in~\cite[\S1.2]{edix-min}; in particular, the completion along~$\infty$ is given by $\Z[[q]]$, and so $\infty$ is a smooth point. Thus the map~$\tau$ factors through $M_0(N)^0$, and so we can define a $q$-expansion map on $H^0(M_0(N)^0, \Omega_{M_0(N)^0/\Z})$ as we did (for~$\Xmu(N)$) above. The usual $q$-expansion map over~$\C$ is just given by extending scalars from~$\Z$ to~$\C$ in the description just above, and hence our $q$-expansion map is compatible with the usual one over~$\C$. Another method, which is more moduli-theoretic, was communicated to us by B.~Conrad, and is as follows: it is shown in~\cite{conrad:modular} that one can merge the ``affine'' moduli-theoretic $\Z$-theory in~\cite{katz-mazur} with the ``proper'' moduli-theoretic $\Z[1/N]$-theory in~\cite{dera}. Using this, one can show that the proper schemes over~$\Z$ in~\cite{katz-mazur} are in fact moduli schemes for generalized elliptic curves with ``Drinfeld structure''. Then, by the moduli interpretation, the Tate curve with its canonical subgroup gives a map~$\tau$ and the cusp~$\infty$ as in the previous paragraph. Next, one can use a deformation theoretic argument to show that the cusp~$\infty$ is a smooth point, i.e., that $\tau$ factors through~$M_0(N)^0$. As in the previous paragraph, one can now pullback via~$\tau$ to get the $q$-expansion map over~$\Z$, which by the moduli interpretation agrees with the usual $q$-expansions over~$\C$. \end{rmk} \begin{rmk}\label{rmk:maninconstnotone} The Manin constants $\ca$ might not equal~$1$. For example, let $\Gamma = \Gamma_0(N)$, and suppose $A=J_0(N)$ is the quotient by the trivial ideal. Let us work in the setting of Remark~\ref{x0nremark}, using the model~$M_0(N)^0$ over~$\Z$ of~$X_0(N)$. Then, since $A = J_0(N)$, the map~$\phia^*$ is just $$ H^0(J_0(N)_\Z, \Omega^1_{J_0(N)/\Z}) \ra H^0(M_0(N)^0, \Omega^1_{M_0(N)^0/\Z}), $$ which is an isomorphism. Let us identify $S_2(\Z)$ with its image in~$\Z[[q]]$. Then using the argument in the proof of Proposition~\ref{subgroupprop} we see that $\ca$ is the order of the cokernel of the map \begin{eqnarray}\label{qexp1} H^0(M_0(N)^0, \Omega^1_{M_0(N)^0/{\Z}}) \xrightarrow{\sqe} S_2(\Z), \end{eqnarray} where $\qe$ is the $q$-expansion map discussed in Remark~\ref{x0nremark}. The map~(\ref{qexp1}) need not be surjective, and the order of its cokernel can be calculated by using methods in \cite[VII.3.17]{dera} (see~\cite{edix:comparison}). For example, B.~Edixhoven observed that for~$N=33$ the cokernel has order~$3$, so $c_{\scriptscriptstyle{J_0(33)}} = 3$. B.~Edixhoven also informed us that if~$N$ is square free, then the map~(\ref{qexp1}) is surjective if and only if there are no old spaces in $S_2(\Gamma_0(N),\C)$ (cf.~\cite{edix:comparison}). See also Remark~\ref{rem:joyce} for an example of a quotient of $J_0(N)$, with~$N$ prime, and with Manin constant~$2$. Note that $H^0(M_0(N)^0_\Z, \Omega^1_{M_0(N)^0/\Z})$ is precisely the subgroup of $S_2(\Q) = H^0(X_0(N), \Omega^1_{X_0(N)/\Q})$ of elements that have integral Fourier expansion at all the cusps (this follows from the interpretation in~\cite{edix:comparison} of the integrality condition in terms of a differential having no pole along along any irreducible component of~$M_0(N)^0$). Whereas $S_2(\Z)$ consists of differentials that are required only to have integral Fourier expansion at the cusp~$\infty$. If one assumes the BSD conjecture, then a comparison of formulas for the ratio $L(J_e,1)/\Omega_{J_e}$, where $J_e$ is the winding quotient of prime level, and the corresponding formulas for winding quotients of level a product of two distinct primes (see \cite[Thm.~3.2.2 and Thm.~4.2.1]{mythesis}) suggests that the Manin constant of such winding quotients is not~$1$ when there are old forms involved (see \cite[\S4.2.1]{mythesis} for details). \end{rmk} \subsection{Generalizations of theorems of Mazur and Raynaud} \label{proofofstein} In this section, we prove the following two theorems: \begin{thm} \label{thm:stein} Let $A$ be a quotient of $J=J_0(N)$ by an ideal of the Hecke algebra such that the quotient map factors through~$J_0(N)^{\rm new}$. If $p$ is a prime such that $p\mid \ca$, then $p^2 \mid 4N$. \end{thm} \begin{thm} \label{thm:stein-raynaud} Let $f$ be a newform on~$\Gamma_0(N)$, and let $A_f$ be the associated newform quotient. If $4 \nmid N$, then ${\rm ord}_2(\caf) \leq \dim A_f$. \end{thm} Theorem~\ref{thm:stein} generalizes Mazur's Theorem~\ref{mazman}, while Theorem~\ref{thm:stein-raynaud} generalizes Raynaud's Theorem~\ref{thmofraynaud}. The proofs of the theorems are similar. Suppose $p\mid\mid N$. The reduction $X_0(N)_{\Fp}$ is a union of two copies of $X_0(N/p)_{\Fp}$, identified at the supersingular points. A differential on $X_0(N)_{\Fp}$ has $q$-expansion~$0$ if and only if it vanishes on the component~$X$ of $X_0(N)_{\Fp}$ that contains~$\infty$. Since there can be differentials that vanish on~$X$, but not on the other component, the $q$-expansion map on differentials on $X_0(N)_{\Fp}$ need not be injective. However, as Mazur observed in \cite{maziso}, if a differential is an eigenvector for the Atkin-Lehner involution $W_p$, then it is $0$ on one component if and only if it is $0$ on both components, since $W_p$ swaps the two components. That the $q$-expansion map {\em is} injective on each eigenspace for $W_p$ is one of the key ideas behind the proofs of Theorems \ref{thm:stein}, \ref{thm:stein-raynaud}, and~\ref{agn}. %\subsubsection{Proof of Theorem~\ref{thm:stein}} \label{stein-mazur} \begin{proof}[Proof of Theorem~\ref{thm:stein}] We want to prove that that~$\ca$ is a unit in $\Z[\frac{1}{2m}]$, where~$m$ is the largest square dividing~$N$. We do this by generalizing the proof of \cite[Prop.~3.1]{maziso}. Let $R=\Z[\frac{1}{2m}]$, and let $J_R$ denote the N\'eron model of $J_0(N)$ over~$R$. Let~$\cX$ be the smooth locus of a minimal proper regular model for $X_0(N)$ over~$R$, and let $\Omega_\cX$ denote the sheaf of ``regular differentials'', denoted~$\Omega$ in~\cite[\S2(e)]{maziso}. Let~$\pi$ denote the map $J_0(N)\ra A$. Consider the diagram \begin{equation}\label{eqn:qexp} H^0(A_R,\Omega_{A_R}) \xrightarrow{\pi^*} H^0({J_R},\Omega_{{J_R}}) \isom H^0(\cX,\Omega_\cX) \xrightarrow{\text{ $q$-exp }} R[[q]], \end{equation} where the map $q$-exp is as in~\cite[\S2(e)]{maziso}. (Note that we defined a different $q$-expansion map in Section~\ref{section:integrality}.) The composite of the maps in~(\ref{eqn:qexp}) must be an inclusion %The map~$\pi^*$ must be an inclusion because $H^0(A_R,\Omega_{A_R})$ is torsion free and %~$\pi^*$ the composite is an inclusion after tensoring with~$\C$. To show that the generalized Manin constant is a unit in~$R$, it suffices to check that the image of $H^0(A_R,\Omega_{A_R})$ in $R[[q]]$ is {\em saturated}, in the sense that the cokernel is torsion free. This is because the image of $S_2(\Gamma_0(N);R)[I]$ is saturated in $R[[q]]$ and $S_2(\Gamma_0(N);R)[I]\tensor\Q = H^0(A_R,\Omega_{A_R})\tensor\Q$. For the image of $H^0(A_R,\Omega_{A_R})$ in $R[[q]]$ to be saturated means that the quotient~$D$ is torsion free. Let~$\ell$ be a prime not dividing~$2m$. Tensoring $$0\ra H^0(A_R,\Omega_{A_R})\xrightarrow{\text{$q$-exp}} R[[q]]\ra D\ra 0$$ with~$\Fell$, we obtain $$0 \ra D[\ell] \ra H^0(A_R,\Omega_{A_R})\tensor\Fell\ra \Fell[[q]] \ra D\tensor\Fell \ra 0.$$ Here we have used either the snake lemma applied to multiplication-by-$\ell$ or that $\Tor^1(D,\Fell)$ is the $\ell$-torsion in~$D$, and that $\Tor^1(-,\Fell)$ vanishes on the torsion-free group $R[[q]]$. To show $D[\ell]=0$, it suffices to prove injectivity of $$ \Phi:H^0(A_R,\Omega_{A_R})\tensor\Fell\lra \Fell[[q]]. $$ Since $A$ is optimal, $J$ has good or semistable reduction at~$\ell$, and $\ell\neq 2$, \cite[Cor 1.1]{maziso} gives an exact sequence $$0 \ra H^0(A_{\Zell},\Omega_{A_{\Zell}}) \ra H^0(J_{\Zell},\Omega_{J_{\Zell}}) \ra H^0(B_{\Zell},\Omega_{B_{\Zell}}) \ra 0$$ where $B=\ker(J\to A)$. Since $H^0(B_{\Zell},\Omega_{B_{\Zell}})$ is torsion free, the map $$H^0({A_R}_{\Zell},\Omega_{{A_R}_{\Zell}})\tensor\Fell \ra H^0({J_R}_{\Zell},\Omega_{{J_R}_{\Zell}})\tensor\Fell \isom H^0(\cX_{\Fell},\Omega_{\cX_{\Fell}})$$ is injective. We also remark that $$H^0({A_R},\Omega_{{A_R}})\tensor\Fell\isom H^0(A_{\Zell},\Omega_{{A_{\Zell}}})\tensor\Fell,$$ because~$\Zell$ is torsion free, hence flat over~$R$. This proves injectivity of $$H^0({A_R},\Omega_{{A_R}})\tensor\Fell\ra H^0(\cX_{\Fell},\Omega_{\cX_{\Fell}}).$$ If $\ell\nmid N$, then injectivity of~$\Phi$ now follows from the $q$-expansion principle, which asserts that the $q$-expansion map $H^0(\cX_{\Fell},\Omega_{\cX_{\Fell}})\ra \Fell[[q]]$ is injective. (This part of the argument does not assume that~$A$ is new.) Next suppose that~$\ell\mid{}N$; note that $\ell \mid\mid N$ because $\ell\nmid m$. As mentioned above, the reduction $\cX_{\Fell}$ is a union of two copies of $X_0(N/\ell)_{\Fell}$ identified transversely at the supersingular points, and these two copies are swapped under the action of the Atkin-Lehner involution $W_\ell$. If $\omega\in\ker(\Phi)$, then the $q$-expansion principle implies that~$\omega$ vanishes on the irreducible component containing the cusp~$\infty$. The action of~$W_\ell$ on $H^0({A_R},\Omega_{{A_R}}) \tensor \Fell$ is diagonalizable since its minimal polynomial divides $X^2 -1$, and $X^2-1$ has distinct roots since $\ell \neq 2$, and the eigenvalues $\pm~1$ are in~$\F_\ell$. %However, since~$A$ corresponds to a single %Galois-conjugacy class of newforms, $W_\ell$ acts as the scalar $\pm %1$ on $H^0({A_R},\Omega_{{A_R}})$. Let $\omega\in \ker(\Phi)$ be in the $+1$ eigenspace for the action of~$W_\ell$. If~$\omega$ is also nonzero on the component that does not contain~$\infty$, then $\omega = W_\ell(\omega)$ is nonzero when restricted to the component that contains~$\infty$, which is a contradiction. Therefore $\omega=0$. A similar argument shows that if $\omega\in \ker(\Phi)$ is in the $-1$ eigenspace for the action of~$W_\ell$, then $\omega=0$. Hence~$\Phi$ is injective, as required. \end{proof} %We generalize a result of Mazur and Raynaud (see \cite[Prop.~3.1, %pg.~274]{abbull}) to arbitrary dimension. \begin{proof}[Proof of Theorem~\ref{thm:stein-raynaud}] Recall that we want to prove that if $A=A_f$ is a quotient of $J= J_0(N)$ attached to a newform~$f$, and $4\nmid N$, then $\ord_2(\ca \leq \dim(A)$. The proof closely follows the one in \cite{abbull}, except at the end we argue using indexes instead of multiples. Let $B$ denote the kernel of the quotient map $J \ra A$. Consider the exact sequence $0\to B\to J\to A\to 0$, and the corresponding complex $ B_{\Z_2}\to J_{\Z_2} \to A_{J_{\Z_2}}$ of N\'eron models. Because $J_{\Z_2}$ has semiabelian reduction (since $4\nmid N$), Theorem A.1 of the appendix of \cite[pg.~279--280]{abbull}, due to Raynaud, implies that there is an integer~$r$ and an exact sequence \[ 0 \to \Tan(B_{\Z_2}) \to \Tan(J_{\Z_2}) \to \Tan(A_{\Z_2}) \to (\Z/2\Z)^r \to 0. \] Here $\Tan$ is the tangent space at the $0$ section; it is a free abelian group of rank equal to the dimension. %(it gives an integral %structure on the usual tangent space, just as differentials on the %N\'eron model give an integral structure on the differentials on the %abelian variety). Note that $\Tan$ is $\Z_2$-dual to the cotangent space, and the cotangent space is isomorphic to the global differential $1$-forms. The theorem of Raynaud mentioned above is the generalization to $e=p-1$ of \cite[Cor.~1.1]{maziso}, which we used above in the proof of Theorem~\ref{thm:stein}. Let $C$ be the cokernel of $\Tan(B_{\Z_2})\to \Tan(J_{\Z_2})$. We have a diagram \begin{equation}\label{eqn:insert_C} \xymatrix@=0.15in{ 0 \ar[r] & {\Tan(B_{\Z_2})}\ar[r]& {\Tan(J_{\Z_2})}\ar[rr]\ar@{->>}[dr]&& {\Tan(A_{\Z_2})}\ar[r]& {(\Z/2\Z)^r}\ar[r]& 0.\\ & & & C\ar@{^(->}[ru]\\ % & & 0\ar[ur] & &0\ar[ul] } \end{equation} Note that $C\subset \Tan(A_{\Z_2})$, so $C$ is torsion free, hence~$C$ is a free $\Z_2$-module of rank $d=\dim(A)$. Let $C^* = \Hom_{\Z_2}(C,\Z_2)$ be the $\Z_2$-linear dual of~$C$. Applying the $\Hom_{\Z_2}(-,\Z_2)$ functor to the two short exact sequences in (\ref{eqn:insert_C}), we obtain exact sequences \[ 0 \to C^* \to \H^0(J_{\Z_2},\Omega_{J/\Z_2}) \to \H^0(B_{\Z_2},\Omega_{B/\Z_2}) \to 0, \] and \begin{equation}\label{eqn:dualc} 0 \to \H^0(A_{\Z_2},\Omega_{A/\Z_2}) \to C^* \to (\Z/2\Z)^r\to 0. \end{equation} Note that the $(\Z/2\Z)^r$ on the right in (\ref{eqn:dualc}) is really $\Ext^1_{\Z_2}((\Z/2\Z)^r,\Z_2)$, which is isomorphic to $(\Z/2\Z)^r$. Also, (\ref{eqn:dualc}) implies that $r\leq d=\dim(A)$. Let $\cX'$ be the smooth locus a minimal proper regular model for $X_0(N)$ over~$\Z[1/m]$, where~$m$ is the largest square dividing~$N$, and let $\Omega_{\cX'}$ denote the sheaf of ``regular differentials'' on $\cX'$ (denoted~$\Omega$ in~\cite[\S2(e)]{maziso}). Arguing as in the last two paragraphs of the proof of Theorem~\ref{thm:stein} above (note that since $A$ is attached to a single newform, the Atkin-Lehner involution~$W_2$ acts either as~$+1$ or as~$-1$), we see that the composition \[ C^*\tensor \F_2 \to \H^0(J_{\Z_2},\Omega_{J/\Z_2}) \tensor \F_2 \cong \H^0(\cX'_{\F_2}, \Omega_{\cX'_{\F_2}}) \xrightarrow{\text{ $q$-exp }} \F_2[[q]] \] is injective. Thus, just as in the proof of Theorem~\ref{thm:stein}, we see that the image of $C^*$ in $\Z_2[[q]]$ is saturated. The Manin constant for~$A$ at~$2$ is the index of the image via $q$-expansion of $\H^0(A_{\Z_2},\Omega)$ in $\Z_2[[q]]$ in its saturation. Since the image of $C^*$ in $\Z_2[[q]]$ is saturated, the image of $C^*$ is the saturation of the image of $\H^0(A_{\Z_2},\Omega)$, so the Manin index at~$2$ is the index of $\H^0(A_{\Z_2},\Omega)$ in $C^*$, which is~$2^r$ by (\ref{eqn:insert_C}), hence is at most~$2^d$. \end{proof} \subsection{The Manin constant and congruence primes} \label{proofsofman} In this section, we prove the following theorem, %which is a generalization of Theorem~\ref{agell} whose proof builds on techniques of~\cite{abbull}: \begin{thm} \label{agn} Let $A=A_f$ be a quotient of $J=J_0(N)$ attached to a newform~$f$. If $p\mid \caf$ is a prime, then $p^2 \mid N$ or $p \mid \rAfe$. \end{thm} The key idea is to project the ``Manin index'' to the differentials on the dual of~$A$ and to use a ``conjugate isogeny'' to bring it back to differentials on a model of~$X_0(N)$, and then use an argument similar to the one in the last two paragraphs of the proof of Theorem~\ref{thm:stein}. Note that the techniques of the proof of this theorem can be used to prove that if the quotient map $J_0(N) \ra A$ factors through~$J_0(N)^{\rm new}$, and if $p\mid \ca$, then $p^2 \mid N$ or $p=2$ or $p \mid \rAe$ (see Remark~\ref{finalrmk}). However, this does not add anything new, in light of Theorem~\ref{thm:stein}. \comment{ Recall that we want to prove that if $p\mid \ca$, then $p^2 \mid N$ or $p \mid \rAe$. } We will use notation consistent with~\cite{abbull} since we will follow their techniques closely. If $G$ is a finite group, we denote its order by~$\# \, G$. Suppose $A_1$ and~$A_2$ are abelian varieties such that there is an isogeny $f: A_1 \ra A_2$. If $n$ is a positive integer which annihilates ${\rm ker}f$, then the multiplication by~$n$ map on~$A_1$ factors through $A_1/{\rm ker}f \cong A_2$, thus giving an isogeny $f': A_2 \ra A_1$ such that $f' \circ f$ is the multiplication by~$n$ map on~$A_1$. Also one sees that $f \circ f'$ is the multiplication by~$n$ map on~$A_2$. We apply this to our situation as follows. Recall that $\phi_2$ denotes the quotient map $J \ra A$, and $\po$ denotes the composition of the dual map $\Adual \ra \Jdual$ with the canonical polarization $\Jdual \cong J$. By Proposition~\ref{modular:isogeny0}, the composite \begin{eqnarray} \label{modular:isogeny} \Adual \stackrel{\po}{\lra} J \stackrel{\pt}{\lra} A \end{eqnarray} is an isogeny. As in Definition~\ref{defi:modular}, we denote the exponent of the kernel of this isogeny by~$\nAe$. %Applying the argument in the previous paragraph %to the composite map in~(\ref{modular:isogeny}), %Recall that we defined maps $\phi_1$ and $\phi_2$ that gave %the sequence %$\Adual \stackrel{\phi_1}{\lra} J \stackrel{\phi_2}{\lra} A$ %whose composite was an isogeny %and %(Definition~\ref{modular:exponent}) %that $\na$ is the smallest integer that annihilates %the kernel of this composite isogeny. %\begin{eqnarray*} \label{modular:isogeny2} %\Adual \stackrel{\po}{\lra} J \stackrel{\pt}{\lra} A. %\end{eqnarray*} There is an isogeny $\phi': A \ra \Adual$ such that the composite \begin{eqnarray}\label{modular:isogeny2} \Adual \stackrel{\po}{\lra} J \stackrel{\pt}{\lra} A \stackrel{\phi'}{\lra} \Adual \end{eqnarray} is the multiplication by~$\nAe$ map on~$\Adual$, and the composite \begin{eqnarray}\label{modular:isogeny3} A \stackrel{\phi'}{\lra} \Adual \stackrel{\po}{\lra} J \stackrel{\pt}{\lra} A \end{eqnarray} is the multiplication by~$\nAe$ map on~$A$. %(see \cite[\S8]{milne:abvar}). Pulling back differentials along~$\pt$ then~$\po$ in~(\ref{modular:isogeny}), we obtain maps: $$H^0(A_{\C}, \Omega_{A/{\C}}^1) \stackrel{\phi_2^*}{\lra} H^0(J_{\C}, \Omega_{J/{\C}}^1) \stackrel{\phi_1^*}{\lra} \GS{\Adual}{\C}.$$ Let~$m$ denote the largest square that divides the level~$N$ and let $S = \Spec \Zm$. Let $M_0(N)$ be as in Remark~\ref{x0nremark}. Then $M_0(N)_S$ is semistable over~$S$. Let $\Omega$ be the relative dualizing sheaf of $M_0(N)_S$ over~$S$. Consider the map $$ \qe : H^0(M_0(N)_S,\Omega) \hookrightarrow \Z[1/m][[q]] $$ in~\cite[\S2.1]{abbull} (cf. Remark~\ref{x0nremark}). Note that we are abusing notation slightly since we had defined a different $q$-expansion map in Section~\ref{section:integrality}. As mentioned in~\cite[\S2.1]{abbull} we have an inclusion $$ \qe : H^0(M_0(N)_S,\Omega) \hookrightarrow S_2(\tZm)\ $$ (this really follows from the discussion in~Section~\ref{section:integrality}). This map is not an isomorphism in general, but it induces an isomorphism \begin{eqnarray} \label{qexpmodp} \qe: H^0(M_0(N)_{{\F}_p},\Omega) \stackrel{\cong}{\lra} S_2({\F}_p) \end{eqnarray} for each prime~$p$ that does not divide~$N$ (see~\cite[\S2.1]{abbull}, and the arguments in the proof of Theorem~\ref{thm:stein}). We have $$H^0(M_0(N)_S,\Omega) \hookrightarrow S_2(\tZm) \hookrightarrow S_2({\C}) \cong H^0(J_{\C}, \Omega_{J/{\C}}^1).$$ Applying~$\po^*$ to the first two groups, we get an injection $$\po^*(\HM) \hra \po^*(S_2(\tZm)),$$ where the source and target are viewed as sitting in $H^0(\Adual_\C, \Omega_{\Adual/\C})$. Denote the cokernel of the above map by~$C$. It is a finite group and, by~(\ref{qexpmodp}), the only primes that can divide its order are the primes that divide~$N$. An easy generalization of~\cite[Prop.~$3.2$]{abbull} gives $$\po^*(\HM) = \GS{\Adual}{S},$$ so we have an exact sequence $$0 \ra \GS{\Adual}{S} \ra \po^*(S_2(\tZm)) \ra C \ra 0.$$ On considering the quotient of the middle group above by the pullback of~$\GS{A}{S}$ under $\pt \circ \po$, we obtain \begin{eqnarray} \label{3} 0 \ra \frac{\GS{\Adual}{S}} {\po^*\pt^*\GS{A}{S} } \ra \frac{\po^*(S_2(\Zm))} { \po^*\pt^*\GS{A}{S}} \ra C \ra 0. \end{eqnarray} Now $\po^*$ is injective when restricted to $\pt^*\GS{A}{\C}$, because the pullback of the composite of the maps in (\ref{modular:isogeny3}) is injective, since it is multiplication by $\nAe$ on a vector space over~$\C$. So, since $$S_2({\Z})[I] \subseteq \pt^*\GS{A}{\C},$$ we have a natural isomorphism $$\frac{S_2({\Z})[I] \tensor \Zm}{\pt^*\GS{A}{S}} \stackrel{\cong}{\lra} \frac{\po^*(S_2({\Z})[I] \tensor \Zm)}{\po^*(\pt^*\GS{A}{S})}.$$ If $n$ and $m$ are positive integers, let $n_m$ denote the largest divisor of~$n$ that is coprime to~$m$. By the discussion in Section~\ref{section:integrality}, $$(\ca)_m = \# \left( \frac{S_2({\Z})[I] \tensor \Zm}{\pt^*\GS{A}{S}} \right).$$ \noindent So $$(\ca)_m = \# \left( \frac{\po^*(S_2({\Z})[I] \tensor \Zm)}{\po^*(\pt^*\GS{A}{S})} \right). $$ Hence \begin{eqnarray} \label{4} \# \left( \frac{\po^*(S_2(\Zm))} {\po^*\pt^*\GS{A}{S}} \right) = (\ca)_m \cdot \; \# \left( \frac{\po^*(S_2(\Zm))}{\po^*(S_2({\Z})[I] \tensor \Zm)} \right). \end{eqnarray} As in the proof of~\cite[Prop.~$3.3$]{abbull}, we have isomorphisms \begin{eqnarray*} \left( \frac{S_2({\Z})} { S_2({\Z})[I] \oplus W(I} \right) \tensor \tZm & \stackrel{\cong}{\lra}& \frac{S_2(\Zm)}{ (S_2({\Z})[I] \tensor \Zm) \oplus (W(I)\tensor \Zm) } \\ & \stackrel{\cong}{\lra} & \frac{\phi_1^*(S_2(\Zm)))}{ \phi_1^* (S_2({\Z})[I] \tensor{\Z}[\frac{1}{m}])}. \end{eqnarray*} Thus $$\# \left( \frac{\phi_1^*(S_2(\Zm)))} { \phi_1^* (S_2({\Z})[I] \tensor \Zm)} \right) = (\rA)_m.$$ %where $\rA$ is as in Definition~\ref{congruence:number}. Putting this in~(\ref{4}) and then using~(\ref{3}), we get \begin{eqnarray} \label{1} (\ca)_m\cdot (\rA)_m = \# \left( \frac{\GS{\Adual}{S}} { \po^* \pt^* \GS{A}{S}} \right) \cdot \# C. \end{eqnarray} Since the composite of the maps in (\ref{modular:isogeny2}) is multiplication by~$\nAe$, we see that multiplication by some power of~$\nAe$ kills $\left( \frac{\GS{\Adual}{S}} { \po^* \pt^* \GS{A}{S}} \right)$. %$\GS{\Adual}{S} / \po^* \pt^* \GS{A}{S}$. Thus we obtain the following lemma: \begin{lem} \label{n1} If $p\,\,\Big\vert\,\, \# \left( \frac{\GS{\Adual}{S}} { \po^* \pt^* \GS{A}{S}} \right)$ is a prime, then $p \mid \nAe$. \end{lem} We already remarked that a prime can divide~$\# C$ only if it divides~$N$. The main addition to the techniques of~\cite{abbull} is the following result, which further controls the primes that can divide~$\# C$: \begin{prop} \label{n2} If $A=A_f$ is a newform quotient of~$J_0(N)$ and $p \mid \# C$ is a prime, then $p^2 \mid N$ or $p \mid \rAe$. \end{prop} Before proving Proposition~\ref{n2} we use it to prove Theorem~\ref{agn}. \begin{proof}[Proof of Theorem~\ref{agn}] Suppose $p^2 \nmid N$ and $p \mid \ca$. Then $p \mid (\ca)_m$, and so by equation~(\ref{1}), $p \mid \# \left( \frac{\GS{\Adual}{S}} { \po^* \pt^* \GS{A}{S}} \right)$ or $p \mid \# C$. In the former case, by Lemma~\ref{n1}, $p \mid \nAe$, and hence by Proposition~\ref{ndivsm}, $p \mid \rAe$ and in the latter case, by Proposition~\ref{n2}, $p \mid \rAe$. \end{proof} \begin{rmk} \label{rem:diff_to_generalize} The obstruction to proving a generalization of Theorem~\ref{abulman} to dimension greater than~$1$ lies in equation~(\ref{1}). When $A$ is an elliptic curve, Abbes-Ullmo \cite{abbull} prove that the quotient of differentials on the right hand side of (\ref{1}) divides $(\rA)_m$. Thus $(\ca)_m\mid \#C$, which proves Theorem~\ref{abulman}, since the prime divisors of $\#C$ divide~$N$. When $A$ has dimension bigger than~$1$, the relationship between the quotient of differentials and $(\rA)_m$ is unclear. For example, Remark~\ref{rem:24} suggests that divisibility might sometimes fail when multiplicity one fails. \end{rmk} \begin{proof}[Proof of Proposition~\ref{n2}] We have the exact sequence \begin{eqnarray}\label{exseqofC} 0 \ra \po^*(\HM) \ra \po^*(S_2(\tZm)) \ra C \ra 0. \end{eqnarray} Suppose $p$ is a prime such that $p^2 \nmid N$ and $p \nmid \rAe$. We want to show that $p \nmid \# C$. We already know that the only primes that can divide~$\# C$ are those that divide~$N$; so we may assume that~$p$ exactly divides~$N$. Then considering the multiplication by~$p$ map applied to each term of the sequence~(\ref{exseqofC}) and using the snake lemma, we get: $$0 \ra C[p] \ra \po^*(\HM) \tensor \Fp \xrightarrow{\sqe} \po^*(S_2(\tZm)) \tensor \Fp \ra C \tfp \ra 0$$ (note the similarity to the situation in the proof of Theorem~\ref{thm:stein}). Then to show that $p \nmid \# C$, i.e., that $C[p]$ is trivial, all we have to show is that the map \begin{equation} \label{2} \po^*(\HM) \tensor \Fp \xrightarrow{\,\,\,\sqe\,\,\,} \po^*\left(S_2\left(\Z[1/m]\right)\right) \tensor \Fp \end{equation} is injective. The key idea is to use the isogeny~$\phi'$\label{keyconjiso} defined at the beginning of this section. We have maps \begin{eqnarray} \label{aja} \Adual \stackrel{\po}{\ra} J \stackrel{\pt}{\ra} A \stackrel{\phi'}{\ra} \Adual \end{eqnarray} such that the composite is multiplication by~$\nAe$. Let $\phi''=\phi' \circ \pt$. Pulling back differentials, we get \begin{eqnarray} \label{pullback} \GS{\Adual}{{\C}} \stackrel{{\phi''}^*}{\lra} H^0(J_{\C}, \Omega_{J/{\C}}^1) \stackrel{\phi_1^*}{\lra} \GS{\Adual}{{\C}}, \end{eqnarray} where the composite is again multiplication by~$\nAe$. By the N\'eron mapping property, the maps~(\ref{aja}) extend to the corresponding N\'eron models, and we see that $$\phi''^* (\po^*(H^0(J_S,\Omega_{J/S}))) \subseteq H^0(J_S,\Omega_{J/S}).$$ By~\cite[p.271]{abbull}, the canonical morphism $X_0(N) \ra J_0(N)$ induces a canonical isomorphism $$ H^0(J_S,\Omega_{J/S}) \stackrel{\isom}{\ra} H^0(M_0(N)^0_S,\Omega) = H^0(M_0(N)_S,\Omega). $$ Thus we see that the image of $$ \po^*(\HM) = \po^*(H^0(J_S,\Omega_{J/S})) $$ under ${\phi''}^*$ lands in $\HM = H^0(J_S,\Omega_{J/S})$. Also, since $p \nmid \rAe$, we have $$ S_2(\Zm) \tfp = S_2(\Zm)[I] \tfp \oplus (W(I) \cap S_2(\Zm)[I]) \tfp. $$ Thus if $f \in S_2(\Zm) \tfp$, then there exist unique $f_1 \in S_2(\Zm)[I] \tfp$ and $f_2 \in (W(I) \cup S_2(\Zm)[I]) \tfp$ such that $f = f_1 + f_2$. It then follows that $\po^* f = \po^* f_1$, and so ${\phi''}^* (\po^* f) = \nAe f_1 \in S_2(\Zm) \tfp$. Thus the image of $\po^*(S_2(\Zm)) \tfp$ under ${\phi''}^*$ lands in $S_2(\Zm) \tfp$. Hence, applying the maps in~(\ref{pullback}) to the groups in~(\ref{2}), which are subgroups of~$\GS{\Adual}{{\C}}$, we get the following commutative diagram: $$\xymatrix@=1.95em{ {\po^*(\HM)_{\Fp}}\ar[r]^{\quad{\phi''}^*}\ar[d]^{\sqe} & {\HM_{\Fp}}\ar[r]^{\po^*\quad}\ar[d]^{\sqe} & {\po^*(\HM)_{\Fp}} \ar[d]^{\sqe} \\ {\po^*(S_2(\Zm))_{\Fp}}\ar[r]^{{\quad\phi''}^*} & {S_2(\Zm)_{\Fp}}\ar[r]^{\po^*\quad} & {\po^*(S_2(\Zm))_{\Fp}.} }$$ The Atkin-Lehner involution~$W_p$ acts on $\po^*(\HM) \tfp$ and since~$A$ is attached to a newform, $W_p$ acts as either $+1$ or $-1$. Suppose $x$ is an element of $\po^*(\HM) \tfp$ that is in the $+1$ eigenspace for the action of~$W_p$ %in the top left corner of the diagram above that and in the kernel of the map in~(\ref{2}), i.e., the left-most~$\qe$ map above. Then its image $y = ({\phi''}^*)(x)$ in $\HM \tfp$ above maps to~zero in~$S_2(\Zm) \tfp$ under the middle~$\qe$ map, by commutativity of the first square. But we have $\HM \tfp \cong H^0(M_0(N)_{\Fp},\Omega)$. Since $p^2 \nmid N$, $M_0(N)_{\Fp}$ is a union of two irreducible components. Now $\qe(y)=0$ means that $y \in H^0(M_0(N)_{\Fp},\Omega)$ is zero on the component that contains the cusp~$\infty$. But $x$ is an eigenvector for~$W_p$, and hence so is~$y$. But $W_p$ is an involution that swaps the two components of~$M_0(N)_{\Fp}$. Hence~$y$ is zero on all of~$M_0(N)_{\Fp}$; i.e., $y=0$. Note that this part of the argument is very similar to the one towards the end of the proof of Theorem~\ref{thm:stein}. Looking at the top line in the diagram above, we find that $x$ maps to~zero under the composite. But its image under this composite is~$\nAe x$, and so $\nAe x=0$. Since $p \nmid \rAe$, Proposition~\ref{ndivsm} %an easy generalization of~\cite[Lem~3.2]{abbull} shows that $p \nmid \nAe$, and so $x=0$. A similar argument shows that if $x$ is an element of $\po^*(\HM) \tfp$ in the $-1$ eigenspace for the action of~$W_p$ and in the kernel of the map in~(\ref{2}), then $x=0$. This shows that the map~(\ref{2}) is injective, which is what was left to prove. \end{proof} \begin{rmk} \label{finalrmk} Note that the fact that $A$ is associated to a single newform was used only in the last two paragraphs of the proof above. We could have used the fact that the action of~$W_p$ on~$\po^*(\HM) \tfp$ is diagonalizable if $p \neq 2$ (e.g., see the last paragraph of the proof of Theorem~\ref{thm:stein}; the paragraph at the beginning of Section~\ref{sec:firstpart} is also relevant here), to prove that if $A$ is a quotient of $J_0(N)$ by an ideal of the Hecke algebra such that the quotient map factors through~$J_0(N)^{\rm new}$, and if $p\mid \#C$, then $p^2 \mid N$ or $p=2$ or $p \mid \rAe$. 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