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\def\evenhead{{\protect\centerline{\textsl{\large{Amod Agashe,
Kenneth Ribet and William A. Stein}}}\hfill}}
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\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}
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\footnotetext{Received January 25, 2006. \\ Stein was supported by
the National Science Foundation by Grant No. 0400386.}
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%Kenneth Ribet\\
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\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}
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%Amod Agashe
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%Kenneth A. Ribet Insert Current Address \Address William A. Stein
%Department of Mathematics Harvard University Cambridge, MA 02138
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\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