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Stein}}}\hfill}} \def\oddhead{{\protect\centerline{\textsl{\large{The Manin Constant}}}\hfill}} \pagestyle{myheadings} \markboth{\evenhead}{\oddhead} \thispagestyle{empty} \noindent{{\small\rm Pure and Applied Mathematics Quarterly\\ Volume 2, Number 2\\ (\textit{Special Issue: In honor of \\ John H. Coates, Part 2 of 2})\\ 617---636, 2006} \vspace*{1.5cm} \normalsize \begin{center} {\bf{\Large The Manin Constant}} \end{center} \begin{center} \large{Amod Agashe, Kenneth Ribet and William A. Stein} %\end{center} %\begin{center} \end{center} \footnotetext{Received January 25, 2006. \\ Stein was supported by the National Science Foundation by Grant No. 0400386.} %%%%% ------------- fill in your data below this line ------------------- %%%%% The following lines \Title ... \EndAddress must ALL be present %%%%% and in the given order. %\Title %%%%% Put here the title. Line breaks will be recognized. %The Manin constant %\ShortTitle %The Manin constant %%%%% 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\footnote{Stein was supported by the National Science Foundation by Grant No. 0555776.} %\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. \begin{center} \begin{minipage}{5in} \noindent \textbf{Abstract:} The Manin constant of an elliptic curve is an invariant that arises in connection with the conjecture of Birch and Swinnerton-Dyer. One conjectures that this constant is 1; it is known to be an integer. After surveying what is known about the Manin constant, we establish a new sufficient condition that ensures that the Manin constant is an {\em odd} integer. Next, we generalize the notion of the Manin constant to certain abelian variety quotients of the Jacobians of modular curves; these quotients are attached to ideals of Hecke algebras. We also generalize many of the results for elliptic curves to quotients of the new part of~$J_0(N)$, and conjecture that the generalized Manin constant is~$1$ for newform quotients. Finally an appendix by John Cremona discusses computation of the Manin constant for all elliptic curves of conductor up to $130000$. \end{minipage} \end{center} %\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~$\Q$, and and let $N$ be the conductor of~$E$. By~\cite{breuil-conrad-diamond-taylor}, we may view~$E$ as a quotient of the modular Jacobian~$J_0(N)$. 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 $\omega$ be the unique (up to sign) rational $1$-form on a minimal Weierstrass model of $E$ over $\Z$ that restricts to a nowhere-vanishing $1$-form on the smooth locus. %Let $\omega$ be the unique (up to sign) minimal differential on a %minimal Weierstrass model of~$E$. The pullback of $\omega$ is a rational multiple of the differential associated to the normalized new cuspidal eigenform $f_E\in S_2(\Gamma_0(N))$ associated to~$E$. The Manin constant $c_E$ of is $E$ is the absolute value of this rational multiple. The Manin constant plays a role in the conjecture of Birch and Swinnerton-Dyer (see, e.g.,~\cite[p.~310]{gross-zagier}) and in work on modular parametrizations (see \cite{stevens:stickel, stein-watkins:ns, MR2135139}). It is known that the Manin constant is an integer; this fact is important to Cremona's computations of elliptic curves (see \cite[pg.~45]{cremona:alg}), and algorithms for computing special values of elliptic curve $L$-functions. Manin conjectured that $c_E=1$. In Section~\ref{sec:elliptic}, we summarize known results about $c_E$, and give the new result that $2\nmid c_E$ if~$2$ is not a congruence prime and $4 \nmid N$. \comment{I made some modifications in the paragraph below. --Amod} In Section~\ref{maninmotiv}, we generalize the definition of the Manin constant and many of the results mentioned so far to optimal quotients of~$J_0(N)$ and $J_1(N)$ of any dimension associated to ideals of the Hecke algebra. The generalized Manin constant comes up naturally in studying the conjecture of Birch and Swinnerton-Dyer for such quotients (see~\cite[\S4]{agst-bsd}), which is our motivation for studying the generalization. %In Section~\ref{maninmotiv}, we give the %generalization of the Manin constant to such quotients, We state what we can prove about the generalized Manin constant, and make a conjecture that the constant is also $1$ for quotients associated to newforms. The proofs of the theorems stated in Section~\ref{maninmotiv} are in Section~\ref{sec:proofs}. Section~\ref{sec:appdx} is an appendix written by J.~Cremona about computational verification that the Manin constant is $1$ for many elliptic curves. \vspace{2ex} {\bf \noindent Acknowledgments.} The authors are grateful to A.~Abbes, K.~Buzzard, R.~Coleman, B.~Conrad, B.~Edixhoven, A.~Joyce, L.~Merel, and R.~Taylor for discussions and advice regarding this paper. The authors wish to thank the referee for helpful comments and suggestions. \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 $n\geq 1$. Suppose~$f$ is 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 $a_n(f)$. 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 $x$ to $(\infty) -(x)$ 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~$\Z$. A general reference for N\'{e}ron models is~\cite{neronmodels}; for the special case of elliptic curves, see, e.g., \cite[App.~C,~\S15]{silverman:aec}, and \cite{silverman:aec2}. Let~$\omega$ be a generator for the rank $1$ ${\Z}$-module of invariant differential $1$-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 $X_0(N)\to E$, the differential $\phie^*\omega$ is a $\T$-eigenvector with the same eigenvalues as $f(z)$, so by \cite{atkin-lehner} we have $\phie^*\omega = c \cdot 2 \pi i f(z) dz$ 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 the rational number~$c$ defined above. %\end{defi} The following conjecture is implicit in \cite[\S5]{manin:parabolic}. \begin{conj}[Manin]\label{conjman} We have $\ce = 1$. \end{conj} Significant progress has been made towards this conjecture. In the following theorems,~$p$ denotes a prime and~$N$ denotes the conductor of~$E$. \begin{thm}[Edixhoven~\mbox{\cite[Prop.~2]{edix:manin}}] \label{edixman} The constant $\ce$ 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 theorem to quotients of arbitrary dimension in Theorem~\ref{maninint}. %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 theorem in Corollary~\ref{cor:gen_mazur}. %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 Theorem~\ref{thm:stein-raynaud}. %Section~\ref{maninmotiv} However, it is not clear if Theorem~\ref{abulman} generalizes to dimension greater than~$1$. It would be fantastic if the theorem could be generalized. It would imply that the Manin constant is~$1$ for newform quotients $A_f$ of $J_0(N)$, with~$N$ odd and square free, which be useful for computations regarding the conjecture of Birch and Swinnerton-Dyer. 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 the construction of certain stable integral models for $X_0(p^2)$. See Section~\ref{sec:appdx} for more details about the following computation: \begin{thm}[Cremona] If $E$ is an optimal elliptic curve over $\Q$ with conductor at most $130000$, then $c_E = 1$. \end{thm} \comment{ \begin{defi}[Congruence Number] \label{def:congnum} 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 of theorems we add the following: %, whose 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{thm:moddeg} below. %In fact, Theorem~\ref{thm:moddeg} asserts that if $p\mid \ce$ then %$p^2\mid N$ or $p\mid \re$. %However,~\cite{agashe-ribet-stein} 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 optimal elliptic curve quotients of conductor $\leq 120000$, only~$56$ of them satisfy the hypothesis. \comment{ In the notation of \cite{cremona:onlinetables}, they are \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, 31214c, 38158a, 39086a, 40366a, 41774a, 42638a, 45134a, 48878a, 50894b, 53678a, 54286a, 56558f, 58574b, 59918a, 61454b, 63086a, 63694a, 64366b, 64654b, 65294a, 65774b, 71182b, 80942a, 83822a, 93614a Each of the curves in this list has conductor $2p$ with $p\equiv 3\pmod{4}$ prime. The situation is similar to that of \cite[Conj.~4.2]{stein-watkins:ns}, which suggests there are infinitely many such curves. See also \cite{calegari-emerton:odd} for a classification of elliptic curves with odd modular degree. } \section{Quotients of arbitrary dimension} \label{maninmotiv} For $N\geq 4$, let $\Gamma$ a subgroup of~$\Gamma_1(N)$ that contains~$\Gamma_0(N)$, %\edit{I included intermediate subgroups since this is needed %to discuss integrality of Manin constant for such subgroups %(Theorem~\ref{maninint}). --Amod} 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. For a newform $f = \sum a_n(f) q^n \in S_2(\Gamma)$, consider the ring homomorphism $\T \to \Z[\ldots,a_n(f),\ldots]$ that sends $T_n$ to $a_n(f)$. The kernel $I_f\subset \T$ of this homomorphism is a saturated prime ideal of $\T$. %\begin{defi}[Newform quotient] The {\em newform quotient} $A_f$ associated to~$f$ is the quotient $J/I_f J$. %\end{defi} Shimura introduced $A_f$ in \cite{shimura:factors} where he proved that $A_f$ is an abelian variety over~$\Q$ of dimension equal to the degree of the field $\Q(\ldots,a_n(f),\ldots)$. He also observed that there is a natural map $\T \to \End(A_f)$ with kernel $I_f$. For the rest of this section, fix a quotient $A$ associated to a saturated ideal~$I$ of $\T$; note that $A$ may or may not be attached to a newform. \comment{ The {\em modular degree} $m_A$ of an optimal quotient $A$ of $J$ is the (positive) square root of the degree of the induced composite $A^{\vee} \to J^{\vee}\isom J \to A$. } \subsection{Generalization to quotients of arbitrary dimension} \label{sec:genman} %\edit{I added some stuff between here and two lemmas below. --Amod} If~$R$ is a subring of~$\C$, let $S_2(R)=S_2(\Gamma;R)$ denote the $\T$-submodule of~$S_2(\Gamma;\C)$ of modular forms whose Fourier expansions have all coefficients in~$R$. \begin{lem} \label{lem:Hecke-stable} The Hecke operators leave $S_2(R)$ stable. \end{lem} \begin{proof} If $\Gamma = \Gamma_0(N)$, then by the explicit description of the Hecke operators on Fourier expansions (e.g., see~\cite[Prop.~3.4.3]{diamond-im}), it is clear that the Hecke operators leave $S_2(R)$ stable. For a general~$\Gamma$, by \cite[(12.4.1)]{diamond-im}, one just has to check in addition that the diamond operators also leave $S_2(R)$ stable, which in turn follows from~\cite[Prop.~12.3.11] {diamond-im}. \end{proof} %The following lemma is not used in this paper, but we mention %it for the convenience of the reader. \begin{lem} We have $S_2(R)\isom S_2(\Z)\tensor R$. \end{lem} \begin{proof} This is \cite[Thm.~12.3.2]{diamond-im} when our spaces $S_2(R)$ and $S_2(\Z)$ are replaced by their algebraic analogues (see \cite[pg.~111]{diamond-im}). Our spaces and their algebraic analogues are identified by the natural $q$-expansion maps according to \cite[Thm.~12.3.7]{diamond-im}. \end{proof} % Note that this Lemma provides another proof of Lemma~\ref{lem:Hecke-stable}. %Let~$R = \Zell$, with $\ell$ prime, %and let $S_2(R)=S_2(\Gamma;R)$ denote the submodule of~$S_2(\Gamma;\C)$ %consisting of cuspforms whose Fourier expansions at~$\infty$ %have coefficients in~$R$. (That $S_2(R)$ is a $\T$-module %follows from the discussion in \cite[\S12]{diamond-im} and %the following lemma.) %\begin{lem} % The map $R\tensor_\Z S_2(\Z)\to S_2(R)$ that sends $a\tensor f$ to % $af$ is surjective. %In particular the image of $R \cdot S_2(\Z)$ in % $R[[q]]$ is saturated (since $S_2(R)$ is by definition saturated). %\end{lem} %\begin{proof} % By \cite[Thm.~3.52]{shimura:intro} there is a basis for $S_2(\Q_\ell)$ % with $q$-expansion coefficients in~$\Z$, i.e., $\Q_\ell\cdot % S_2(\Z)=S_2(\Q_\ell)$. Thus if $f\in S_2(\Zell)\subset S_2(\Q_\ell)$, %then there exists $u\in\Zell^*$, $n\geq 0$, and $g\in % S_2(\Z)$ such that % $$ % f = \frac{u}{\ell^n} g, % $$ % with $g \not\in \ell\Z[[q]]$. % But if $n>0$, then $g=\frac{\ell^n}{u} f \in \ell\Z[[q]]$, since the % coefficients of~$f$ are $\ell$-integral. % Thus $n=0$, hence $f \in \Zell^*\cdot S_2(\Z)$, which % proves that the map is surjective. %\end{proof} If~$B$ is an abelian variety over~$\Q$ and $S$ is a Dedekind domain with field of fractions~$\Q$, then we denote by $B_S$ the N\'{e}ron model of~$B$ over~$S$; also, for ease of notation, %\edit{Will: This was my response to the referee's comment: ``Observe also %that in the lower left corner (as well as in Definition 3.2), %$\Omega^1_{A/\Z}$ should really be $\Omega^1_{A_{\Z}/\Z}$. %This kind of "missing subscript" abuse of notation in $\Omega^1$'s %completely pervades %the paper. I will not list all of them, but will leave it to the %authors to correct all such %abuses of notation.'' It looks horrible if we follow the ref. %But feel free to make changes as per the ref if you think it is OK. --Amod %} we will abbreviate $H^0(B_S, \Omega^1_{B_S/S})$ by~$H^0(B_S, \Omega^1_{B/S})$. The inclusion $X \hra J$ that sends the cusp~$\infty$ to~$0$ induces an isomorphism $$H^0(X,\Omega^1_{X/{\Q}}) \isom H^0(J,\Omega^1_{J/{\Q}}).$$ Let $\phi_2$ be the optimal quotient map $J \ra A$. Then $\phi_2^*$ induces an inclusion $\psi: H^0(A_{\Z},\Omega^1_{A/{\Z}}) \hookrightarrow H^0(J,\Omega^1_{J/{\Q}})[I] \isom S_2(\Q)[I]$, and we have the following commutative diagram: $$ \xymatrix{ {H^0(A,\Omega^1_{A/{\Q}})\,\,}\ar@{^(->}[r]^{\quad \isom} & {H^0(J,\Omega^1_{J/{\Q}})[I]} \ar[r]^{\quad \isom} &{S_2(\Q)[I]}\\ {H^0(A_{\Z},\Omega^1_{A/\Z})}\ar@{^(->}[u]\ar@{^(->}[urr]_{\psi} & & {S_2(\Z)[I]}\ar@{^(->}[u] } $$ \begin{defi} \label{def:maninconst} The {\em Manin constant} of $A$ is the (lattice) index $$ \ca = [S_2(\Z)[I]: \psi(H^0(A_{\Z},\Omega^1_{A/\Z}))]. $$ \end{defi} Theorem~\ref{maninint} below asserts that $\ca\in\Z$, so we may also consider the Manin module of $A$, which is the quotient $M = S_2(\Z)[I] / \psi(H^0(A_{\Z},\Omega^1_{A/\Z}))$, and the Manin ideal of~$A$, which is the annihilator of~$M$ in~$\T$. If~$A$ is an elliptic curve, then $\ca$ is the usual Manin constant. % as in Definition~\ref{def:ce}. The constant~$c$ as defined above was also considered by Gross~\cite[2.5, p.222]{gross} and Lang~\cite[III.5, p.95]{lang}. The constant~$\ca$ was defined for the winding quotient in~\cite{aginv}, where it was called the generalized Manin constant. A Manin constant is defined in the context of $\Q$-curves in \cite{gonz-lario:manin}. \subsection{Motivation: connection with the conjecture of Birch and Swinnerton-Dyer} On a N\'eron model, the global differentials are the same as the invariant differentials, so $H^0(A_{\Z},\Omega^1_{A/\Z})$ is a free $\Z$-module of rank~$d=\dim(A)$. The {\em real measure}~$\Omega_A$ of~$A$ is the measure of~$A(\R)$ with respect to the volume given by a generator of~$\bigwedge^d H^0(A_{\Z},\Omega^1_{A/\Z}) \simeq \H^0(A_{\Z}, \Omega^d_{A_{\Z}/\Z})$. This quantity is of interest because it appears in the conjecture of Birch and Swinnerton-Dyer, which expresses the ratio $L^{(r)}(A,1)/\Omega_A$, in terms of arithmetic invariants of~$A$, where $r=\ord_{s=1} L(A,s)$ (see, e.g.,~\cite[Chap.~III, \S5]{lang} and \cite[\S2.3]{agst-bsd}). If we take a $\Z$-basis of $S_2(\Z)[I]$ and take the inverse image via the top chain of arrows in the commutative diagram above, we get a $\Q$-basis of~$H^0(A,\Omega^1_{A/\Q})$; let $\Omega_A'$ denote the volume of~$A(\R)$ with respect to the wedge product of the elements in the latter basis (this is independent of the choice of the former basis). In doing calculations or proving formulas regarding the ratio in the Birch and Swinnerton-Dyer conjecture mentioned above, it is easier to work with the volume~$\Omega_A'$ instead of working with~$\Omega_A$. 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 conjecture of Birch and Swinnerton-Dyer, since $\Omega_A = \ca \cdot \Omega_{A}'$. For example, see \cite[\S4.2]{agst-bsd} when $r=0$ and \cite[p.~310--311]{gross-zagier} when $r=1$; in each case, one gets a formula for computing the Birch and Swinnerton-Dyer conjectural order of the Shafarevich-Tate group, and the formula contains the Manin constant (see, e.g., \cite{mccallum:kolyvaginw}). %\edit{Will: this sentence is my response to the referee's comment: %``Section 3.2: Is there a single example of this stuff being applied to %computations with $r > 0$? You should comment on it one way or the other.''. %I think Mark Watkins has done computations of Shah for rank 1 using %the Gross-Zagier formula; is there a convenient reference to his %work? I suppose the Manin constant does intervene in his work. --Amod} The method of Section~\ref{sec:appdx} for verifying that $\ca=1$ for specific elliptic curves is of little use when applied to general abelian varieties, since there is no simple analogue of the minimal Weierstrass model (but see \cite{gonz-lario:manin} for $\Q$-curves). This emphasizes the need for general theorems regarding the Manin constant of quotients of dimension bigger than one. \subsection{Results and a conjecture} We start by giving several results regarding the Manin constant for quotients of arbitrary dimension. The proofs of most of the theorems are given in Section~\ref{sec:proofs}. Let $\Gamma$ be a subgroup of $\Gamma_0(N)$ that contains $\Gamma_1(N)$. %Analogous to Definition~\ref{def:maninconst}, %we have a notion of Manin constant for quotients~$A$ %of Jacobians of the modular curve associated to~$\Gamma$ by %saturated ideals of suitable Hecke algebras. We have the following generalization of Edixhoven's Theorem~\ref{edixman}. %for such quotients. %we give its proof in Section~\ref{section:integrality}. \begin{thm}\label{maninint} The Manin constant $\ca$ is an integer. (In the notation of Section~\ref{sec:genman} we even have that $\psi(H^0(A_{\Z},\Omega^1_{A/\Z})) \subseteq S_2(\Z)[I]$.) \end{thm} \begin{proof} %Suppose $\Gamma$ is a subgroup of $\Gamma_0(N)$ that contains %$\Gamma_1(N)$. Let $J = \Jac(X_\Gamma)$ and $J'=J_1(N)$. Suppose $A$ is an optimal quotient of~$J$. We have natural maps $\H^0(J'_{\Z}, \Omega^1_{J'/\Z}) \hookrightarrow \H^0(J', \Omega^1_{J'/\Q}) \stackrel{\isom}{\ra} S_2(\Gamma_1(N);\Q)$; from the proof of Lemma~6.1.6 of~\cite{ces}, the image of the composite is contained in~$S_2(\Gamma_1(N);\Z)$. The maps $J' \to J \to A$ induce a chain of inclusions $$ \H^0(A_{\Z},\Omega^1_{A/\Z}) \hra \H^0(J_{\Z}, \Omega^1_{J/\Z}) \hra \H^0(J'_{\Z}, \Omega^1_{J'/\Z}) \hra S_2(\Gamma_1(N);\Z) \hra \Z[[q]]. $$ Combining this chain of inclusions with commutativity of the diagram $$ \xymatrix{ & {S_2(\Gamma_1(N))}\ar[dr]^{\text{$F$-exp}}\\ {S_2(\Gamma)} \ar[ur]^{f(q)\mapsto f(q)}\ar[rr]^{\text{$F$-exp}} & & {\C[[q]]}, } $$ where $F$-exp is the Fourier expansion map, we see that the image of $\H^0(A_{\Z},\Omega^1_{A/\Z})$ lies in $S_2(\Z)[I]$, as claimed. \end{proof} For the rest of the paper, we take $\Gamma = \Gamma_0(N)$. For each prime $\ell \mid N$ with $\ord_{\ell}(N)=1$, let $W_\ell$ be the $\ell$th Atkin-Lehner operator. Let $J=J_0(N)$ and $A=A_I=J/IJ$ be an optimal quotient of $J$ attached to a saturated ideal~$I$. If $\ell$ is a prime, then as usual, $\Zell$ will denote the localization of~$\Z$ at~$\ell$. %\comment{ %For the rest of the article, we define %$S_2(\Zl) = S_2(\Z) \tensor \Zl$, and %$S_2(\Q_\ell) = S_2(\Z) \tensor \Q_\ell$. As an alternative, %one may replace $\Zell$ by $\Z_{(\ell)}$ (the localization of~$\Z$ %at~$\ell$) and $\Q_\ell$ by~$\Q$ throughout the rest of the article %(in which case the definition of $S_2(\Z_{(\ell)})$ is as given %earlier for subrings of~$\C$) and the results would still hold %(we are not using~$\Z_{(\ell)}$ since the notation becomes %too cumbersome). %} %\edit{Will: This was my response to the ref's comment: %``In the first paragraph of section 3.1, you have not defined the concept %$S_2(R)$ for rings such as l-adic integers. So in all later places %where you have $S_2(Z_l)$ you should replace $Z_l$ with the %algebraic localization $Z_{(l)}$ %(and replace $Q_l$ with $Q$). You could of course opt to make a definition %for general $R$, though nowhere later do you actually use $Z_l$ in %a way that $Z_{(l)}$ %wouldn't work just as well. Take your pick for which route to follow.'' %} \begin{thm} \label{thm:stein} Suppose that $\ell$ is an odd prime such that $\ell^2 \nmid N$, and that if $\ell \mid N$, then $A^{\vee}\subset J$ is stable under $W_{\ell}$. Then $\ell \mid \ca$ if and only if $\ell \mid N$ and $S_2(\Zell)[I]$ is not stable under the action of $W_\ell$. \end{thm} We will prove this theorem in Section~\ref{thm:proofstein}. \begin{rmk} The condition that $S_2(\Zell)[I]$ is stable under $W_{\ell}$ can be verified using standard algorithms. Thus in light of Theorem~\ref{thm:stein}, if $A$ is stable under all Atkin-Lehner operators and $N$ is square free, then one can compute the set of odd primes that divide $c_A$. It would be interesting to refine the arguments of this paper to find an algorithm to compute $c_A$ exactly. \end{rmk} Let $J_{\rm old}$ denote the abelian subvariety of~$J$ generated by the images of the degeneracy maps from levels that properly divide~$N$ (see, e.g.,~\cite[\S2(b)]{maziso}) and let $J^{\rm new}$ denote the quotient of~$J$ by~$J_{\rm old}$. A {\em new quotient} is a quotient $J \to A$ that factors through the map $J \to J^{\new}$. The following corollary generalizes Mazur's Theorem~\ref{mazman}: \begin{cor} \label{cor:gen_mazur} If $A=A_f$ is an optimal newform quotient of $J_0(N)$ and $\ell \mid \ca$ is a prime, then $\ell = 2$ or $\ell^2 \mid N$. \end{cor} \begin{proof} Since $f$ is a newform, $W_{\ell}$ acts as either $1$ or $-1$ on $A$ hence on $S_2(\Zell)[I]$. Thus $S_2(\Zell)[I]$ is $W_{\ell}$-stable. \end{proof} \begin{cor} If $A=J_0(N)_{\new}$ is the new subvariety of $J_0(N)$ and $\ell \mid \ca$ is a prime, then $\ell=2$ or $\ell^2\mid N$. (In particular, if~$N$ is prime then the Manin constant of $J_0(N)$ is a power of $2$, since $A=J_0(N)[I]$ for $I=0$.) \end{cor} \begin{proof} We have $W_{\ell} = -T_{\ell}$ on~$A$ (e.g., see the end of \cite[\S6.3]{diamond-im}). Also the new subspace $S_2(\Z)_{\new}$ of $S_2(\Gamma_0(N))$ is $T_{\ell}$-stable. \end{proof} \begin{rmk} \label{rmk:wl} If $A=J_0(33)$, then $$W_{3} = \left(\begin{array}{rrr} 1&0&0\\ \frac{1}{3}&\frac{1}{3}&-\frac{4}{3}\\ \frac{1}{3}&-\frac{2}{3}&-\frac{1}{3} \end{array}\right) $$ with respect to the basis \begin{align*} f_1 &= q - q^{5} - 2q^{6} + 2q^{7} + \cdots,\\ f_2 &= q^{2} - q^{4} - q^{5} - q^{6} + 2q^{7} + \cdots,\\ f_3 &= q^{3} - 2q^{6} + \cdots \end{align*} for $S_2(\Z)$. Thus $W_3$ does not preserve $S_2(\Z_{(3)})$. In fact, the Manin constant of $J_0(33)$ is not $1$ in this case (see Section~\ref{sec:counter}). \comment{ Note that Theorem~\ref{thm:stein} implies that the only primes that can divide the Manin constant of any optimal quotient of~$J_0(33)$ are~$2$ and~$3$. } The hypothesis of Theorem~\ref{thm:stein} sometimes holds for non-new $A$. For example, take $J = J_0(33)$ and $\ell=3$. Then $W_3$ acts as an endomorphism of $J$, and a computation shows that the characteristic polynomial of $W_3$ on $S_2(33)_{\new}$ is $x-1$ and on $S_2(33)_{\old}$ is $(x-1)(x+1)$, where $S_2(33)_{\old}$ is the old subspace of $S_2(33)$. %\edit{Will: can you address the referee's comment: %``In paragraph 2: operators on an abelian variety don't have a "characteristic %polynomial on" the abelian variety. Perhaps their action on $H_1$ or on a Tate %module has a characteristic polynomial? Also, putting parentheses around $x-1$ %doesn't make sense when it is on its own.'' Perhaps you mean the action %on the new and old subspace of cuspforms? --Amod} Consider the optimal elliptic curve quotient $A = J/(W_3+1)J$, which is isogenous to $J_0(11)$. Then~$A$ is an optimal old quotient of~$J$, and $W_3$ acts as~$-1$ on~$A$, so~$W_3$ preserves the corresponding spaces of modular forms. Thus Theorem~\ref{thm:stein} implies that $3\nmid \ca$. \end{rmk} The following theorem generalizes Raynaud's Theorem~\ref{thmofraynaud} (see also \cite{gonz-lario:manin} for generalizations to $\Q$-curves). \begin{thm} \label{thm:stein-raynaud} If $f \in S_2(\Gamma_0(N))$ is a newform and $\ell$ is a prime such that $\ell^2 \nmid N$, then $\ord_\ell(\caf) \leq \dim A_f$. \end{thm} Note that in light of Theorem~\ref{thm:stein}, this theorem gives new information only at $\ell=2$, when $2 \parallel N$. We prove the theorem in Section~\ref{proof:stein-raynaud} %Theorem~\ref{thm:stein} generalizes Mazur's Theorem~\ref{mazman}, %while Theorem~\ref{thm:stein-raynaud} generalizes Raynaud's %Theorem~\ref{thmofraynaud} (see \cite{gonz-lario:manin} for %generalizations to $\Q$-curves). \comment{ Let $S_2(\Z)[I]^{\perp}$ be the orthogonal complement of $S_2(\Z)[I]$ in $S_2(\Z)$ with respect to the Petersson inner product. \begin{defi}[Congruence exponent and number] The {\em congruence number}~$\rA$ of $A$ is the order of the quotient group \begin{equation}\label{eqn:congexp} S_2(\Z)/ (S_2(\Z)[I] + S_2(\Z)[I]^{\perp}). \end{equation} \end{defi} This definition of~$\rA$ agrees with Definition~\ref{def:congnum} when~$A$ is an elliptic curve (see \cite[p.~276]{abbull}). } Let $\pi$ denote the natural quotient map $J \ra A$. When we compose $\pi$ with its dual $\Adual \ra \Jdual$ (identifying $\Jdual$ with~$J$ using the inverse of the principal polarization of~$J$), we get an isogeny $\phi: \Adual \ra A$. % (for details, see~\cite{agashe-ribet-stein}). %\begin{defi}[Modular exponent] The {\em modular exponent}~$\mA$ of $A$ is the exponent of the group $\ker(\phi)$. %\end{defi} When $A$ is an elliptic curve, the modular exponent is just the modular degree of~$A$ (see, e.g.,~\cite[p.~278]{abbull}). \begin{thm} \label{thm:moddeg} If $f \in S_2(\Gamma_0(N))$ is a newform and $\ell \mid \caf$ is a prime, then $\ell^2 \mid N$ or $\ell \mid \ma$. \end{thm} Again, in view of Corollary~\ref{cor:gen_mazur}, this theorem gives new information only at $\ell=2$, when $\ord_2(N) \le 1$. We prove the theorem in Section~\ref{sec:moddeg}. The theorems above suggest that the Manin constant is~$1$ for quotients associated to newforms of square-free level. 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 $\ca=1$ for~$28$ two-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 suggest the following conjecture, which generalizes Conjecture~\ref{conjman}: \begin{conj} \label{conjmannew} If $f$ is a newform on $\Gamma_0(N)$ then $\caf=1$. \end{conj} It is plausible that $\caf=1$ for any newform on any congruence subgroup %$\Gamma_H(N)$ between $\Gamma_0(N)$ and $\Gamma_1(N)$. However, we do not have enough data to justify making a conjecture in this context. %\edit{Will: the referee asks ``Are there examples to show that %Conjecture 3.10 shouldn't be made for %all groups between $\Gamma_0(N)$ and $\Gamma_1(N)$?'' I was wondering %earlier: on what basis are we making a conjecture for $\Gamma_1(N)$? %I realized just now that all of our results can probably be %generalized to quotients of~$J_1(N)$, replacing $W_\ell$ in %the key lemma with~$U_\ell$ as Wiles does, and one can then %probably do it for intermediate congruence subgroups as well. %But we should leave all this for later...what about the conjecture %for now though? --Amod} \subsection{Examples of nontrivial Manin constants}\label{sec:counter} \comment{Earlier this was a section; I made it into a subsection, since it is short. --Amod} We present two sets of examples in which the Manin constant is not~$1$. %\subsubsection{Joyce's example} \comment{I rewrote this subsection, and kept a copy of William's original version after it. Feel free to pick the one you like. --Amod} Using results of \cite{kilford}, Adam Joyce~\cite{joyce} proves that there is a new optimal quotient of $J_0(431)$ with Manin constant $2$. Joyce's methods also produce examples with Manin constant~$2$ at levels $503$ and $2089$. For the convenience of the reader, we breifly discuss his example at level $431$. There are exactly two elliptic curves~$E_1$ and~$E_2$ of prime conductor $431$, and $E_1\cap E_2 = 0$ as subvarieties of $J_0(431)$, so $A=E_1 \times E_2$ is an optimal quotient of $J_0(431)$ attached to a saturated ideal $I$. If $f_i$ is the newform corresponding to $E_i$, then one finds that $f_1\equiv f_2\pmod{2}$, and so $g = (f_1 - f_2)/2 \in S_2(\Z)[I]$. However~$g$ is not in the image of $\H^0(A_{\Z},\Omega^1_{A/\Z})$. Thus the Manin constant of~$A$ is divisible by~$2$. %%\subsubsection{The Atkin-Lehner obstruction} \label{sec:atleh} %\comment{I rewrote this one too, since I think one does not need %to introduce $W_{\ell^r}$ to give an example. Again I put %William's original version after the modified one. --Amod} %Let $\Gamma = \Gamma_0(N)$ or $\Gamma_1(N)$ and $J=\Jac(X_\Gamma)$. %\begin{prop}\label{prop:alob} % If the Atkin-Lehner operator $W_{\ell}$ does not preserve $S_2(\Zell)$, % then $\ell \mid c_{\scriptscriptstyle{J}}$. %\end{prop} %\begin{proof} %If $\ell\nmid c_{\scriptscriptstyle{J}}$, then % the image of % $\H^0(J_{\Zell},\Omega^1_{J/\Zell})$ in $S_2(\Zell)$ equals % $S_2(\Zell)$. By the N\'eron mapping property, $W_\ell$ %preseves $\H^0(J_{\Zell},\Omega^1_{J/\Zell})$, i.e., it preserves %$S_2(\Zell)$. This contradicts the hypothesis. %\end{proof} As another class of examples, one finds by computation for each prime $\ell\leq 100$ that $W_\ell$ does not leave $S_2(\Gamma_0(11\ell);\Zell)$ stable. Theorem~\ref{thm:stein} (with $I=0$) then implies that the Manin constant of $J_0(11\ell)$ is divisible by~$\ell$ for these values of~$\ell$. %\edit{Will: I added this back. If you want, you can remove it, %but then the first line of the section will have to change. --Amod} %Edixhoven explained to us that this order can be understood using methods %of \cite[VII.3.17]{dera} and \cite{edix:comparison}. %More precisely, the image of~$i$ is the subgroup of %$S_2(\Z)$ of elements that have integral %Fourier expansion at all the cusps. %When $N=33$, Edixhoven observes that the cokernel has order divisible %by $3$. \section{Proofs of some of the Theorems}\label{sec:proofs} %\subsection{Proof of Theorem~\ref{maninint}} %\label{section:integrality} In Sections~\ref{thm:proofstein}, \ref{sec:moddeg}, and~\ref{proof:stein-raynaud}, we prove Theorems~\ref{thm:stein}, \ref{thm:moddeg}, and~\ref{thm:stein-raynaud} respectively. In Section~\ref{sec:keylem}, we state two lemmas that will be used in these proofs. The proofs of the theorems may be read independently of each other, after reading Section~\ref{sec:keylem}. \subsection{Two lemmas} \label{sec:keylem} The following lemma is a standard fact; we state it as a lemma merely because it is used several times. \begin{lem}\label{lem:torker} Suppose $i: A\hra B$ is an injective homomorphism of torsion-free abelian groups. If $p$ is a prime, then $B/i(A)$ has no nonzero $p$-torsion if and only if the induced map $A\tensor\F_p \to B\tensor\F_p$ is injective. \end{lem} \begin{proof} Let $Q$ denote the quotient $B/i(A)$. Tensor the exact sequence $0 \to A \to B \to Q \to 0$ with $\F_p$. The associated long exact sequences reveal that $\ker(A\tensor\F_p \to B\tensor\F_p) \isom Q_{\tor}[p]$. \end{proof} Suppose $\ell$ is a prime such that $\ell^2 \nmid N$. In what follows, we will be stating some standard facts taken from~\cite[\S2(e)]{maziso} (which in turn relies on~\cite{dera}). Let~${\cX_{\Zell}}$ be the minimal regular resolution of the coarse moduli scheme associated to~$\Gamma_0(N)$ (as in~\cite[\S~VI.6.9]{dera}) over~$\Z_{(\ell)}$, %the minimal proper regular model for $X_0(N)$ over~${\Zell}$, and let $\Omega_{\cX/{\Zell}}$ denote the relative dualizing sheaf of~${\cX_{\Zell}}$ over~$\Zell$. %(it is the sheaf of regular differentials as in~\cite[\S7]{mazur-ribet}). The Tate curve over~$\Zell[[q]]$ gives rise to a morphism from ${\rm Spec}\ \Zell[[q]]$ to the smooth locus of ${\cX_{\Zell}} \ra {\rm Spec}\ \Zell$. Since the module of completed Kahler differentials for~$\Zell[[q]]$ over $\Zell$ is free of rank~$1$ on the basis $dq$, we obtain a map $\text{ $q$-exp }: H^0({\cX_{\Zell}},\Omega_{\cX/{\Zell}}) \ra \Zell[[q]]$. %\edit{I moved the definition of q-exp up here in response to the referee's %comment that the fancy stuff below should not be introduced till needed. %--Amod} The natural morphism ${\rm Pic}^0_{\cX/{\Zell}} \ra J_{\Zell}$ identifies ${\rm Pic}^0_{\cX/\Zell}$ with the identity component of~$J_{\Zell}$ (see, e.g., \cite[\S9.4--9.5]{neronmodels}). Passing to tangent spaces along the identity section over~$\Zell$, we obtain an isomorphism $H^1(\cX_{\Zell}, \OO_{\cX_{\Zell}}) \isom {\rm Tan}(J_{\Zell})$. Using Grothendieck duality, one gets an isomorphism ${\rm Cot}(J_{\Zell}) \stackrel{\isom}{\ra} H^0({\cX_{\Zell}},\Omega_{\cX/{\Zell}})$, where ${\rm Cot}(J_{\Zell})$ is the cotangent space at the identity section. On the N\'eron model~$J_{\Zell}$, the group of global differentials is the same as the group of invariant differentials, which in turn is naturally isomorphic to~${\rm Cot}(J_{\Zell})$. Thus we obtain an isomorphism $H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}}) \isom H^0({\cX_{\Zell}},\Omega_{\cX/{\Zell}})$. %Recall that $\pi$ denotes the quotient map $J_0(N)\ra A$ %and $A=A_I = J_0(N)/IJ_0(N)$. Let $G$ be a $\T$-module equipped with an injection $G \hookrightarrow H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}})$ of $\T$-modules such that $G$ is annihilated by~$I$. If $\ell \mid N$, assume moreover that $G$ is a $\T[W_\ell]$-module and that the inclusion in the previous sentence is a homomorphism of $\T[W_\ell]$-modules. As a typical example, $G = H^0(A_{\Zell},\Omega^1_{A/{\Zell}})$, with the injection $\pi^*: H^0(A_{\Zell},\Omega^1_{A/{\Zell}}) \hookrightarrow % \xrightarrow{\,\,\,\,\,\pi^*} H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}})$. Let $\Phi$ be the composition of the inclusions \begin{equation}\label{eqn:qexp1} G \hookrightarrow H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}}) \isom H^0({\cX_{\Zell}},\Omega_{\cX/{\Zell}}) \xrightarrow{\text{ $q$-exp }} {\Zell}[[q]], \end{equation} and let $\psi'$ be the composition of %\begin{equation} \label{eqn:qexppsi} $$G \hookrightarrow H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}})[I] %H^0(J,\Omega^1_{J/{\Zell}})[I] \isom \hookrightarrow S_2(\Zell)[I], $$ %\end{equation} where the last inclusion follows from a ``local'' version of Theorem~\ref{maninint}. The maps $\Phi$ and $\psi'$ are related by the commutative diagram \begin{equation} \label{qandFexp} \xymatrix{ & {S_2({\Zell})[I]}\ar[dr]^{\text{$F$-exp}}\\ G \ar[ur]^{\psi'} \ar[rr]^{\Phi} & & {{\Zell}[[q]]}, }\, \end{equation} where $F$-exp is the Fourier expansion map (at infinity), as before. \comment{ where the map $q$-exp is the $q$-expansion map on differentials as in~\cite[\S2(e)]{maziso} (actually, Mazur works over~$\Z$; our map is obtained by tensoring with~$\Zell$). } We say that a subgroup~$B$ of an abelian group~$C$ is {\em saturated} (in~$C$) if the quotient~$C/B$ is torsion free. \begin{lem} \label{lem:keylem} Recall that $\ell$ is a prime such that $\ell^2 \nmid N$. If $\ell$ divides $N$, suppose that $S_2(\Zell)[I]$ is stable under the action of $W_\ell$; if $\ell=2$ assume moreover that $W_{\ell}$ acts as a scalar on $A$. %If $\ell \mid N$, then %suppose either that $A$ is a newform quotient, or %that $\ell$ is odd and $W_\ell \cdot \psi'(G) \subseteq S_2(\Zell)$. Consider the map $$G \tensor \Fell \ra H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}}) \tensor \Fell,$$ which is obtained by tensoring the inclusion $G \hookrightarrow H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}})$ %(\ref{eqn:qexppsi}) with $\Fell$. If this map is injective, then the image of~$G$ under the map $\Phi$ of (\ref{qandFexp}) is saturated in~${\Zell}[[q]]$. \end{lem} \begin{proof} By Lemma~\ref{lem:torker}, it suffices to prove that the map $$ \Phi_\ell: G \tensor \Fell \ra {\Zell}[[q]] \tensor \Fell = \Fell[[q]] $$ obtained by tensoring (\ref{eqn:qexp1}) with $\Fell$ is injective. Let $\cX_{\F_\ell}$ denote the special fiber of~$\cX_{\Zell}$ and let $\Omega_{\cX/{\F_\ell}}$ denote the relative dualizing sheaf of~$\cX_{\Fell}$ over~$\Fell$. {\em First suppose that $\ell$ does not divide $N$.} Then ${\cX_{\Zell}}$ is smooth and proper over $\Zell$. Thus the formation of $\H^0({\cX_{\Zell}}, \Omega_{{\cX_{\Zell}}})$ is compatible with any base change on~$\Zell$ (such as reduction modulo $\ell$). %Now the map $q$-exp is injective and has %torsion-free cokernel, hence by Lemma~\ref{lem:torker}, the induced map %Then by the $q$-expansion principle, the %$q$-expansion map %$H^0(\cX_{\Fell},\Omega_{\cX/{\Fell}}) %\isom H^0(\cX_{\Zell},\Omega_{\cX/{\Zell}}) \tensor \Fell %\ra \Fell[[q]]$ is injective. The injectivity of~$\Phi_\ell$ now follows since by hypothesis %the following map is injective: the induced map $ G \tensor \Fell \ra H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}}) \tensor \Fell$ is injective, and $$ H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}}) \tensor \Fell \isom H^0(\cX_{\Zell},\Omega_{\cX/{\Zell}}) \tensor \Fell \isom H^0(\cX_{\Fell},\Omega_{\cX/{\Fell}}) \ra \F_\ell[[q]] $$ is injective by the $q$-expansion principle (which is easy in this setting, since $\cX_{\Fell}$ is a smooth and geometrically connected curve). {\em Next suppose that $\ell$ divides $N$.} First we verify that $\ker(\Phi_\ell)$ is stable under~$W_\ell$. %\edit{Will: As I said before, this is the same as your proof, but %I rewrote it to convince myself. You may revert back to what %you wrote earlier. --Amod} Suppose $\omega \in \ker(\Phi_\ell)$. Choose $\omega' \in G$ such that the image of~$\omega'$ in $G \tensor \F_\ell$ is~$\omega$, and let $f = \psi'(\omega')$. %denote by $f(q) \in \Zell[[q]]$ the $q$-expansion at %infinity of $f$. Because $\Phi_\ell(\omega) = 0$ in~$\F_\ell[[q]]$, there exists $h \in \Zell[[q]]$ such that $\ell h = {\text{$F$-exp}}(f)$. Let $f' = f/\ell \in S_2(\Q)$; then $f'$ is actually in~$S_2(\Zell)$ (since ${\text{$F$-exp}}(f/\ell) = h \in \Zell[[q]]$). Now $\ell f' = f$ is annihilated by every element of~$I$, hence so is $f'$; thus $f' \in S_2(\Zell)[I]$. By hypothesis, $W_{\ell}(f') \in S_2(\Zell)[I]$. Then $$\Phi(W_\ell \omega') = {\text{$F$-exp}}( W_\ell f) = \ell \cdot {\text{$F$-exp}}(W_\ell f') \in \ell\Zell[[q]].$$ %Since by definition $\omega' \in G$ reduces to $\omega \in G \tensor \F_\ell$, Reducing modulo~$\ell$, we get $\Phi_\ell (W_\ell \omega) = 0$ in~$\F_\ell[[q]]$. Thus $W_\ell \omega \in \ker(\Phi_\ell)$, which proves that $\ker(\Phi_\ell)$ is stable under $W_{\ell}$. %{\bf [[FIX!!!!!]]} %By hypothesis, $\psi(G)+W_\ell \cdot \psi(G) \subseteq S_2({\Zell})$ %(note that if $A$ is a newform quotient, this follows since $W_\ell$ %acts as~$1$ or~$-1$). Hence, in the commutative diagram~(\ref{qandFexp}), %the map %$\Phi_\ell$ factors as $G \tensor \Fell \stackrel{\psi}{\ra} %(\psi(G)+W_\ell \cdot \psi(G)) \tensor \Fell %\stackrel{\text{$F$-exp}}{\longrightarrow} \Fell[[q]]$, %Since $\Phi_{\ell} %and so $\ker(\Phi_\ell)$ %is invariant under~$W_\ell$. %If the characteristic %$\ell$ is odd, then since $W_\ell$ is an involution, %$\ker(\Phi_\ell)$ is a %direct sum of $+1$ and $-1$ eigenspaces for $W_\ell$. %If $A$ is associated to a single newform, then %$W_\ell$ acts either as~$+1$ or as~$-1$. Thus in either case, %it suffices to prove that if %$\omega\in \ker(\Phi_\ell)$ is in either the $+1$ or~$-1$ %eigenspace for the action of~$W_\ell$ on~$\ker(\Phi_\ell)$, then $\omega=0$. Since $W_\ell$ is an involution, and by hypothesis either $\ell$ is odd or $W_\ell$ is a scalar, the space $\ker(\Phi_\ell)$ breaks up into a direct sum of eigenspaces under~$W_\ell$ with eigenvalues~$\pm 1$. It suffices to show that if $\omega \in \ker(\Phi_\ell)$ is an element of either eigenspace, then $\omega = 0$. For this, we use a standard argument that goes back to Mazur (see, e.g., the proof of Prop.~22 in~\cite{mazur-ribet}); we give some details to clarify the argument in our situation. \edit{I added this line so the the referee or the reader is more at ease. The argument first seems to have appeared clearly in Mazur's ``Rational isogenies...'' paper, though I think he made the mistake of assuming that the minimal regular model has only two irreducible components. The description in~\cite{mazur-ribet} does not have this problem. --Amod} %Note that in our context, %the formation of the relative dualizing sheaf commutes %with any base change~\cite[\S~I.2.1]{dera}. Following the proof of Prop.~3.3 on p.~68 of~\cite{mazur:eisenstein}, we have \edit{I reworded this a bit; the earlier reason (``Since $\Zell$ is flat over~$\Z$'') may not work due to the confusing wording of Prop.~3.3 on p.~68 of~\cite{mazur:eisenstein}. --Amod} $$H^0(\cX_{\Zell},\Omega_{\cX/{\Zell}}) \tensor \Fell \isom H^0(\cX_{\Fell},\Omega_{\cX/{\Fell}}).$$ In the following, we shall think of $G \tensor \Fell$ as a subgroup of~$H^0(\cX_{\Fell},\Omega_{\cX/{\Fell}})$, which we can do by the hypothesis that the induced map $G \tensor \Fell \ra H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}}) \tensor \Fell$ is injective and that $$H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}}) \tensor \Fell \isom H^0(\cX_{\Zell},\Omega_{\cX/{\Zell}}) \tensor \Fell \isom H^0(\cX_{\Fell},\Omega_{\cX/{\Fell}}).$$ \edit{Everything below in this proof has been modified. --Amod} Suppose $\omega \in \ker(\Phi_\ell)$ is in the $\pm 1$ eigenspace (we will treat the cases of $+1$ and~$-1$ eigenspaces together). We will show that $\omega$ is trivial over $\cX_\Fellbar$, the base change of~$\cX_\Fell$ to an algebraic closure~$\Fellbar$, which suffices for our purposes. Since $\ell^2 \nmid N$, we have $\ell \mid\mid N$, and so the special fiber $\cX_{\Fellbar}$ is as depicted on p.~177 of~\cite{mazur:eisenstein}: it consists of the union of two copies of~$X_0(N/\ell)_{\Fellbar}$ identified transversely at the supersingular points, and some copies of~$\PP^1$, each of which intersects exactly one of the two copies of~$X_0(N/\ell)_{\Fellbar}$ and perhaps another~$\PP^1$, all of them transversally. All the singular points are ordinary double points, and the cusp~$\infty$ lies on one of the two copies of~$X_0(N/\ell)_{\Fellbar}$. In particular, $\cX_{\Fellbar} \ra \Spec{\Fellbar}$ is locally a complete intersection, hence Gorenstein, and so by~\cite[\S~I.2.2, p.~162]{dera}, the sheaf~$\Omega_{\cX/{\Fellbar }} = \Omega_{\cX/{\Fell}} \tensor \Fellbar$ is invertible. \edit{I picked this from~\cite[p.~63]{mazur:eisenstein}. Phew! --Amod} Since $\omega\in\ker(\Phi_\ell)$, the differential~$\omega$ vanishes on the copy of~$X_0(N/\ell)_{\Fellbar}$ containing the cusp~$\infty$ by the $q$-expansion principle (which is easy in this case, since all that is being invoked here is that on an integral curve, the natural map from the group of global sections of an invertible sheaf into the completion of the sheaf's stalk at a point is injective). The two copies of~$X_0(N/\ell)_{\Fellbar}$ are swapped under the action of the Atkin-Lehner involution $W_\ell$, and thus $W_\ell(\omega)$ vanishes on the other copy that does not contain the cusp~$\infty$. %the other irreducible component. Since $W_\ell(\omega) = \pm \omega$, we see that $\omega$ is zero on both copies of~$X_0(N/\ell)_{\Fellbar}$. Also, by the description of the relative dualizing sheaf in~\cite[\S~I.2.3, p.~162]{dera}, if $\pi: \widetilde{\cX}_{\Fellbar} \ra \cX_{\Fellbar}$ is a normalization, then $\omega$ correponds to a meromorphic differential~$\widetilde{\omega}$ on~$\widetilde{\cX}_{\Fellbar}$ which is regular outside the inverse images (under~$\pi$) of the double points on~$\cX_{\Fellbar}$ and has at worst a simple pole at any point that lies over a double point on~$\cX_{\Fellbar}$. Moreover, on the inverse image of any double point on~$\cX_{\Fellbar}$, the residues of~$\widetilde{\omega}$ add to zero. For any of the~${\PP}^1$'s, above a point of intersection of the~$\PP^1$ with a copy of~$X_0(N/\ell)_{\Fellbar}$, the residue of~$\widetilde{\omega}$ on the inverse image of the copy of~$X_0(N/\ell)_{\Fellbar}$ is zero (since $\omega$ is trivial on both copies of~$X_0(N/\ell)_{\Fellbar}$), and thus the residue of~$\widetilde{\omega}$ on the inverse image of~$\PP^1$ is zero. Thus $\widetilde{\omega}$ restricted to the inverse image of~$\PP^1$ is regular away from the inverse image of any point where the~$\PP^1$ meets another~$\PP^1$ (recall that there can be at most one such point). Hence the restriction of $\widetilde{\omega}$ to the inverse image of the~$\PP^1$ is either regular everywhere or is regular away from one point where it has at most a simple pole; in the latter case, the residue is zero by the residue theorem. Thus in either case, $\widetilde{\omega}$ restricted to the inverse image of the~$\PP^1$ is regular, and therefore is zero. Thus $\omega$ %is determined on %a copy of~$\PP^1$ by the residue of~$\widetilde{\omega}$ above %a point of intersection of the~$\PP^1$ with %any of the copies of~$X_0(N/\ell)_{\Fellbar}$; %this residue is zero %and hence $\omega$ is trivial on all the copies of~$\PP^1$ as well. Hence $\omega=0$, as was to be shown. %A similar argument shows that if %$\omega\in \ker(\Phi_\ell)$ is in the $+1$ eigenspace for the action %of~$W_\ell$, then $\omega=0$. %Finally, $\ker(\Phi_\ell)$ is a direct sum of its $+1$ %and $-1$ eigenspaces since by hypothesis either $\ell$ is odd %or $W_\ell$ is a scalar, so $\ker(\Phi_\ell)=0$. \end{proof} \subsection{Proof of Theorem~\ref{thm:stein}}\label{thm:proofstein} We continue to use the notation of Section~\ref{sec:keylem}. First suppose that $\ell \mid N$ and $S_2(\Zell)[I]$ is not stable under the action of $W_\ell$. Relative differentials and N\'eron models are functorial, so $H^0(A_{\Zell},\Omega^1_{A/{\Zell}})$ is $W_{\ell}$-stable. Thus the map $H^0(A_{\Zell},\Omega^1_{A/{\Zell}}) \to S_2(\Zell)[I]$ is not surjective. But $c_A$ is the order of the cokernel, so $\ell \mid c_A$. Next we prove the other implication, namely that if $\ell \mid c_A$, then $\ell\mid N$ and $S_2(\Zell)[I]$ is not stable under~$W_\ell$. We will prove this by proving the contrapositive, i.e., that if either $\ell\nmid N$ or $S_2(\Zell)[I]$ is stable under~$W_\ell$, then $\ell \nmid c_A$. We now follow the discussion preceding Lemma~\ref{lem:keylem}, taking $G = H^0(A_{\Zell},\Omega^1_{A/{\Zell}})$. To show that $\ell \nmid \ca$, we have to show that $\ca$ is a unit in~${\Zell}$. For this, it suffices to check that in diagram~(\ref{qandFexp}), the image of $H^0(A_{\Zell},\Omega^1_{A/{\Zell}})$ in ${\Zell}[[q]]$ under~$\Phi$ is saturated, since the image of $S_2(\Gamma_0(N);{\Zell})[I]$ under $F$-exp is saturated in ${\Zell}[[q]]$. In view of Lemma~\ref{lem:keylem}, it suffices to show that the map $$H^0({A_{\Zell}},\Omega^1_{{A/{\Zell}}})\tensor\Fell\ra H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}})\tensor\Fell$$ is injective. Since~$A$ is an optimal quotient, $\ell\neq 2$, and~$J$ has good or semistable reduction at~$\ell$, \cite[Cor 1.1]{maziso} yields an exact sequence $$0 \ra H^0(A_{\Zell},\Omega^1_{A/{\Zell}}) \ra H^0(J_{\Zell},\Omega^1_{J/{\Zell}}) \ra H^0(B_{\Zell},\Omega^1_{B/{\Zell}}) \ra 0$$ where $B=\ker(J\to A)$. Since $H^0(B_{\Zell},\Omega^1_{B/{\Zell}})$ is torsion free, by Lemma~\ref{lem:torker} the map $H^0({A_{\Zell}},\Omega^1_{{A/{\Zell}}})\tensor\Fell \ra H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}})\tensor\Fell$ is injective, as was to be shown. \subsection{Proof of Theorem~\ref{thm:moddeg}} \label{sec:moddeg} We continue to use the notation and hypotheses of Section~\ref{sec:keylem} (so $\ell^2 \nmid N$) and assume in addition that $A$ is a newform quotient, and that $\ell \nmid \ma$. We have to show that then $\ell \nmid \ca$. Just as in the previous proof, %before, it suffices to check that the image of $H^0(A_{\Zell},\Omega^1_{A/{\Zell}})$ in ${\Zell}[[q]]$ is saturated. %\edit{Will: I repeated parts of the previous proof. I had earlier %deleted it following the referee's suggestion, but now the previous %proof has changed, since it goes in both directions. So it seemed %appropriate to add some more lines for clarity. --Amod} %since the image of $S_2(\Gamma_0(N);{\Zell})[I]$ is %saturated in ${\Zell}[[q]]$. Since $A$ is a newform quotient, if $\ell \mid N$, then $W_\ell$ acts as a scalar on~$A$ and on~$S_2(\Gamma_0(N);{\Zell})[I]$. So again, using Lemma~\ref{lem:keylem}, %with $G = H^0(A_{\Zell},\Omega^1_{A/{\Zell}})$, %; thus it suffices to show that the map $H^0({A_{\Zell}},\Omega^1_{{A/{\Zell}}})\tensor\Fell\ra H^0({J_{\Zell}},\Omega^1_{{J/{\Zell}}})\tensor\Fell$ is injective. The composition of pullback and pushforward in the following diagram is multiplication by the modular exponent of $A$: $$\xymatrix{ & {H^0(J_{{\Zell}}, \Omega^1_{J/{\Zell}})}\ar[dr]^{\pi_*} & \\ {H^0(A_{{\Zell}}, \Omega^1_{A/{\Zell}})}\ar[ur]^{\pi^*}\ar[rr]^{m_A} & & {H^0(A_{{\Zell}}, \Omega^1_{A/{\Zell}})} }$$ Since $m_A \in \Zell^{\times}$, the map $\pi^*$ is a section to the map $\pi_*$ up to a unit and hence its reduction modulo~$\ell$ is injective, which is what was left to be shown. \comment{ Let $\overline{\pi}_*$ and $\overline{\pi}^*$ denote the maps obtained by tensoring the diagram above with $\F_\ell$. Then $\overline{\pi}_*\circ \overline{\pi}^*$ is multiplication by an integer coprime to~$\ell$ from the finite dimension $\F_\ell$-vector space $\H^0(A_{{\Zell}}, \Omega^1_{A/{\Zell}})\tensor \F_\ell$ to itself, hence an isomorphism. In particular, $\overline{\pi}^*$ is injective, which is what was left to show. } \comment{ \begin{rmk} Adam Joyce observed that one can also obtain injectivity of $\overline{\pi}^*$ as a consequence of Prop.~7.5.3(a) of \cite{neronmodels}. \end{rmk} } \subsection{Proof of Theorem~\ref{thm:stein-raynaud}} \label{proof:stein-raynaud} Theorem~\ref{thm:stein-raynaud} asserts that if $A=A_f$ is a quotient of $J= J_0(N)$ attached to a newform~$f$, and $\ell$ is a prime such that $\ell^2 \nmid N$, then $\ord_{\ell}(\ca) \leq \dim(A)$. Our proof follows \cite{abbull}, except at the end we argue using lattice indices 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_{\Zell}\to J_{\Zell} \to A_{J_{\Zell}}$ of N\'eron models. Because $J_{\Zell}$ has semiabelian reduction (since $\ell^2 \nmid N$), 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 \[ 0 \to \Tan(B_{\Zell}) \to \Tan(J_{\Zell}) \to \Tan(A_{\Zell}) \to (\Z/{\ell}\Z)^r \to 0. \] Here $\Tan$ is the tangent space at the $0$ section; it is a finite free $\Zell$-module of rank equal to the dimension. In particular, we have $r \leq d=\dim(A)$. %(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 $\Zell$-dual to the cotangent space, and the cotangent space is isomorphic to the space of global differential $1$-forms. The theorem of Raynaud mentioned above is the generalization to $e=\ell -1$ of \cite[Cor.~1.1]{maziso}, which we used above in the proof of Theorem~\ref{thm:stein}. Let $C$ be the cokernel of $\Tan(B_{\Zell})\to \Tan(J_{\Zell})$. We have a diagram \begin{equation}\label{eqn:insert_C} \xymatrix@=0.15in{ 0 \ar[r] & {\Tan(B_{\Zell})}\ar[r]& {\Tan(J_{\Zell})}\ar[rr]\ar@{->>}[dr]&& {\Tan(A_{\Zell})}\ar[r]& {(\Z/{\ell}\Z)^r}\ar[r]& 0.\\ & & & C\ar@{^(->}[ru]\\ % & & 0\ar[ur] & &0\ar[ul] } \end{equation} Since $C\subset \Tan(A_{\Zell})$, so $C$ is torsion free, we see that~$C$ is a free $\Zell$-module of rank $d$. Let $C^* = \Hom_{\Zell}(C,\Zell)$ be the $\Zell$-linear dual of~$C$. Applying the $\Hom_{\Zell}(-,\Zell)$ functor to the two short exact sequences in (\ref{eqn:insert_C}), we obtain exact sequences \[ 0 \to C^* \to \H^0(J_{\Zell},\Omega^1_{J/\Zell}) \to \H^0(B_{\Zell},\Omega^1_{B/\Zell}) \to 0, \] and \begin{equation}\label{eqn:dualc} 0 \to \H^0(A_{\Zell},\Omega^1_{A/\Zell}) \to C^* \to (\Z/{\ell}\Z)^r\to 0. \end{equation} The $(\Z/{\ell}\Z)^r$ on the right in (\ref{eqn:dualc}) occurs as $\Ext^1_{\Zell}((\Z/{\ell}\Z)^r,\Zell)$. %Also, (\ref{eqn:dualc}) implies that $r\leq d=\dim(A)$. Since $\H^0(B_{\Zell},\Omega^1_{B/\Zell})$ is torsion free, by Lemma~\ref{lem:torker}, the induced map $$ C^* \tensor \F_\ell \to \H^0(J_{\Zell},\Omega^1_{J/\Zell}) \tensor \F_\ell $$ is injective. Since $A$ is a newform quotient, if $\ell \mid N$ then $W_\ell$ acts as a scalar on~$C^*$ and on~$S_2(\Gamma_0(N);{\Zell})[I]$. Using Lemma~\ref{lem:keylem}, with $G = C^*$, we see that the image of $C^*$ in $\Zell[[q]]$ under the composite of the maps in~(\ref{eqn:qexp1}) is saturated. The Manin constant for~$A$ at~${\ell}$ is the index of the image via $q$-expansion of $\H^0(A_{\Zell},\Omega^1_{A/\Zell})$ in $\Zell[[q]]$ in its saturation. Since the image of $C^*$ in~$\Zell[[q]]$ is saturated, the image of $C^*$ is the saturation of the image of~$\H^0(A_{\Zell},\Omega^1_{A/\Zell})$, so the Manin constant at~${\ell}$ is the index of~$\H^0(A_{\Zell},\Omega^1_{A/\Zell})$ in $C^*$, which is~${\ell}^r$ by (\ref{eqn:dualc}), hence is at most~${\ell}^d$. %\end{proof} \section{Appendix by J.~Cremona: Verifying that $c=1$} \label{sec:appdx} Let $f$ be a normalised rational newform for $\Gamma_0(N)$. Let $\Lambda_f$ be its period lattice; that is, the lattice of periods of $2\pi if(z)dz$ over $H_1(X_0(N),\Z)$. We know that $E_f=\C/\Lambda_f$ is an elliptic curve $E_f$ defined over $\Q$ and of conductor~$N$. This is the optimal quotient of $J_0(N)$ associated to~$f$. Our goal is two-fold: to identify $E_f$ (by giving an explicit Weierstrass model for it with integer coeffients); and to show that the associated Manin constant for $E_f$ is~$1$. In this section we will give an algorithm for this; our algorithm applies equally to optimal quotients of $J_1(N)$. As input to our algorithm, we have the following data: \begin{enumerate} \item a $\Z$-basis for $\Lambda_f$, known to a specific precision; \item the type of the lattice $\Lambda_f$ (defined below); and \item a complete isogeny class of elliptic curves $\{E_1,\dots,E_m\}$ of conductor~$N$, given by minimal models, all with $L(E_j,s)=L(f,s)$. \end{enumerate} So $E_f$ is isomorphic over~$\Q$ to $E_{j_0}$ for a unique $j_0\in\{1,\dots,m\}$. The justification for this uses the full force of the modularity of elliptic curves defined over $\Q$: we have computed a full set of newforms $f$ at level $N$, and the same number of isogeny classes of elliptic curves, and the theory tells us that there is a bijection between these sets. Checking the first few terms of the $L$-series (i.e., comparing the Hecke eigenforms of the newforms with the traces of Frobenius for the curves) allows us to pair up each isogeny class with a newform. We will assume that one of the $E_j$, which we always label $E_1$, is such that $\Lambda_f$ and $\Lambda_1$ (the period lattice of~$E_1$) are approximately equal. This is true in practice, because our method of finding the curves in the isogeny class is to compute the coefficients of a curve from numerical approximations to the $c_4$ and $c_6$ invariants of $\C/\Lambda_f$; in all cases these are very close to integers which are the invariants of the minimal model of an elliptic curve of conductor $N$, which we call $E_1$. The other curves in the isogeny class are then computed from $E_1$. For the algorithm described here, however, it is irrelevant how the curves $E_j$ were obtained, provided that $\Lambda_1$ and $\Lambda_f$ are close (in a precise sense defined below). Normalisation of lattices: every lattice $\Lambda$ in $\C$ which defined over~$\R$ has a unique $\Z$-basis $\omega_1$, $\omega_2$ satisfying one of the following: \begin{itemize} \item {\bf Type 1:} $\omega_1$ and $(2\omega_2-\omega_1)/i$ are real and positive; or \item {\bf Type 2:} $\omega_1$ and $\omega_2/i$ are real and positive. \end{itemize} For $\Lambda_f$ we know the type from modular symbol calculations, and we know $\omega_1,\omega_2$ to a certain precision by numerical integration; modular symbols provide us with cycles $\gamma_1,\gamma_2\in H_1(X_0(N),\Z)$ such that the integral of $2\pi if(z)dz$ over $\gamma_1,\gamma_2$ give $\omega_1,\omega_2$. For each curve $E_j$ we compute (to a specific precision) a $\Z$-basis for its period lattice $\Lambda_j$ using the standard AGM method. Here, $\Lambda_j$ is the lattice of periods of the N\'eron differential on $E_j$. The type of $\Lambda_j$ is determined by the sign of the discriminant of~$E_j$: type~$1$ for negative discriminant, and type $2$ for positive discriminant. For our algorithm we will need to know that $\Lambda_1$ and $\Lambda_f$ are approximately equal. To be precise, we know that they have the same type, and also we verify, for a specific positive $\varepsilon$, that \[ \left|\frac{\omega_{1,1}}{\omega_{1,f}}-1\right| < \varepsilon \qquad\text{and}\qquad \left|\frac{\im(\omega_{2,1})}{\im(\omega_{2,f})}-1\right| < \varepsilon. \tag{*} \] Here $\omega_{1,j}$, $\omega_{2,j}$ denote the normalised generators of~$\Lambda_j$, and $\omega_{1,f}$, $\omega_{2,f}$ those of~$\Lambda_f$. Pulling back the N\'eron differential on $E_{j_0}$ to $X_0(N)$ gives $c\cdot2\pi if(z)dz$ where $c\in\Z$ is the Manin constant for~$f$. Hence \[ c\Lambda_f = \Lambda_{j_0}. \] Our task is now to \begin{enumerate} \item identify $j_0$, to find which of the $E_j$ is (isomorphic to) the ``optimal'' curve $E_f$; and \item determine the value of~$c$. \end{enumerate} Our main result is that $j_0=1$ and $c=1$, provided that the precision bound $\varepsilon$ in (*) is sufficiently small (in most cases, $\varepsilon<1$ suffices). In order to state this precisely, we need some further definitions. A result of Stevens says that in the isogeny class there is a curve, say $E_{j_1}$, whose period lattice $\Lambda_{j_1}$ is contained in every $\Lambda_j$; this is the unique curve in the class with minimal Faltings height. (It is conjectured that $E_{j_1}$ is the $\Gamma_1(N)$-optimal curve, but we do not need or use this fact. In many cases, the $\Gamma_0(N)$- and $\Gamma_1(N)$-optimal curves are the same, so we expect that $j_0=j_1$ often; indeed, this holds for the vast majority of cases.) For each $j$, we know therefore that $a_j=\omega_{1,j_1}/\omega_{1,j}\in\N$ and also $b_j=\im(\omega_{2,j_1})/\im(\omega_{2,j})\in\N$. Let $B$ be the maximum of $a_1$ and $b_1$. \begin{prop} Suppose that (*) holds with $\epsilon=B^{-1}$; then $j_0=1$ and $c=1$. That is, the curve $E_1$ is the optimal quotient and its Manin constant is~$1$. \end{prop} \begin{proof} Let $\varepsilon=B^{-1}$ and $\lambda=\frac{\omega_{1,1}}{\omega_{1,f}}$, so $|\lambda-1|<\varepsilon$. For some $j$ we have $c\Lambda_f=\Lambda_j$. The idea is that $\lcm(a_1,b_1)\Lambda_1\subseteq\Lambda_{j_1}\subseteq\Lambda_j=c\Lambda_f$; if $a_1=b_1=1$, then the closeness of $\Lambda_1$ and $\Lambda_f$ forces $c=1$ and equality throughout. To cover the general case it is simpler to work with the real and imaginary periods separately. Firstly, \[ \frac{\omega_{1,j}}{\omega_{1,f}} =c\in\Z. \] Then \[ c= \frac{\omega_{1,1}}{\omega_{1,f}} \frac{\omega_{1,j}}{\omega_{1,1}} = \frac{a_1}{a_j}\lambda. \] Hence \[ 0 \le |\lambda-1| = \frac{|a_jc-a_1|}{a_1} <\varepsilon. \] If $\lambda\not=1$, then $\varepsilon>|\lambda-1|\ge a_1^{-1}\ge B^{-1}=\varepsilon$, contradiction. Hence $\lambda=1$, so $\omega_{1,1}= \omega_{1,f}$. Similarly, we have \[ \frac{\im(\omega_{2,j})}{\im(\omega_{2,f})} =c\in\Z \] and again we can conclude that $\im(\omega_{2,1})= \im(\omega_{2,f})$, and hence $\omega_{2,1}= \omega_{2,f}$. Thus $\Lambda_1=\Lambda_f$, from which the result follows. \end{proof} \begin{thm}\label{thm:cremona60000} For all $N<60000$, every optimal elliptic quotient of $J_0(N)$ has Manin constant equal to~$1$. Moreover, the optimal curve in each class is the one whose identifying number on the tables \cite{cremona:onlinetables} is~$1$ (except for class $990h$ where the optimal curve is $990h3$). \end{thm} \begin{proof} For all $N<60000$ we used modular symbols to find all newforms~$f$ and their period lattices, and also the corresponding isogeny classes of curves. In all cases we verified that (*) held with the appropriate value of $\varepsilon$. (The case of $990h$ is only exceptional on account of an error in labelling the curves several years ago, and is not significant.) \end{proof} \begin{rmk} In the vast majority of cases, the value of $B$ is~$1$, so the precision needed for the computation of the periods is very low. For $N<60000$, out of $258502$ isogeny classes, only $136$ have $B>1$: we found $a_1=2$ in $13$ cases, $a_1=3$ in $29$ cases, and $a_1=4$ and $a_1=5$ once each (for $N=15$ and $N=11$ respectively); $b_1=2$ in $93$ cases; and no larger values. Class $17a$ is the only one for which both $a_1$ and $b_1$ are greater than~$1$ (both are~$2$). \end{rmk} Finally, we give a slightly weaker result for $600001$ we also have $a_j=1$ and $\Lambda_j$ of type $2$. This occurs 28 times for $600001$. There are only three of these for $60000