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Author: William A. Stein
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\begin{document}
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\Title
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The Modular Degree, Congruence Primes and Multiplicity One
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\ShortTitle
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The Modular Degree and Congruences
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\Author
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Amod Agashe\\
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Kenneth A. Ribet\\
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William A. Stein
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Agashe, Ribet, Stein
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\Abstract
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The modular degree and congruence number are two fundamental
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invariants of an elliptic curve over the rational field. Frey and
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M{\"u}ller have asked whether these invariants coincide. Although
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this question has a negative answer, we prove a theorem about the
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relation between the two invariants: one divides the other, and the
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ratio is divisible only by primes whose squares divide the conductor
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of the elliptic curve. We discuss the ratio even in the case where
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the square of a prime does divide the conductor, and we study
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analogues of the two invariants for modular abelian varieties of
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arbitrary dimension.
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\EndAbstract
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\MSC
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\Address
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Amod Agashe
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Insert Current Address
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\Address
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Kenneth A. Ribet
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\Address
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William A. Stein
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Department of Mathematics
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Harvard University
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Cambridge, MA 02138
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{\tt was@math.harvard.edu}
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%%---------------------Here the prologue ends---------------------------------
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%%--------------------Here the manuscript starts------------------------------
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\section{Introduction}
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Let~$E$ be an elliptic curve over~$\Q$. By
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\cite{breuil-conrad-diamond-taylor}, we may view~$E$ as an abelian
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variety quotient over $\Q$ of the modular Jacobian $J_0(N)$, where $N$
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is the conductor of~$E$. After possibly replacing $E$ by an isogenous
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curve, we may assume that the kernel of the map $J_0(N)\to E$ is
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connected, i.e., that~$E$ is an {\em optimal quotient} of $J_0(N)$.
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Let $f_E = \sum a_n q^n \in S_2(\Gamma_0(N))$ be the newform attached
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to $E$. The {\em congruence number}~$\re$ of~$E$ is the largest
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integer such that there is an element $g =\sum b_n q^n \in
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S_2(\Gamma_0(N))$ with integer Fourier coefficients $b_n$ that is
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orthogonal to~$f_E$ with respect to the Peterson innner product, and
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congruent to~$f_E$ modulo~$\re$ (i.e., $a_n \equiv b_n\pmod{\re}$ for
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all~$n$). The {\em modular
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degree}~$\me$ is the degree of the composite map $X_0(N)\to
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J_0(N)\to E$, where we map $X_0(N)$ to $J_0(N)$ by sending $P\in
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X_0(N)$ to $[P]-[\infty] \in J_0(N)$.
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Section~\ref{congintro} is about relations between~$\re$ and~$\me$.
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For example, $\me \mid \re$. In \cite[Q.~4.4]{frey-muller}, Frey and
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M{\"u}ller~ asked whether $\re = \me$. We give examples in which $\re
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\neq \me$, then conjecture that for any prime $p$, $\ord_p(\re/\me)
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\leq \frac{1}{2}\ord_p(N)$. We prove this conjecture when
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$\ord_p(N)\leq 1$.
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In Section~\ref{sec:quotients}, we consider analogues of congruence
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primes and the modular degree for optimal quotients that are not
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necessarily elliptic curves; these are quotients of~$J_0(N)$ and
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$J_1(N)$ of any dimension associated to ideals of the relevant Hecke
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algebras. In Section~\ref{sec:main} we prove the main theorem of this
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paper, and in Section~\ref{sec:mult1} we give some new examples of
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failure of multiplicity one motivated by the arguments in
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Section~\ref{sec:main}.
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% For an introduction and the motivation for studying
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% the objects in the title of the paper, the reader
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% may read Sections~\ref{sec:elliptic}
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% and~\ref{sec:quotients}, skipping the proofs.
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\smallskip
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{\bf \noindent Acknowledgment.} The authors are grateful to A.~Abbes,
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R.~Coleman, B.~Conrad, J.~Cremona, H.~Lenstra, E.~de Shalit,
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B.~Edixhoven, L.~Merel, and R.~Taylor for several discussions and
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advice regarding this paper.
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\section{Congruence Primes and the Modular Degree}
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\label{sec:elliptic}
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Let~$N$ be a positive integer and let $X_0(N)$ be the modular curve
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over~$\Q$ that classifies isomorphism classes of elliptic curves with
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a cyclic subgroup of order~$N$. The Hecke algebra~$\T$ of level~$N$
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is the subring of the ring of endomorphisms of $J_0(N)=\Jac(X_0(N))$
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generated by the Hecke operators $T_n$ for all $n\geq 1$. Let~$f$ be
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a newform of weight~$2$ for~$\Gamma_0(N)$ with integer Fourier
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coefficients, and let $I_f$ be kernel of the
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homomorphism $\T\to \Z[\ldots, a_n(f), \ldots]$ that sends $T_n$ to
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$a_n$. Then the quotient $E = J_0(N)/I_f J_0(N)$ is an elliptic curve
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over~$\Q$. We call~$E$ the {\em optimal quotient} associated to~$f$.
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Composing the embedding $X_0(N)\hra J_0(N)$ that sends $\infty$ to~$0$
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with the quotient map $J_0(N) \ra E$, we obtain a surjective morphism
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of curves $\phie: X_0(N) \ra E$.
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\begin{defi}
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The {\em modular degree} $\me$ of~$E$ is the degree of~$\phie$.
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\end{defi}
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\label{congintro}
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%The congruence number $\re$ and the modular degree $\me$
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%are of great interest.
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Congruence primes have been studied by Doi, Hida, Ribet,
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Mazur and others (see, e.g.,~\cite[\S1]{ribet:modp}),
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and played an important role in Wiles's work~\cite{wiles}
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on Fermat's last theorem. Frey and Mai-Murty have
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observed that an
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appropriate asymptotic bound on the modular degree is equivalent to
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the $abc$-conjecture (see~\cite[p.544]{frey:ternary}
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and~\cite[p.180]{murty:congruence}).
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Thus, results that relate congruence primes and the modular degree
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are of great interest.
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\begin{thm}\label{thm:ribet_au}
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\label{ddivsr}
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Let $E$ be an elliptic curve over $\Q$ of conductor~$N$, with modular
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degree $\me$ and congruence number $\re$.
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Then $\me \mid \re$ and if $\ord_p(N)\leq 1$ then $\ord_p(\re) = \ord_p(\me)$.
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\end{thm}
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We will prove a generalization of Theorem~\ref{thm:ribet_au}
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in Section~\ref{sec:main} below.
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The divisibility $\me\mid \re$ was first discussed
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in~\cite[Th.~3]{zagier}, where it is attributed to the second author
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(Ribet); however in \cite{zagier} the divisibility was mistakenly
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written in the opposite direction. For some other expositions of the
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proof, see~\cite[Lem~3.2]{abbull} and~\cite{cojo-kani}. We generalize
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this divisibility in Proposition~\ref{ndivsm}. The second part of
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Theorem~\ref{thm:ribet_au}, i.e., that if $\ord_p(N) \leq 1$ then
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$\ord_p(\re) = \ord_p(\me)$, follows from the more general
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Theorem~\ref{thm:ribet_gen} below.
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%\edit{I made this change. --Amod}
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%in more generality in in Section~\ref{sec:proof_ribet} below.
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Note that \cite[Prop.~3.3--3.4]{abbull} implies the weaker
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statement that if $p\nmid N$ then $\ord_p(\re)=\ord_p(\me)$,
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since \cite[Prop.~3.3]{abbull}
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implies $$\ord_p(\re) - \ord_p(\me) = \ord_p(\#\mathcal{C})
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- \ord_p(\ce) - \ord_p(\#\mathcal{D}),$$ and by \cite[Prop.~3.4]{abbull}
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$\ord_p(\#\mathcal{C}) =0$. (Here $\ce$ is the Manin
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constant of $E$, which is an integer by results of
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Edixhoven and Katz-Mazur; see e.g., \cite{ars} for more details.)
498
499
Frey and M{\"u}ller~\cite[Ques.~4.4]{frey-muller} asked whether $\re =
500
\me$ in general. After implementing an algorithm to compute $\re$ in
501
Magma \cite{magma}, we quickly found that the answer is no. The
502
counterexamples at conductor $N\leq 144$ are given in Table~\ref{table:moddeg},
503
where the curve
504
is given using the notation of \cite{cremona:alg}:
505
\begin{table}\caption{Differing Modular Degree and
506
Congruence Number\label{table:moddeg}}
507
\begin{center}
508
\begin{tabular}{|l|l|l|}\hline
509
Curve & $\me$ & $\re$\\\hline
510
54B1 & 2 & 6\\\hline
511
64A1 & 2 & 4 \\\hline
512
72A1 & 4 & 8 \\\hline
513
80A1 & 4 & 8 \\\hline
514
88A1 & 8 & 16 \\\hline
515
92B1 & 6 & 12\\\hline
516
96A1 & 4 & 8 \\\hline
517
96B1 & 4 & 8 \\\hline
518
\end{tabular}
519
\begin{tabular}{|l|l|l|}\hline
520
Curve & $\me$ & $\re$\\\hline
521
99A1 & 4 & 12\\\hline
522
108A1 & 6 & 18\\\hline
523
112A1 & 8 & 16\\\hline
524
112B1 & 4 & 8\\\hline
525
112C1 & 8 & 16\\\hline
526
120A1 & 8 & 16\\\hline
527
124A1 & 6 & 12\\\hline
528
126A1 & 8 & 24\\\hline
529
\end{tabular}
530
\begin{tabular}{|l|l|l|}\hline
531
Curve & $\me$ & $\re$\\\hline
532
128A1 & 4 & 32\\\hline
533
128B1 & 8 & 32\\\hline
534
128C1 & 4 & 32\\\hline
535
128D1 & 8 & 32\\\hline
536
135A1 & 12 & 36\\\hline
537
144A1 & 4 & 8 \\\hline
538
144B1 & 8 & 16 \\\hline
539
&&\\\hline
540
\end{tabular}
541
\end{center}
542
\end{table}
543
%$$54, 64, 72, 80, 88, 92, 96, 99, 108, 112, 120, 124, 126, 128, 135,
544
%\text{ and } 144.$$
545
546
For example, the elliptic curve 54B1, given by the equation $y^2 + xy +
547
y = x^3 - x^2 + x - 1$, has $\re=6$ and $\me=2$. To see explicitly
548
that $3 \mid \re$, observe that the newform corresponding to~$E$ is
549
$f=q + q^2 + q^4 - 3q^5 - q^7 + \cdots$ and the newform corresponding
550
to $X_0(27)$ if $g=q - 2q^4 - q^7 + \cdots$, so $g(q) + g(q^2)$
551
appears to be congruent to~$f$ modulo~$3$. To prove this congruence,
552
we checked it for $18$ Fourier coefficients, where the
553
sufficiency of precision to degree $18$
554
was determined using \cite{sturm:cong}.
555
556
% In accord with Theorem~\ref{thm:ribet_au},
557
%since $\ord_3(\re) \neq \ord_3(\ce)$, we have $\ord_3(54)\geq 2$.
558
559
In our computations, there appears to be no absolute bound on the~$p$
560
that occur. For example, for the curve 242B1 of conductor $N=2\cdot 11^2$
561
we have\footnote{The curve 242a1 in ``modern notation.''}
562
$$
563
\me = 2^4 \neq \re = 2^4\cdot 11.
564
$$
565
We propose the following replacement for Question~4.4 of
566
\cite{frey-muller}:
567
\begin{conj}\label{conj:rm}
568
Let~$E$ be an optimal elliptic curve of conductor~$N$
569
and~$p$ be any prime.
570
Then
571
$$
572
\ord_p\left(\frac{\re}{\me}\right) \leq \frac{1}{2}\ord_p(N).
573
$$
574
\end{conj}
575
We verified Conjecture~\ref{conj:rm} using Magma for every optimal
576
elliptic curve quotient of $J_0(N)$, with $N\leq 539$.
577
578
If $p\geq 5$ then $\ord_p(N)\leq 2$, so a special case
579
of the conjecture is
580
$$
581
\ord_p\left(\frac{\re}{\me}\right) \leq 1\qquad\text{ for any }p\geq 5.
582
$$
583
584
585
\begin{rmk}
586
It is often productive to parametrize elliptic curves by $X_1(N)$
587
instead of $X_0(N)$ (see, e.g., \cite{stevens:param} and
588
\cite{MR2135139}). Suppose $E$ is an optimal quotient of $X_1(N)$,
589
let $m_E'$ be the degree of the modular parametrization, and let
590
$r_E'$ be the $\Gamma_1(N)$-congruence number, which is defined as
591
above but with $S_2(\Gamma_0(N))$ replaced by $S_2(\Gamma_1(N))$.
592
For the optimal quotient of $X_1(N)$ isogenous to 54B1, we find
593
using Magma that $m_E' = 18$ and $r_E'=6$. Thus the equality
594
$m_E'=r_E'$ fails, and the analogous divisibility $m_E'\mid
595
r_E'$ no longer holds. Also, for a curve of conductor $38$ we have
596
$m_E'=18$ and $r_E'=6$, so equality need not hold even if the level
597
is square free. We hope to investigate this in a future paper.
598
%> N := 38; D := ND(NS(CS(ModularSymbols(Gamma1(N)))));
599
%> ModularDegree(D[1]); CongruenceModulus(D[1]);
600
%18
601
%6
602
%
603
%> N := 54; D := ND(NS(CS(ModularSymbols(Gamma1(N)))));
604
%> ModularDegree(D[1]); CongruenceModulus(D[1]);
605
%18
606
%18
607
%> ModularDegree(D[2]); CongruenceModulus(D[2]);
608
%18
609
%6
610
\end{rmk}
611
612
613
%\subsection{Proof of Theorem~\ref{thm:ribet_au}}\label{sec:proof_ribet}
614
615
\section{Modular abelian varieties of arbitrary dimension}
616
\label{sec:quotients}
617
For $N\geq 4$, let~$\Gamma$ be a fixed choice of either~$\Gamma_0(N)$
618
or~$\Gamma_1(N)$, let~$X$ be the modular curve over~$\Q$ associated
619
to~$\Gamma$, and let~$J$ be the Jacobian of~$X$. Let~$I$ be a {\em
620
saturated} ideal of the corresponding Hecke algebra
621
$\T\subset\End(J)$, so $\T/ I$
622
is torsion free. Then $A = A_I = J/IJ$ is an optimal quotient of~$J$
623
since $IJ$ is an abelian subvariety.
624
625
\begin{defi}
626
If~$f=\sum a_n(f)q^n \in S_2(\Gamma)$ and $I_f=\ker(\T\to
627
\Z[\ldots,a_n(f),\ldots])$, then $A=A_f=J/I_f J$ is the {\em newform
628
quotient} associated to~$f$. It is an abelian variety over~$\Q$
629
of dimension equal to the degree of the field
630
$\Q(\ldots,a_n(f),\ldots)$.
631
\end{defi}
632
633
In this section, we generalize the notions of the congruence number
634
and the modular degree to quotients~$A=A_I$, and state a theorem
635
relating the two numbers, which we prove in
636
Sections~\ref{sec:firstpart}--\ref{sec:secondpart}.
637
638
Let $\phi_2$ denote the quotient map $J \ra A$. By Poincare
639
reducibility over $\Q$ there is a unique abelian subvariety $A^{\vee}$
640
of $J$ that projects isogenously to the quotient $A$ (equivalently,
641
which has finite intersection with $\ker(\phi_2)$), and so by Hecke
642
equivariance of $J \to A$ it follows that $A^{\vee}$ is $\T$-stable.
643
Let $\phi$ be the composite isogeny
644
$$
645
\phi: \Adual \stackrel{\po}{\lra} J \stackrel{\pt}{\lra} A.
646
$$
647
648
\begin{rmk}
649
Note that $A^{\vee}$ is the dual abelian variety of $A$. More
650
generally, if~$C$ is any abelian variety, let $C^{\vee}$ denote the
651
dual of~$C$. There is a canonical principal polarization $J \cong
652
\Jdual$, and dualizing $\phi_2$, we obtain a map $\phi_2^\vee: \Adual
653
\ra \Jdual$, which we compose with $\theta^{-1}: \Jdual \cong J$ to
654
obtain a map $\po: \Adual \ra J$. Note also that $\vphi$
655
is a polarization (induced by pullback of the theta divisor).
656
\end{rmk}
657
658
%\begin{prop} \label{modular:isogeny0}
659
%The map $\phi$ is a polarization.
660
%\end{prop}
661
% \begin{proof}
662
% Let $i$ be the injection $\phi_2^{\vee}:\Adual \ra \Jdual$, and let
663
% $\Theta$ denote the theta divisor. From the definition of the
664
% polarization attached to an ample divisor, we see that the map~$\phi$
665
% is induced by the pullback $i^*(\Theta)$ of the theta divisor. The
666
% theta divisor is effective, and hence so is $i^*(\Theta)$.
667
%By~\cite[\S6, Application~1, p. 60]{mumford:av}, $\ker \phi$ is
668
%finite. Since the dimensions of $A$ and~$\Adual$ are the same, $\phi$
669
%is an isogeny.
670
% Since $\Theta$ is ample, some power of it is
671
% very ample. Then the pullback of this very ample power by~$i$ is again
672
% very ample, and hence a power of $i^*(\Theta)$ is very ample, so
673
% $i^*(\Theta)$ is ample (by~\cite[II.7.6]{hartshorne:ag}).
674
% \end{proof}
675
676
The {\em exponent} of a finite group~$G$ is the smallest positive
677
integer~$n$ such that every element of~$G$ has order dividing~$n$.
678
679
\begin{defi}\label{defi:modular}
680
The {\em modular exponent} of~$A$ is the exponent of the kernel
681
of the isogeny~$\phi$, and the {\em modular number} of~$A$ is
682
the degree of~$\phi$.
683
\end{defi}
684
685
We denote the modular exponent of~$A$ by~$\nAe$ and
686
the modular number by~$\nA$.
687
When~$A$ is an elliptic curve, the modular
688
exponent is equal to the modular degree of~$A$,
689
and the modular number is the square of the modular degree
690
(see, e.g.,~\cite[p.~278]{abbull}).
691
%(see \cite[p.~276]{abbull}).
692
%When~$A$ is an elliptic curve, $\na$ is just the
693
%modular degree of~$A$.
694
695
If~$R$ is a subring of~$\C$,
696
let $S_2(R)=S_2(\Gamma;R)$ denote the subgroup of~$S_2(\Gamma)$
697
consisting of cups forms whose Fourier expansions at the cusp~$\infty$
698
have coefficients in~$R$. (Note that $\Gamma$ is fixed for this whole
699
section.)
700
Let $S_2(\Gamma;\Z)[I]^{\perp}$ denote the orthogonal complement of
701
$S_2(\Gamma;\Z)[I]$ in $S_2(\Gamma;\Z)$ with respect to the Petersson inner
702
product.
703
704
The following is well known, but we had difficulty finding
705
a good reference.
706
\begin{prop}
707
The group $S_2(\Gamma;\Z)$ is of finite rank as a $\Z$-module.
708
\end{prop}
709
\begin{proof}
710
Using the standard pairing between $\T$ and $S_2(\Gamma,\Z)$ (see
711
also~\cite[Theorem~2.2]{ribet:modp}) we see that $S_2(\Gamma,\Z)
712
\isom \Hom(\T,\Z)$. Thus $S_2(\Gamma,\Z)$ is finitely generated
713
over~$\Z$ if and only if~$\T$ is finitely generated over~$\Z$. But
714
the action of~$\T$ on $\H_1(J,\Z)$ is a faithful representation that
715
embeds~$\T$ into $\Mat_{2d}(\Z) \isom \Z^{(2d)^2}$. But~$\Z$ is
716
Noetherian, so~$\T$ is finitely generated over~$\Z$.
717
\end{proof}
718
719
\begin{defi}\label{def:congexp}
720
The exponent of the quotient group
721
\begin{equation}\label{eqn:congexp}
722
\frac{S_2(\Gamma; \Z)} { S_2(\Gamma; \Z)[I] + S_2(\Gamma;\Z)[I]^{\perp}}
723
\end{equation}
724
is the {\em congruence exponent} $\rAe$ of~$A$ and its
725
order is the {\em congruence number} $\rA$.
726
\end{defi}
727
728
\begin{rmk}
729
Note that $S_2(\Gamma,\Z)\tensor_\Z R = S_2(\Gamma,R)$; see, e.g.,
730
the discussion in \cite[\S12]{diamond-im}. Thus the analogue of
731
Definition~\ref{def:congexp} with $\Z$ replaced by an algebraic
732
integer ring (or even $\Zbar$) gives a torsion module whose
733
annihilator ideal meets~$\Z$ in the ideal generated by the
734
congruence exponent.
735
\end{rmk}
736
737
Our definition of~$\rA$ generalizes the definition in
738
Section~\ref{congintro} when~$A$ is an elliptic curve (see
739
\cite[p.~276]{abbull}), and the following generalizes
740
Theorem~\ref{thm:ribet_au}:
741
\begin{thm}\label{thm:ribet_gen}
742
If $f \in S_2(\C)$ is a newform, then
743
\begin{itemize}
744
\item[(a)] We have $\nAfe \mid \rAfe$, and
745
\item[(b)] If $p^2 \nmid N$, then $\ord_p(\rAfe) = \ord_p(\nAfe)$.
746
\end{itemize}
747
% $p \nmid \frac{\rAfe}{\nAfe}$.
748
\end{thm}
749
%We give the proof of this theorem in the next two sections.
750
%The rest of the section is devoted to proving Proposition~\ref{ndivsm}
751
%below, which asserts that if~$f$ is a newform, then $\nAfe \mid
752
%\rAfe$.
753
754
\begin{rmk}\label{rem:24}
755
When $A_f$ is an elliptic curve, Theorem~\ref{thm:ribet_gen} implies
756
that the modular degree divides the congruence number (since for an
757
elliptic curve the modular degree and modular exponent are the
758
same), i.e., $\sqrt{\nAf} \mid \rAf$. In general, the divisibility
759
$\nAf\mid r^2_{A_f}$ need not hold. For example, there is a newform
760
of degree $24$ in $S_2(\Gamma_0(431))$ such that
761
$$\nAf = (2^{11}\cdot 6947)^2 \,\,\nmid\,\, r^2_{A_f} = (2^{10}\cdot
762
6947)^2.$$
763
Note that $431$ is prime and mod~$2$ multiplicity one fails for $J_0(431)$ (see
764
\cite{kilford}).
765
%The following Magma session illustrates how to verify the above
766
%assertion about $\nAf$ and $\rAf$. The commands are parts of Magma
767
%V2.11 or greater. \vspace{-1ex}
768
%{\small
769
%\begin{verbatim}
770
% > A := ModularSymbols("431F");
771
% > Factorization(ModularDegree(A));
772
% [ <2, 11>, <6947, 1> ]
773
% > Factorization(CongruenceModulus(A));
774
% [ <2, 10>, <6947, 1> ]
775
%\end{verbatim}
776
%}
777
\end{rmk}
778
779
780
\section{Proof of the Main Theorem}\label{sec:main}
781
In this section we prove Theorem~\ref{thm:ribet_gen}.
782
We continue using the notation introduced so far.
783
784
\subsection{Proof of Theorem~\ref{thm:ribet_gen} (a)}
785
\label{sec:firstpart}
786
787
We begin with a remark about compatibilities. In general, the
788
polarization of~$J$ induced by the theta divisor need not be Hecke
789
equivariant, because if~$T$ is a Hecke operator on~$J$, then
790
on~$\Jdual$ it acts as $W_N T W_N$, where $W_N$ is the Atkin-Lehner
791
involution (see e.g.,~\cite[Rem.~10.2.2]{diamond-im}). However,
792
on~$J^{\rm new}$ the action of the Hecke operators commutes with that
793
of~$W_N$, so if the quotient map $J \ra A$ factors through~$J^{\rm new}$,
794
then the Hecke action on~$\Adual$ induced by the embedding $\Adual \to
795
J^{\vee}$ and the action on $\Adual$ induced by $\phi_1:\Adual\to{}J$
796
are the same. Hence $\Adual$ is isomorphic to $\po(\Adual)$
797
as a $\T$-module.
798
799
Recall that $f$ is a newform, $I_f = {\rm Ann}_\T (f)$, and
800
$J=J_0(N)$. Let $B = I_fJ$, so that $\Adual+B=J$, and $J/B\isom A$.
801
The following lemma is well known, but we prove it here for the
802
convenience of the reader.
803
804
\begin{lem}\label{lem:homzero}
805
$\Hom_\Q(\Adual,B)=0$.
806
\end{lem}
807
\begin{proof}
808
% Suppose there were a nonzero element of $\Hom_\Q(\Adual,B)$. Since
809
% $A$ is simple, for all~$\ell$ the Tate module
810
% $V_{\ell}(\Adual)=\Q\tensor\varprojlim_n \Adual[\ell^n]$ would
811
% be a factor of $V_{\ell}(B)$.
812
% Thus the characteristic polynomial
813
% The Eichler-Shimura relation then implies that the characteristic
814
% polynomial of each
815
% One could then extract almost all
816
% prime-indexed coefficients of the corresponding eigenforms from the
817
% Tate modules, which would violate multiplicity one for systems of
818
% Hecke eigenvalues .
819
820
Pick a prime $\ell$. Then $\Qbar_{\ell} \tensor V_{\ell} (J)^{\ss}$
821
as a $\Qbar_{\ell}[G_\Q]$-module is a direct sum of copies of the
822
representations $\rho_g$ as $g$ ranges through all normalized
823
eigenforms of weight $2$ and level $N$ with coefficients in $\Qbar$;
824
by a well-known result of the second author, these representations
825
are absolutely irreducible. Now since~$f$ is a newform and
826
$A^{\vee} \to A$ is an isogeny, $\Qbar_{\ell} \tensor
827
V_{\ell}(A^{\vee})^{\ss}$ is a direct sum of copies of
828
$\rho_{\sigma(f)}$ as $\sigma$ ranges over all embeddings of $K_f$
829
into $\Qbar$. Thus, by the analytic theory of multiplicity one (see
830
\cite[Cor.~3, pg.~300]{winnie:newforms}), the Galois modules
831
$V_{\ell}(A^{\vee})$ and $V_{\ell}(B) =
832
V_{\ell}(J)/V_{\ell}(A^{\vee})$ share no common Jordan-H\"older
833
factors even when coefficients are extended to $\Qbar_{\ell}$, so
834
$\Hom_\Q(A',B) = 0$.
835
\end{proof}
836
837
838
Let $\T_1$ be the image of~$\T$ in $\End(\Adual)$,
839
and let $\T_2$ be the image of $\T$ in $\End(B)$.
840
We have the following commutative diagram with exact rows:
841
\begin{equation}\label{eqn:diagram}
842
\[email protected]=2em{
843
0\ar[r] & {\T} \ar[r]\ar[d] & {\T_1\oplus \T_2} \ar[r]\ar[d] &
844
{\displaystyle \frac{\T_1 \oplus \T_2}{\T}}\[email protected]{.>}[d]\ar[r] & 0\\
845
0\ar[r] & {\End(J)} \ar[r] & {\End(\Adual)\oplus\End(B)} \ar[r] &
846
{\displaystyle \frac{\End(\Adual)\oplus\End(B)}{\End(J)}}\ar[r] & 0.\\
847
}
848
\end{equation}
849
Let
850
$$
851
e=(1,0)\in \T_1 \oplus \T_2,
852
$$
853
and let $e_1$ and $e_2$ denote the images of~$e$ in the groups $(\T_1
854
\oplus \T_2)/\T$ and $(\End(\Adual) \oplus \End(B))/\End(J)$,
855
respectively. It follows from Lemma~\ref{lem:homzero} that the two
856
quotient groups on the right hand side of (\ref{eqn:diagram}) are
857
finite, so~$e_1$ and~$e_2$ have finite order. Note that because $e_2$
858
is the image of $e_1$, the order of $e_2$ is a divisor of the order of
859
$e_1$.
860
861
%this will be used in the proof of Proposition~\ref{ndivsm}
862
%below.
863
864
865
The {\em denominator} of any $\vphi\in\End(J)\tensor\Q$ is the
866
smallest positive integer~$n$ such that $n\vphi\in\End(J)$.
867
% Explicitly, the denominator of~$\vphi$ is the least common multiples
868
% of the denominators of the entries of any matrix that represents the
869
% action of $\vphi$ on the lattice $\H_1(J,\Z)$.
870
871
Let $\piAd, \piB \in \End(J)\tensor\Q$ be projection onto
872
$\Adual$ and $B$, respectively. Note that the denominator of
873
$\piAd$ equals the denominator of $\piB$, since $\piAd
874
+ \piB = 1_J$, so that $\piB = 1_J - \piAd$.
875
876
\begin{lem}\label{lem:ord_e2}
877
The element $e_2\in (\End(\Adual) \oplus \End(B))/\End(J)$
878
defined above has order $\nAe$.
879
\end{lem}
880
\begin{proof}
881
Let $n$ be the order of $e_2$, so~$n$ is the denominator
882
of $\piAd$, which, as mentioned above, is also the
883
denominator of $\piB$. We want to show that $n$ is equal
884
to~$\nAe$, the exponent of $\Adual\cap B$.
885
886
Let $i_{\Adual}$ and $i_B$
887
be the embeddings of $\Adual$ and $B$ into $J$, respectively.
888
Then $$\vphi = (n\piAd,n\piB)\in\Hom(J,\Adual\times B)$$
889
and $\vphi\circ (i_{\Adual} + i_B) = [n]_{\Adual\times B}.$
890
We have an exact sequence
891
$$
892
0\to \Adual\cap B\xra{x\mapsto (x,-x)}\Adual\times B \xra{i_{\Adual} + i_B} J \to 0.
893
$$
894
Let $\Delta$ be the image of $\Adual\cap B$. Then by exactness,
895
$$
896
[n]\Delta = (\vphi\circ (i_{\Adual} + i_B))(\Delta) =
897
\vphi\circ ((i_{\Adual} + i_B)(\Delta)) = \vphi(\{0\}) = \{0\},
898
$$
899
so $n$ is a multiple of
900
the exponent~$\nAe$ of $\Adual\cap B$.
901
902
To show the opposite divisibility, consider the
903
commutative diagram
904
$$
905
\[email protected]=4em{
906
0 \ar[r] & {\Adual \cap B} \ar[r]^{x\mapsto (x,-x)}\ar[d]^{[\nAe]}&
907
{\Adual \times B}\ar[d]^{([\nAe],0)}
908
\ar[r]& J \ar[r]\[email protected]{.>}[d]^{\psi} & 0\\
909
0 \ar[r] & {\Adual \cap B} \ar[r]^{x\mapsto (x,-x)}& {\Adual \times B}
910
\ar[r]& J \ar[r] & 0,
911
}
912
$$
913
where the middle vertical map is $(a,b)\mapsto (\nAe a,0)$
914
and the map~$\psi$ exists because $[\nAe](\Adual\cap B)=0$.
915
But $\psi = \nAe \piAd$ in $\End(J)\tensor\Q$.
916
This shows that $\nAe \piAd \in \End(J)$, i.e.,
917
that $\nAe$ is a multiple of the
918
denominator~$n$ of $\piAd$.
919
920
\end{proof}
921
922
Let $\Ext^1 = \Ext^1_{\Z}$ denote the first $\Ext$ functor
923
in the category of $\Z$-modules.
924
925
\begin{lem}\label{lem:compare_with_dual}
926
The group $(\T_1 \oplus \T_2)/\T$ is isomorphic to
927
the quotient (\ref{eqn:congexp})
928
in Definition~\ref{def:congexp}, so
929
$\rA = \#((\T_1 \oplus \T_2)/\T)$ and $\rAe$ is the
930
exponent of $(\T_1 \oplus \T_2)/\T$.
931
More precisely, $\Ext^1((\T_1 \oplus \T_2)/\T,\Z)$ is isomorphic as a
932
$\T$-module to the quotient (\ref{eqn:congexp}).
933
\end{lem}
934
\begin{proof}
935
Apply the $\Hom(-,\Z)$ functor to the first row of (\ref{eqn:diagram})
936
to obtain a three-term exact sequence
937
\begin{equation}\label{eqn:dualseq}
938
0 \to \Hom(\T_1\oplus \T_2,\Z) \to \Hom(\T,\Z)
939
\to \Ext^1((\T_1\oplus\T_2)/\T,\Z) \to 0.
940
\end{equation}
941
%The term $\Ext^1(\T_1\oplus \T_2,\Z)$ is $0$ is because
942
%$\Ext^1(M,\Z)=0$ for any finitely generated free abelian group. Also,
943
%$\Hom((\T_1\oplus\T_2)/\T,\Z)=0$ since $(\T_1\oplus\T_2)/\T$ is
944
%torsion.
945
There is a $\T$-equivariant bilinear pairing $\T\times
946
S_2(\Z)\to\Z$ given by $(t,g)\mapsto a_1(t(g))$, which is perfect by
947
\cite[Lemma~2.1]{abbull} (see also~\cite[Theorem~2.2]{ribet:modp}).
948
Using this pairing, we transform (\ref{eqn:dualseq}) into an exact
949
sequence
950
$$
951
0 \to S_2(\Z)[I_f] \oplus S_2(\Gamma;\Z)[I_f]^{\perp} \to S_2(\Z) \to
952
\Ext^1((\T_1\oplus\T_2)/\T,\Z) \to 0
953
$$
954
of $\T$-modules.
955
Here we use that $\Hom(\T_2,\Z)$ is the unique saturated
956
Hecke-stable complement of $S_2(\Z)[I_f]$ in $S_2(\Z)$, hence
957
must equal $S_2(\Z)[I_f]^{\perp}$.
958
Finally note that if~$G$ is any finite abelian group, then
959
$\Ext^1(G,\Z)\approx G$ as groups, which gives the desired result.
960
\end{proof}
961
962
\begin{lem}\label{lem:ord_e1}
963
The element $e_1 \in (\T_1 \oplus \T_2)/\T$ has order $\rAe$.
964
\end{lem}
965
\begin{proof}
966
By Lemma~\ref{lem:compare_with_dual}, the lemma is equivalent to the
967
assertion that the order~$r$ of~$e_1$ equals the exponent of
968
$M=(\T_1 \oplus \T_2)/\T$. Since $e_1$ is an element of~$M$, the
969
exponent of~$M$ is divisible by~$r$.
970
971
To obtain the reverse divisibility, consider any element $x$ of~$M$.
972
Let $(a,b)\in\T_1\oplus \T_2$ be such that its image in~$M$ is~$x$.
973
By definition of $e_1$ and~$r$, we have $(r,0)\in\T$, and since
974
$1=(1,1)\in\T$, we also have $(0,r)\in\T$. Thus $(\T{}r,0)$ and
975
$(0,\T{}r)$ are both subsets of $\T$ (i.e., in the image of $\T$
976
under the map $\T\to\T_1\oplus \T_2$), so $r(a,b)
977
=(ra,rb)=(ra,0)+(0,rb)\in \T$. This implies that the order of~$x$
978
divides~$r$. Since this is true for every $x \in M$, we conclude
979
that the exponent of~$M$ divides~$r$.
980
\end{proof}
981
982
983
\begin{prop} \label{ndivsm}
984
If $f \in S_2(\C)$ is a newform, then
985
$\nAfe \mid \rAfe$.
986
\end{prop}
987
\begin{proof}
988
Since~$e_2$ is the image of~$e_1$ under the right-most
989
vertical homomorphism in (\ref{eqn:diagram}), the order
990
of~$e_2$ divides that of~$e_1$. Now
991
apply Lemmas~\ref{lem:ord_e2} and \ref{lem:ord_e1}.
992
\end{proof}
993
994
This finishes the proof of the first statement in
995
Theorem~\ref{thm:ribet_gen}.
996
997
998
999
1000
\subsection{Proof of Theorem~\ref{thm:ribet_gen} (b)}
1001
\label{sec:secondpart}
1002
Let $\T'$ be the saturation of $\T=\Z[\ldots, T_n,\ldots]$ in
1003
$\End(J_0(N))$, i.e., the set of elements of $\End(J_0(N)) \tensor \Q$
1004
some positive multiple of which lie in~$\T$.
1005
%so
1006
%$$
1007
% \T' = (\T\tensor\Q) \cap \End(J_0(N)),
1008
%$$
1009
%where the intersection is taken inside $\End(J_0(N))\tensor\Q$.
1010
The
1011
quotient $\T'/\T$ is a finitely generated abelian group because both
1012
$\T$ and $\End(J_0(N))$ are finitely generated over~$\Z$. Since
1013
$\T'/\T$ is also a torsion group, it is finite.
1014
1015
In Section~\ref{sec:multone}, we will give some conditions under
1016
which $\T$ and~$\T'$ agree locally at maximal ideal of~$\T$.
1017
In Section~\ref{sec:degrees}, we will explain how the ratio of
1018
the congruence number to the modular degree is closely related
1019
to the order of~$\T'/\T$, and finally deduce that this ratio is $1$
1020
(for quotients associated to newforms) locally at a prime~$p$
1021
such that $p^2 \nmid N$.
1022
1023
\subsubsection{Multiplicity One} \label{sec:multone}
1024
1025
%Suppose for the moment that $M=1$, so $p=pM$.
1026
Fixt an integer $N$ and a prime $p\mid N$.
1027
Suppose for a moment that $N$ is prime, so $p=N$.
1028
In \cite{mazur:eisenstein},
1029
Mazur proves that $\T=\T'$; he combines this result with
1030
the equality
1031
$$
1032
\T\tensor\Q = \End(J_0(p)) \tensor\Q,
1033
$$
1034
to deduce that $\T=\End(J_0(p))$.
1035
This result, combined with Ribet's result
1036
\cite{ribet:endo} or \cite{ribet:endalg}
1037
to the effect that $\T\tensor\Q = (\End_{\Qbar} J_0(N)) \tensor \Q$,
1038
shows that~$\T$ is the full ring of endomorphisms of $J_0(N)$ over $\Qbar$.
1039
When $N$ is no
1040
longer necessarily prime,
1041
the method of \cite{mazur:eisenstein} shows
1042
that $\T$ and $\T'$ agree locally at a maximal ideal $\m$ of $\T$
1043
that satisfies a simple condition involving
1044
differentials form mod $\ell$, where $\ell$ is the residue
1045
characteristic of $\m$.
1046
%has dimension at most one.
1047
\comment{
1048
$\Supp_{\T}(\T'/\T)$ contains no maximal ideal $\m$ of~$\T$
1049
for which his space $\H^0(X_0(pM)_{\Fell},\Omega)[\m]$ has
1050
dimension $\leq 1$. (Here $\ell$ is the residue characteristic
1051
of $\m$.) In other words, multiplicity one for
1052
$\H^0(X_0(pM)_{\Fell}, \Omega)[\m]$ implies that $\T$ and
1053
$\T'$ agree at~$\m$.
1054
Mazur's argument (see \cite[pg.~95]{mazur:eisenstein}) is quite
1055
general; it relies on a multiplicity $1$ statement for spaces
1056
of differentials in positive characteristic (see
1057
\cite[Prop.~9.3, pg.~94]{mazur:eisenstein}).
1058
}
1059
1060
For the sake of completeness, we state and prove a lemma
1061
that can be easily extracted from~\cite{mazur:eisenstein}.
1062
Let $m$ be the largest square dividing $N$ and
1063
let $R = \Z[\frac{1}{m}]$. Let $X_0(N)_{R}$ denote
1064
the minimal regular model of $X_0(N)$ over~$R$.
1065
%Let $\m$ be a maximal ideal of the Hecke algebra of
1066
%residue characteristic~$\ell$ and suppose $\ell^2 \nmid N$.
1067
Let $\Omega=\Omega_{X_0(N)/R}$ denote the sheaf of regular
1068
differentials on $X_0(N)_{R}$, as in~\cite[\S2(e)]{mazur:rational}.
1069
If~$\ell$ is a prime such that $\ell^2 \nmid N$, then
1070
$X_0(N)_{\F_\ell}$ denotes the special fiber of $X_0(N)_{R}$ at the
1071
prime~$\ell$.
1072
1073
\comment{
1074
His method shows in
1075
the general case (where~$M$ is no longer constrained to be~$1$)
1076
that $\Supp_{\T}(\T'/\T)$ contains no maximal ideal $\m$ of~$\T$
1077
for which his space $\H^0(X_0(pM)_{\Fell},\Omega)[\m]$ has
1078
dimension $\leq 1$. (Here $\ell$ is the residue characteristic
1079
of $\m$.) In other words, multiplicity one for
1080
$\H^0(X_0(pM)_{\Fell}, \Omega)[\m]$ implies that $\T$ and
1081
$\T'$ agree at~$\m$. We record this fact as a lemma
1082
(see also Section~\ref{sec:dataind} for related data).
1083
}
1084
1085
\begin{lem}[Mazur]\label{lem:m1}
1086
Let $\m$ be a maximal ideal of $\T$ of residue characteristic~$\ell$
1087
such that $\ell^2 \nmid N$.
1088
Suppose that
1089
$$
1090
\dim_{\T/\m} \H^0(X_0(N)_{\Fell},\Omega)[\m] \leq 1.
1091
$$
1092
Then $\T$ and $\T'$ agree locally at~$\m$.
1093
%$\m$ is not in the support of $\T'/\T$.
1094
\end{lem}
1095
1096
\begin{proof}
1097
Let $M$ denote the group
1098
$H^1(X_0(N)_R, \OO_{X_0(N)})$,
1099
where $\OO_{X_0(N)}$ is the structure sheaf of $X_0(N)$.
1100
As explained in~\cite[p.~95]{mazur:eisenstein},
1101
we have an action of $\EJ$ on~$M$, and
1102
the action of $\T$ on~$M$ via the inclusion $\T \subseteq \EJ$
1103
is faithful, so likewise for the action by $\T'$. Hence we have an injection
1104
$\phi: \T' \hookrightarrow {\rm End}_{\T} M$.
1105
% is a free module over~$\T$.
1106
Suppose~$\m$ is a maximal ideal of~$\T$ that satisfies the hypotheses
1107
of the lemma.
1108
To prove that $\T_\m=\T_\m'$ it suffices to
1109
prove the following claim:\\
1110
1111
\noindent {\em Claim:} The map~$\phi|_{\T}$ is surjective locally at~$\m$.
1112
\begin{proof}
1113
By Nakayama's lemma, to show that $M$ is generated
1114
as a single element over~$\T$ locally at~$\m$,
1115
%to show that $\phi$ is surjective, i.e.,
1116
%to show that $M \tensor \T_\m$ is generated by a single element over
1117
%$\T\tensor \T_\m$,
1118
it suffices to check that the dimension of the ${\T/\m}\,$-vector space
1119
$M / \m M$ is at most one.
1120
% for each maximal ideal~$\m$ dividing~$\ell$ in~$\T$.
1121
Since \mbox{$\ell^2 \nmid N$},
1122
%$H^1(X_0(N)_{\F_\ell}, \OO) / \m H^1(X_0(N)_{\F_\ell}, \OO)$
1123
$M/ \m M$ is dual to
1124
$H^0(X_0(N)_{\F_\ell}, \Omega) [\m]$ (see, e.g.,~\cite[\S2]{mazur:rational}).
1125
Since we are assuming that
1126
${\rm dim}_{\T/\m} H^0(X_0(N)_{\F_\ell}, \Omega) [\m] \leq 1$, we have
1127
${\rm dim}_{\T/\m} (M/ \m M) \leq 1$, which proves the claim.
1128
\end{proof}
1129
1130
%We shall use the subscript~$(\m)$ to denote localization at~$\m$.
1131
%Thus $\Mm$ is free of rank one over~$\Tm$. The
1132
%composite $\psi: \Tpm \ra {\rm End}_{\Tm} (\Mm)
1133
%\stackrel{{\phi}^{-1}}{\ra} \Tm$ gives a section of the inclusion
1134
%$\Tm \hookrightarrow \Tpm$.
1135
%Let $x \in \Tpm$, and let $n$ be an integer such that $nx \in \Tm$.
1136
%Let $y = \psi(x) \in \Tm$. Then $nx = \psi ( \phi (nx) ) = \psi(nx)
1137
%= n \psi(x) = ny$. Since $\Tm$ is torsion-free, this means that
1138
%$x = y \in \Tm$. Thus $\Tm = \Tpm$, as was to be shown.
1139
1140
\comment{
1141
Thus $M \tensor \T_\m$ is free of rank one over~$\T_\m$. The
1142
composite $\EJ \tensor \T_\m
1143
\ra {\rm End}_{\T_\m} (M \tensor \T_\m)
1144
\stackrel{{\phi}^{-1}}{\ra} \T_\m$ gives a section of the inclusion
1145
$\T_\m \hookrightarrow \EJ \tensor \T_\m$. This shows
1146
that $\T_\m$ is saturated in $\EJ \tensor \T_\m$, i.e.,
1147
that $\T$ and $\T'$ agree locally at~$\m$.
1148
}
1149
\end{proof}
1150
1151
If $\m$ is a maximal ideal of the Hecke algebra~$\T$
1152
of residue characteristic~$\ell$, we say that
1153
$\m$ satisfies {\em multiplicity one for differentials} if
1154
$$
1155
\dim (\H^0(X_0(N)_{\F_\ell},\Omega)[\m]) \leq 1.
1156
$$
1157
By Lemma~\ref{lem:m1}, multiplicity one for
1158
$\H^0(X_0(N)_{\Fell}, \Omega)[\m]$ implies that $\T$ and
1159
$\T'$ agree at~$\m$.
1160
1161
There is quite a bit of literature on the question of multiplicity~$1$
1162
for $\H^0(X_0(N)_{\Fell},\Omega)[\m]$.
1163
The easiest case is that~$\ell$ is prime to the level $N$:
1164
1165
\begin{lem}[Mazur]\label{lem:m_ell}
1166
If $\m$ is a maximal ideal of $\T$ of residue characteristic~$\ell$
1167
such that $\ell \nmid N$, then
1168
$$\dim_{\T/\m} \H^0(X_0(N)_{\Fell},\Omega)[\m] \leq 1.$$
1169
%If $\ell \nmid pM$, then~$\ell\nmid \#(\T'/\T)$.
1170
\end{lem}
1171
1172
\begin{proof}
1173
Mazur deduces this lemma from injectivity of the $q$-expansion map.
1174
The reader may find the following alternative approach to part of
1175
the argument easier to follow than the one on p.~95 of
1176
\cite{mazur:eisenstein}. We have an $\Fell$-vector space that
1177
embeds in $\Fell[[q]]$, for example a space~$V$ of differentials
1178
that is killed by a maximal ideal $\m$. This space is a
1179
$\T/\m$-vector space, and we want to see that its dimension over
1180
$\T/\m$ is at most~$1$. Mazur invokes tensor products and
1181
eigenvectors; alternatively, we note that~$V$ embeds in
1182
$\Hom_{\Fell}(\T/\m,\Fell)$ via the standard duality that
1183
sends~$v\in V$ to the linear form whose value on a Hecke
1184
operator~$T$ is the $q$th coefficient of $v{|T}$. The group
1185
$\Hom_{\Fell}(\T/\m,\Fell)$ has the same size as $\T/\m$, which
1186
completes the argument because $\Hom_{\Fell}(\T/\m,\Fell)$ has
1187
dimension $1$ as a $\T/\m$-vector space.
1188
\end{proof}
1189
% proves that
1190
%$$\dim_{\T/\m} \H^0(X_0(pM)_{\Fell},\Omega)[\m] \leq 1$$ for
1191
%all $\m\mid \ell$. Now apply Lemma~\ref{lem:m1}
1192
%\edit{Is there a problem if $\ell=2$? How do Lloyd Kilford's examples
1193
%fit into this, where I guess $N=1$ and $\ell=2$ and multiplicity
1194
%one in $J_0(p)$ fails. Is it still OK in Mazur's
1195
%differentials? -WAS}
1196
1197
In the context of Mazur's paper, where the level~$N$ is prime, we see
1198
from Lemma~\ref{lem:m_ell} that $\T$ and $\T'$ agree away from~$N$.
1199
Locally at $N$, Mazur proved that $\T=\T'$ by an analogue of the
1200
arguments that he used away from $N$; see Chapter II of
1201
\cite{mazur:eisenstein} (and especially Prop.~9.4 and 9.5 of that
1202
chapter) as well as \cite{mazur-ribet}, where these arguments are
1203
taken up in a context where the level is no longer necessarily prime
1204
(and where one works locally at a prime whose square does not divide
1205
the level).
1206
%At~$N$, we can still use the $q$-expansion principle
1207
%because of the arguments in \cite[Ch.II~\S4]{mazur:eisenstein}.
1208
Thus in the prime level case, $\T=\T'$, as we asserted above.
1209
1210
1211
Now\label{NPnotation}
1212
let $p$ be a prime such that $p \parallel N$, and let $M = N/p$.
1213
The question of multiplicity $1$ at $p$ for $\H^0(X_0(pM)_{\Fp},
1214
\Omega)[\m]$ is discussed in \cite{mazur-ribet}, where the authors
1215
establish multiplicity~$1$ for maximal ideals $\m\mid p$ for which the
1216
associated mod~$p$ Galois representation is irreducible and {\em not}
1217
$p$-old. (A representation of level $pM$ is $p$-old if it arises from
1218
$S_2(\Gamma_0(M))$.)
1219
1220
If~$\m$ is a maximal ideal of~$\T$ of residue characteristic~$\ell$,
1221
then we say that~$\m$ is ordinary
1222
if $T_\ell \not\in \m$ (note that $T_\ell$ is often denoted $U_\ell$
1223
if $\ell \mid N$). For our purposes, the following lemma is convenient:
1224
1225
\begin{lem}[Wiles]\label{lem:wiles}
1226
If $\m$ is an ordinary maximal ideal of $\T$ of characteristic~$p$, then
1227
$$
1228
\dim_{\T/\m} \H^0(X_0(pM)_{\Fp},\Omega)[\m] \leq 1.
1229
$$
1230
%and $\ord_{\ell}(pM)=1$, then $\m$ is not in the support of $\T'/\T$.
1231
\end{lem}
1232
1233
This is essentially Lemma~2.2 in~\cite[pg.~485]{wiles};
1234
\comment{, which
1235
proves, under a suitable hypothesis, that $\H^0(X_0(pM)_{\F_p},\Omega)[\m]$
1236
is $1$-dimensional if $\m$ is a maximal ideal of~$\T$ that divides~$p$.
1237
The ``suitable hypothesis'' is that $\m$ is ordinary, in the sense that
1238
$T_p \not\in\m$. (Note that $T_p$ is often denoted $U_p$ in this context.)
1239
It follows from Wiles's lemma that $\T'=\T$ locally at~$\m$ whenever
1240
$\m$ is an ordinary prime whose residue characteristic exactly
1241
divides the level (which is $pM$ here).
1242
}
1243
we make a few comments about how it applies on our situation:
1244
\begin{enumerate}
1245
\item
1246
Wiles considers $X_1(M,p)$ instead of $X_0(pM)$, which means that he is
1247
using $\Gamma_1(M)$-structure instead of $\Gamma_0(M)$-structure.
1248
This surely has no relevance to the issue at hand.
1249
1250
\item Wiles assumes (on page 480) that $p$ is an odd prime, but again
1251
this assumption is not relevant to our question.
1252
1253
\item
1254
The condition that $\m$ is ordinary does not appear
1255
explicitly in the statement of Lemma~2.2 in~\cite{wiles};
1256
instead it is a reigning
1257
assumption in the context of his discussion.
1258
1259
\item We see by example that Wiles's ``ordinary'' assumption is less
1260
stringent than the assumption in \cite{mazur-ribet}; note that
1261
\cite{mazur-ribet} rule out cases where $\m$ is both old and new at
1262
$p$, whereas Wiles is happy to include such cases. (On the other
1263
hand, Wiles's assumption is certainly nonempty, since it rules out
1264
maximal ideals $\m$ that arise from non-ordinary (old) forms of
1265
level~$M$. Here is an example with $p=2$ and $M=11$, so $N=22$:
1266
There is a unique newform $f=\sum a_n q^n$ of level~$11$, and
1267
$\T=\Z[T_2] \subset \End(J_0(22))$, where $T_2^2-a_2 T_2 + 2 =0$.
1268
Since $a_2=-2$, we have $\T\isom \Z[\sqrt{-1}]$. We can choose the
1269
square root of $-1$ to be $T_2+1$. Then $T_2$ is a generator of the
1270
unique maximal ideal $\m$ of $\T$ with residue characteristic~$2$,
1271
and this maximal ideal is not ordinary.)
1272
\end{enumerate}
1273
%\end{proof}
1274
1275
We now summarize the conclusions we can make from the lemmas so far:
1276
1277
\begin{prop} \label{prop:TT'}
1278
The modules~$\T$
1279
and $\T'$ agree locally at each maximal ideal~$\m$ that is either prime
1280
to~$N$ or that satisfies the following supplemental hypothesis: the
1281
residue characteristic of~$\m$ divides~$N$ only to the first power
1282
and $\m$ is ordinary.
1283
\end{prop}
1284
\begin{proof}
1285
This follows easily from Lemmas~\ref{lem:m1}, \ref{lem:m_ell},
1286
and~\ref{lem:wiles}.
1287
\end{proof}
1288
\comment{
1289
Wiles's lemma and the standard $q$-expansion argument
1290
(Lemma~\ref{lem:m_ell} and Lemma~\ref{lem:wiles}) imply that~$\T$
1291
and~$\T'$ agree locally at each rational prime that is prime to the
1292
level $pM$, and also at each maximal ideal~$\m$ dividing~$p$ that is
1293
ordinary, in the sense that $T_p \not\in \m$. A more palatable
1294
description of the situation involves considering the Hecke
1295
algebra~$\T$ and its saturation~$\T'$ at some level $N\geq 1$. Then
1296
$\T=\T'$ locally at each maximal ideal $\m$ that is either prime
1297
to~$N$ or that satisfies the following supplemental hypothesis: the
1298
residue characteristic of~$\m$ divides~$N$ only to the first power
1299
and~$\m$ is ordinary.
1300
}
1301
1302
In Mazur's original context, where the level~$N$ is
1303
prime, we have $T_N^2=1$ because there are no forms of
1304
level~$1$. Accordingly, each~$\m$ dividing~$N$ is ordinary, and we
1305
recover Mazur's equality $\T=\T'$ in this special case.
1306
1307
\subsubsection{Degrees and Congruences} \label{sec:degrees}
1308
1309
%Let $e\in \T\tensor\Q$ be an idempotent, and let $A\subset J_0(pM)$
1310
%be the abelian variety image of $e$, i.e., the image of the homomorphism
1311
%$ne\in \T$, where the integer $n\geq 1$ is a multiple of the denominator of $e$.
1312
%Let~$B$ be the image of the complementary idempotent $1-e$.
1313
%Then $J_0(pM)=A+B$, and $A\cap B$ is a finite group whose exponent
1314
%divides the denominator of $e$.
1315
%\edit{We can just say that $e$ is as in the previous section.
1316
%Note that $A$ was~$\Adual$ in the previous section. --Amod}
1317
1318
Let $e \in \T\tensor\Q$ be as in Section~\ref{sec:firstpart},
1319
and let $p,N,M$ be as before Lemma~\ref{lem:wiles}.
1320
The image of $e$ in $J_0(pM)$ is the $\T$-stable abelian subvariety
1321
denoted $\Adual$ in Section~\ref{sec:firstpart}, but since we shall
1322
now exclusively work with this subvariety rather than the
1323
corresponding optimal quotient of $J_0(pM)$ (which was denoted $A$
1324
earlier), we will now write $A$ to denote the image of $e$ (without
1325
risk of confusion). We also write $B$ to denote the unique
1326
$\T$-stable abelian subvariety of $J_0(pM)$ complementary to~$A$.
1327
1328
For $t \in \T$, let $t_A$ be the restriction of~$t$ to $A$, and
1329
let~$t_B$ be the image of~$t$ in $\End(B)$. Let $\T_A$ be the
1330
subgroup of $\End(A)$ consisting of the various $t_A$, and define
1331
$\T_B$ similarly. As before, we obtain an injection
1332
$
1333
j : \T \hra \T_A \times \T_B
1334
$
1335
with finite cokernel. Because~$j$ is an injection, we
1336
refer to the maps $\pi_A:\T\to \T_A$ and $\pi_B : \T \to \T_B$,
1337
given by $t \mapsto t_A$ and $t\mapsto t_B$, respectively,
1338
as ``projections''.
1339
1340
\begin{defi}
1341
The {\em congruence ideal} associated with the projector~$e$ is
1342
$I=\pi_A(\ker(\pi_B)) \subset \T_A.$
1343
\end{defi}
1344
1345
Viewing $\T_A$ as $\T_A\times \{0\}$, we may view $\T_A$ as a subgroup
1346
of $\T\tensor\Q \isom (\T_A\times \T_B)\tensor\Q$. Also, we may view
1347
$\T$ as embedded in $\T_A\times \T_B$, via the map~$j$.
1348
\begin{lem}\label{lem:i_int}
1349
We have $I=\T_A\cap \T$.
1350
\end{lem}
1351
1352
A larger ideal of $\T_A$ is
1353
$
1354
J = \Ann_{\T_A}(A \cap B);
1355
$
1356
it consists of restrictions to $A$ of Hecke operators that
1357
vanish on $A\cap B$.
1358
1359
\begin{lem}
1360
We have $I\subset J$.
1361
\end{lem}
1362
\begin{proof}
1363
The image in $\T_A$ of an operator that vanishes on $B$ also
1364
vanishes on $A\cap B$.
1365
\end{proof}
1366
1367
\begin{lem}\label{lem:j_int}
1368
We have
1369
$J = \T_A \cap \End(J_0(pM)) = \T_A \cap \T'.$
1370
\end{lem}
1371
\begin{proof}
1372
This is elementary; it is an analogue of Lemma~\ref{lem:i_int}.
1373
\end{proof}
1374
1375
\begin{prop}\label{prop:ji_inc}
1376
There is a natural inclusion
1377
$
1378
J/I \hra \T'/\T
1379
$
1380
of $\T$-modules.
1381
\end{prop}
1382
\begin{proof}
1383
Consider the map $\T\to \T\tensor\Q$ given by $t\mapsto te$.
1384
This homomorphism factors through $\T_A$ and yields an injection
1385
$\iota_A : \T_A \hra \T\tensor\Q$. Symmetrically, we also
1386
obtain $\iota_B : \T_B \hra \T\tensor\Q$. The
1387
map
1388
$(t_A, t_B) \mapsto \iota_A(t_A) + \iota_B(t_B)$
1389
is an injection
1390
$\T_A\times \T_B \hra \T\tensor\Q$.
1391
The composite of this map with the inclusion $j:\T\hra \T_A\times \T_B$
1392
defined above is the natural map $\T\hra \T\tensor\Q$. We thus have
1393
a sequence of inclusions
1394
$$
1395
\T \hra \T_A \times \T_B \hra \T\tensor \Q
1396
\subset \End(J_0(pM))\tensor\Q.
1397
$$
1398
By Lemma~\ref{lem:i_int} and Lemma~\ref{lem:j_int},
1399
we have $I=\T_A\cap \T$ and $J=\T_A\cap \T'$.
1400
Thus $I=J\cap \T$, where the intersection is taken
1401
inside $\T'$. Thus
1402
$$
1403
J/I = J/(J\cap \T) \isom (J+\T)/\T \hra \T'/\T.
1404
$$
1405
\end{proof}
1406
1407
\begin{cor}\label{cor:ji_inc}
1408
If $\m$ is a maximal ideal not in $\Supp_{\T}(\T'/\T)$,
1409
then $\m$ is not in the support of $J/I$, i.e.,
1410
if $\T$ and $\T'$ agree locally at $\m$, then
1411
$I$ and $J$ also agree locally at $\m$.
1412
\end{cor}
1413
1414
Note that the Hecke algebra $\T$ acts on $J/I$ through
1415
its quotient $\T_A$,
1416
since the action of~$\T$ on~$I$ and on~$J$ factors through
1417
this quotient.
1418
1419
Now we specialize to the case where $A$ is ordinary at $p$,
1420
in the sense that the image of $T_p$ in $\T_A$, which we
1421
denote $T_{p,A}$, is invertible modulo every maximal ideal
1422
of $\T_A$ that divides~$p$. (This case occurs when~$A$ is
1423
a subvariety of the $p$-new subvariety of $J_0(pM)$, since
1424
the square of $T_{p,A}$ is the identity.)
1425
1426
1427
If $\m\mid p$
1428
is a maximal ideal of $\T$ that arises by pullback from
1429
a maximal ideal of $\T_A$, then~$\m$ is ordinary in the
1430
sense used above. When $A$ is ordinary at~$p$, it follows
1431
from Proposition~\ref{prop:TT'} and Corollary~\ref{cor:ji_inc}
1432
that $I=J$ locally at~$p$. The reason is simple: regarding~$I$
1433
and~$J$ as $\T_A$-modules, we realize that we need to test
1434
that $I=J$ at maximal ideals of $\T_A$ that divide~$p$.
1435
These ideals correspond to maximal ideals $\m\mid p$
1436
of $\T$ that are automatically ordinary, so we have $I=J$
1437
locally at $\m$ because of Proposition~\ref{prop:TT'}.
1438
By Proposition~\ref{prop:TT'},
1439
we have $\T=\T'$ locally at primes away from the
1440
level $pM$. Thus we conclude that $I=J$
1441
locally at all primes $\ell\nmid pM$ and also at~$p$,
1442
a prime that divides the level $pM$ exactly once.
1443
1444
Suppose, finally, that $A$ is the abelian variety associated to a
1445
newform~$f$ of level~$pM$.
1446
%We then have $\T_A=\Z$.
1447
The ideal $I\subset \T_A$ measures congruences between~$f$ and the space of forms
1448
in $S_2(\Gamma_0(pM))$ that are orthogonal to the space generated
1449
by~$f$. Also, $A\cap B$ is the kernel in~$A$ of the map
1450
``multiplication by the modular element~$e$''.
1451
In this case, the inclusion $I\subset J$ corresponds to the divisibility
1452
$
1453
\tilde{n}_A \mid \tilde{r}_A,
1454
$
1455
and we have equality at primes at which $I=J$ locally.
1456
We conclude that the congruence exponent and the modular exponent
1457
agree both at~$p$ and at primes not dividing $pM$, which completes our
1458
proof of Theorem~\ref{thm:ribet_gen}(b).
1459
1460
\begin{rmk}
1461
The ring
1462
$$
1463
R = \End(J_0(pM)) \cap (\T_A \times \T_B)
1464
$$
1465
is often of interest, where the intersection is taken
1466
in $\End(J_0(pM))\tensor \Q$. We proved above that there
1467
is a natural inclusion $J/I \hra \T'/\T$. This
1468
inclusion yields an isomorphism
1469
$
1470
J/I \xra{\sim} R/\T.
1471
$
1472
Indeed, if $(t_A, u_B)$ is an endomorphism of $J_0(pM)$,
1473
where $t,u \in \T$, then
1474
$(t_A, u_B) - u = (t_A, 0)$ is an element of~$J$.
1475
The ideals~$I$ and~$J$ are equal to the extent that the
1476
rings~$\T$ and $R$ coincide. Even when $\T'$ is bigger than~$\T$,
1477
its subring $R$ may be not far from~$\T$.
1478
\end{rmk}
1479
1480
\section{Failure of Multiplicity One}\label{sec:mult1}
1481
In this section, we discuss examples of failure of multiplicity one
1482
(in two different but related senses). The notion of multiplicity one,
1483
originally due to Mazur~\cite{mazur:eisenstein}, has played an
1484
important role in several places (e.g., in Wiles's proof of Fermat's
1485
last theorem~\cite{wiles}). This notion is closely related to
1486
Gorensteinness of certain Hecke algebras (e.g., see~\cite{tilouine:hecke}).
1487
Kilford~\cite{kilford} found examples of failure of Gorensteinness
1488
(and multiplicity one) at the prime~$2$ for certain prime levels.
1489
Motivated by the arguments in Section~\ref{sec:main}, in this section
1490
we give examples of failure of multiplicity one for primes (including
1491
odd primes) whose square divides the level.
1492
1493
\subsection{Multiplicity One for
1494
Differentials}\label{sec:dataind}
1495
In connection with the arguments in Section~\ref{sec:main}, especially
1496
Lemmas~\ref{lem:m1} and \ref{lem:wiles}, it is of interest
1497
to compute the index $[\T':\T]$ for various $N$.
1498
We can compute this index in Magma, e.g., the following
1499
commands compute the index for $N=54$:
1500
``{\tt J := JZero(54); T := HeckeAlgebra(J); Index(Saturation(T), T);}''
1501
We obtain Table~\ref{table:index}, where the first column
1502
contains $N$ and the second column contains $[\T':\T]$:
1503
\begin{table}\caption{The Index $[\T':\T]$\label{table:index}}
1504
\begin{center}
1505
\begin{tabular}{|l|c|}\hline
1506
11 & 1 \\\hline
1507
12 & 1 \\\hline
1508
13 & 1 \\\hline
1509
14 & 1 \\\hline
1510
15 & 1 \\\hline
1511
16 & 1 \\\hline
1512
17 & 1 \\\hline
1513
18 & 1 \\\hline
1514
19 & 1 \\\hline
1515
20 & 1 \\\hline
1516
21 & 1 \\\hline
1517
22 & 1 \\\hline
1518
23 & 1 \\\hline
1519
24 & 1 \\\hline
1520
25 & 1 \\\hline
1521
26 & 1 \\\hline
1522
27 & 1 \\\hline
1523
28 & 1 \\\hline
1524
29 & 1 \\\hline
1525
30 & 1 \\\hline
1526
31 & 1 \\\hline
1527
32 & 1 \\\hline
1528
33 & 1 \\\hline
1529
34 & 1 \\\hline
1530
35 & 1 \\\hline
1531
36 & 1 \\\hline
1532
37 & 1 \\\hline
1533
38 & 1 \\\hline
1534
39 & 1 \\\hline
1535
40 & 1 \\\hline
1536
41 & 1 \\\hline
1537
42 & 1 \\\hline
1538
43 & 1 \\\hline
1539
44 & 2 \\\hline
1540
45 & 1 \\\hline
1541
46 & 2 \\\hline
1542
47 & 1 \\\hline
1543
48 & 1 \\\hline
1544
49 & 1 \\\hline
1545
50 & 1 \\\hline
1546
\end{tabular}
1547
\,\,\,\,\,\,
1548
\begin{tabular}{|l|c|}\hline
1549
51 & 1 \\\hline
1550
52 & 1 \\\hline
1551
53 & 1 \\\hline
1552
54 & 3 \\\hline
1553
55 & 1 \\\hline
1554
56 & 2 \\\hline
1555
57 & 1 \\\hline
1556
58 & 1 \\\hline
1557
59 & 1 \\\hline
1558
60 & 2 \\\hline
1559
61 & 1 \\\hline
1560
62 & 2 \\\hline
1561
63 & 1 \\\hline
1562
64 & 2 \\\hline
1563
65 & 1 \\\hline
1564
66 & 1 \\\hline
1565
67 & 1 \\\hline
1566
68 & 2 \\\hline
1567
69 & 1 \\\hline
1568
70 & 1 \\\hline
1569
71 & 1 \\\hline
1570
72 & 2 \\\hline
1571
73 & 1 \\\hline
1572
74 & 1 \\\hline
1573
75 & 1 \\\hline
1574
76 & 2 \\\hline
1575
77 & 1 \\\hline
1576
78 & 2 \\\hline
1577
79 & 1 \\\hline
1578
80 & 4 \\\hline
1579
81 & 1 \\\hline
1580
82 & 1 \\\hline
1581
83 & 1 \\\hline
1582
84 & 2 \\\hline
1583
85 & 1 \\\hline
1584
86 & 1 \\\hline
1585
87 & 1 \\\hline
1586
88 & 8 \\\hline
1587
89 & 1 \\\hline
1588
90 & 1 \\\hline
1589
\end{tabular}
1590
\,\,\,\,\,\,
1591
\begin{tabular}{|l|c|}\hline
1592
91 & 1 \\\hline
1593
92 & 16 \\\hline
1594
93 & 1 \\\hline
1595
94 & 4 \\\hline
1596
95 & 1 \\\hline
1597
96 & 8 \\\hline
1598
97 & 1 \\\hline
1599
98 & 1 \\\hline
1600
99 & 9 \\\hline
1601
100 & 1 \\\hline
1602
101 & 1 \\\hline
1603
102 & 1 \\\hline
1604
103 & 1 \\\hline
1605
104 & 4 \\\hline
1606
105 & 1 \\\hline
1607
106 & 1 \\\hline
1608
107 & 1 \\\hline
1609
108 & 54 \\\hline
1610
109 & 1 \\\hline
1611
110 & 2 \\\hline
1612
111 & 1 \\\hline
1613
112 & 8 \\\hline
1614
113 & 1 \\\hline
1615
114 & 1 \\\hline
1616
115 & 1 \\\hline
1617
116 & 4 \\\hline
1618
117 & 1 \\\hline
1619
118 & 2 \\\hline
1620
119 & 1 \\\hline
1621
120 & 32 \\\hline
1622
121 & 1 \\\hline
1623
122 & 1 \\\hline
1624
123 & 1 \\\hline
1625
124 & 16 \\\hline
1626
125 & 25 \\\hline
1627
126 & 18 \\\hline
1628
127 & 1 \\\hline
1629
128 & 64 \\\hline
1630
129 & 1 \\\hline
1631
130 & 1 \\\hline
1632
\end{tabular}
1633
\,\,\,\,\,\,
1634
\begin{tabular}{|l|c|}\hline
1635
131 & 1 \\\hline
1636
132 & 8 \\\hline
1637
133 & 1 \\\hline
1638
134 & 1 \\\hline
1639
135 & 27 \\\hline
1640
136 & 16 \\\hline
1641
137 & 1 \\\hline
1642
138 & 4 \\\hline
1643
139 & 1 \\\hline
1644
140 & 8 \\\hline
1645
141 & 1 \\\hline
1646
142 & 8 \\\hline
1647
143 & 1 \\\hline
1648
144 & 32 \\\hline
1649
145 & 1 \\\hline
1650
146 & 1 \\\hline
1651
147 & 7 \\\hline
1652
148 & 4 \\\hline
1653
149 & 1 \\\hline
1654
150 & 5 \\\hline
1655
151 & 1 \\\hline
1656
152 & 32 \\\hline
1657
153 & 9 \\\hline
1658
154 & 1 \\\hline
1659
155 & 1 \\\hline
1660
156 & 32 \\\hline
1661
157 & 1 \\\hline
1662
158 & 4 \\\hline
1663
159 & 1 \\\hline
1664
160 & 256 \\\hline
1665
161 & 1 \\\hline
1666
162 & 81 \\\hline
1667
163 & 1 \\\hline
1668
164 & 8 \\\hline
1669
165 & 1 \\\hline
1670
166 & 2 \\\hline
1671
167 & 1 \\\hline
1672
168 & 128 \\\hline
1673
169 & 13 \\\hline
1674
170 & 1 \\\hline
1675
\end{tabular}
1676
\,\,\,\,\,\,
1677
\begin{tabular}{|l|c|}\hline
1678
171 & 9 \\\hline
1679
172 & 8 \\\hline
1680
173 & 1 \\\hline
1681
174 & 4 \\\hline
1682
175 & 5 \\\hline
1683
176 & 512 \\\hline
1684
177 & 1 \\\hline
1685
178 & 1 \\\hline
1686
179 & 1 \\\hline
1687
180 & 72 \\\hline
1688
181 & 1 \\\hline
1689
182 & 1 \\\hline
1690
183 & 1 \\\hline
1691
184 & 1024 \\\hline
1692
185 & 1 \\\hline
1693
186 & 4 \\\hline
1694
187 & 1 \\\hline
1695
188 & 256 \\\hline
1696
189 & 243 \\\hline
1697
190 & 8 \\\hline
1698
191 & 1 \\\hline
1699
192 & 4096 \\\hline
1700
193 & 1 \\\hline
1701
194 & 1 \\\hline
1702
195 & 1 \\\hline
1703
196 & 14 \\\hline
1704
197 & 1 \\\hline
1705
198 & 81 \\\hline
1706
199 & 1 \\\hline
1707
200 & 80 \\\hline
1708
201 & 1 \\\hline
1709
202 & 1 \\\hline
1710
203 & 1 \\\hline
1711
204 & 32 \\\hline
1712
205 & 1 \\\hline
1713
206 & 4 \\\hline
1714
207 & 81 \\\hline
1715
208 & 256 \\\hline
1716
209 & 1 \\\hline
1717
210 & 2 \\\hline
1718
\end{tabular}
1719
\end{center}
1720
\end{table}
1721
1722
Let $\m$ be a maximal ideal of the Hecke algebra
1723
$\T\subset\End(J_0(N))$ of residue characteristic~$p$. Recall
1724
that we say that
1725
$\m$ satisfies {\em multiplicity one for differentials} if $\dim
1726
(\H^0(X_0(N)_{\Fp},\Omega)[\m]) \leq 1$.
1727
1728
In each case in which $[\T':\T]\neq 1$, Lemma~\ref{lem:m1} implies
1729
that there is some maximal ideal $\m$ of $\T$ such that
1730
$\dim(\H^0(X_0(N)_{\Fp},\Omega)[\m])>1$, which is an example
1731
of failure of multiplicity one for differentials.
1732
1733
1734
In Table~\ref{table:index}, whenever $p\mid [\T':\T]$, then $p^2\mid
1735
2N$. This is a consequence of Proposition~\ref{prop:TT'}, which
1736
moreover asserts that when $2$ exactly divides $N$ and $2\mid[\T':\T]$
1737
then there is a non-ordinary (old) maximal ideal of characteristic $2$
1738
in the support of $\T'/\T$.
1739
1740
%The first case when $2\mid\mid N$ and $2\mid
1741
%[\T':\T]$ is $N=46$, where we find (via a Magma calculation) that
1742
%$G=\T'/\T \isom \Z/2\Z$, and the Hecke operator~$T_2$ acts as~$0$ on
1743
%$G$, so the annihilator of $G$ in $\T$ is not ordinary, which does not
1744
%ontradict Proposition~\ref{prop:TT'}.
1745
1746
Moreover, notice that
1747
Theorem~\ref{thm:ribet_gen}(b) (whose proof is in
1748
Section~\ref{sec:secondpart})
1749
follows formally from two
1750
key facts: that $A_f$ is new and that multiplicity one for differentials
1751
holds for ordinary maximal ideals with residue characteristic
1752
$p\mid\mid N$ and for all maximal ideals with residue
1753
characteristic $p\nmid N$. The conclusion of
1754
Theorem~\ref{thm:ribet_gen}(b) does not hold for the counterexamples
1755
in Section~\ref{sec:elliptic} (e.g., for~54B1), which are
1756
all new elliptic curves, so multiplicity one for
1757
differentials does not hold for certain
1758
maximal ideals that arise from the new quotient of the Hecke algebra.
1759
Note that in all examples we have $p\mid(r/m)$ with $p^2\mid N$,
1760
which raises the question: are there non-ordinary counterexamples
1761
with $p\mid\mid N$?
1762
1763
%\edit{I think I have been to vague. Basically, I wanted to say
1764
%that one reason why the index $[\T':\T]$ could be nontrivial
1765
%(and hence lead to failure of multiplicity one for differentials)
1766
%is due to old-ness. But even in the new part, one may have failure
1767
%of mult one for diffs, due to the level not being square-free. --Amod}
1768
1769
% [email protected]:~/comps/ind_table$ magma
1770
% Magma V2.11-10 Sun Aug 28 2005 18:34:49 on modular [Seed = 1293693469]
1771
% Type ? for help. Type <Ctrl>-D to quit.
1772
% > J := JZero(46);
1773
% > J;
1774
% Modular abelian variety JZero(46) of dimension 5 and level 2*23 over Q
1775
% > T := HeckeAlgebra(J);
1776
% > S := Saturation(T);
1777
% > m := S/T;
1778
% > m;
1779
% Abelian Group isomorphic to Z/2
1780
% Defined on 1 generator
1781
% Relations:
1782
% 2*m.1 = 0
1783
% >
1784
% > S;
1785
% Sat(HeckeAlg(JZero(46))): Group of homomorphisms from JZero(46) to JZero(46)
1786
% > Basis(S);
1787
% [
1788
% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix),
1789
% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix),
1790
% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix),
1791
% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix),
1792
% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix)
1793
% ]
1794
% > S.1;
1795
% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix)
1796
% > S.1 in T;
1797
% true
1798
% > S.2 in T;
1799
% false
1800
% > t2 := HeckeOperator(J,2);
1801
% > t2 in T;
1802
% true
1803
% > t2*S.2;
1804
% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix)
1805
% > t2*S.2 in T;
1806
% true
1807
1808
\subsection{Multiplicity One for Jacobians}
1809
1810
We say that a maximal ideal~$\m$ of~$\T$ satisfies {\it multiplicity one}
1811
if $J_0(N)[\m]$ is of dimension two over~$\T/\m$. We sometimes use the
1812
phrase ``multiplicity one for~$J_0(N)$'' in order to distinguish this notion
1813
from the notion of multiplicity one for differentials.
1814
%\edit{I added these lines for clarity. --Amod}
1815
1816
\begin{prop}\label{prop:mult1J}
1817
Suppose $E$ is an optimal elliptic curve over $\Q$ of conductor~$N$
1818
and~$p$ is a prime such that $p \mid r_E$ but $p\nmid m_E$. Let
1819
$\m$ be the annihilator in $\T$ of $E[p]$. Then multiplicity one
1820
fails for $\m$, i.e., $\dim_{\T/\m} J_0(N)[\m] > 2$.
1821
%\edit{Earlier the conclusion said $\dim_{\T/\m} J_0(N)[\m] > 1$.
1822
%--Amod}
1823
\end{prop}
1824
\begin{proof}
1825
Using the principal polarization $E \isom E^{\vee}$ we view~$E$ as
1826
an abelian subvariety of $J=J_0(N)$ and consider the complementary
1827
$\T$-stable abelian subvariety $A$ of $E$ (thus $A$ is the kernel of
1828
the modular parametrization map $J\to E$). In this setup, $J = E +
1829
A$, and the intersection of $E$ and $A$ is $E[m_E]$. Here we use
1830
that the composite map
1831
$
1832
E \simeq E^{\vee} \to J^{\vee} \to J \to E
1833
$
1834
is a polarization, and hence is multiplication by a positive integer
1835
$m_E$. Because $p\nmid m_E$, we have $E[p]\cap A = 0$. On the
1836
other hand, let $\m$ be the annihilator of $E[p]$ inside $\T$. Then
1837
$J[\m]$ contains $E[p]$ and also $A[\m]$, and because $p$ is a
1838
congruence prime, the submodule $A[\m]\subset J[\m]$ is nonzero.
1839
Thus the sum $E[p] + A[\m]$ is a direct sum and is larger than
1840
$E[p]$, which is of dimension $2$ over $\T/\m = \Z/p\Z$. Hence the
1841
dimension of $J[\m]$ over $\T/\m$ is bigger than $2$, as claimed.
1842
%\edit{I changed the last two lines. --Amod}
1843
\end{proof}
1844
1845
Proposition~\ref{prop:mult1J} implies that any example in which
1846
simultaneously $p\nmid m_E$ and $\ord_p(r_E)\neq \ord_p(m_E)$ produces
1847
an example in which multiplicity one for $J_0(N)$ fails. For example,
1848
for the curve 54B1 and $p=3$, we have $\ord_3(r_E)=1$ but
1849
$\ord_3(m_E)=0$, so multiplicity one at $3$ fails for $J_0(54)$.
1850
%Also, for 242B1, we have $r_E = 11\cdot 2^4$ and $m_E = 2^4$, so
1851
%multiplicity one for $J_0(242)$ fails at $11$. For $N=242$ we also
1852
%have $[\T':\T]=121$, so multiplicity one at $11$ also fails for
1853
%differentials
1854
%\edit{Doesn't this work for 54B1 as well? Why give another
1855
%example? --Amod}
1856
%(see Section~\ref{sec:dataind} above).
1857
1858
%\edit{I commented out the old section, which had mistakes. --Amod}
1859
\comment{
1860
\subsection{Multiplicity one (old section)}
1861
\edit{william: I think this section should be deleted in light
1862
of the above two new sections.}
1863
%\edit{[This is still very rough, and will need to be cleaned up. --Amod]}
1864
1865
Let $\m$ be a maximal ideal of the Hecke algebra
1866
$\T\subset\End(J_0(N))$ of residue characteristic~$p$. We say that
1867
$\m$ satisfies {\em strong multiplicity one} if $\dim
1868
(\H^0(X_0(N)_{\Fp},\Omega)[\m]) = 1$.
1869
\edit{or should be it be equal to~$1$? Also ``strong'' is just
1870
something I came up with; may want to change the name. In fact, I
1871
may have got strong and weak mixed up, since Tilouine calls
1872
above ``weak multiplicity one''. --Amod}
1873
1874
The proof of Theorem~\ref{thm:ribet_gen}(b) follows formally from two
1875
key facts: that $A_f$ is new and that strong multiplicity one holds
1876
for ordinary maximal ideals if~$p^2 \nmid N$. The conclusion of
1877
Theorem~\ref{thm:ribet_gen}(b) does not hold for the counterexamples
1878
in Section~\ref{sec:elliptic} (at levels $54$,~$64$, etc.), which are
1879
all new elliptic curves, which shows that strong multiplicity one does
1880
not hold for certain ordinary\edit{William: I totally don't get this.
1881
When $p^2\mid N$ the ideals are {\em NOT} ordinary. We only got
1882
ordinary in the proof above because $p\mid\mid N$. For example, for
1883
the curve 54b, we have $a_3=0$, and the ideal is not ordinary.}
1884
maximal ideals for the corresponding
1885
levels (in all of them, $p^2 \mid N$). We record this observation:
1886
1887
\begin{prop}
1888
There are ordinary maximal ideals for which strong
1889
multiplicity one fails.\edit{William: But our examples
1890
are not ordinary!}
1891
\end{prop}
1892
1893
There is another notion of multiplicity one:
1894
suppose~$\rho_{\m}$, the representation attached to~$\m$,
1895
is absolutely irreducible.
1896
Then one
1897
says that $\m$ satisfies {\em weak multplicity one} if
1898
$J_0(N)[\m]$ is isomorphic to a single copy of~$\rho_{\m}$.
1899
By standard arguments, strong multiplicity one implies
1900
weak multiplicity one when $\overline{\rho}_{\m}$ is absolutely irreducible.
1901
\edit{William: I think this is wrong.
1902
In connection with Remark~\ref{rem:24}, we find (again via a Magma
1903
computation) that $\T'=\T$ when $N=431$. This was also already known via
1904
work of Mazur \cite{mazur:eisenstein}, and illustrates that
1905
multiplicity one for differentials need not imply multiplicity one for
1906
$J_0(N)$. So I'm confused.}
1907
%\edit{Will: I just realized that for this implication,
1908
%one also needs the representation~$\rho_{\m}$
1909
%to be absolutely irreducible -- it might be worth checking
1910
%if this holds in our couterexamples}
1911
which in turn implies that the Hecke algebra~$\T$ is Gorenstein.\edit{or
1912
maybe weak multiplicity one is equivalent to Gorensteinness, and both
1913
follow from strong multiplicity one; a good reference might be
1914
Tilouine's article in the FLT conference.} Wiles proves that strong
1915
multiplicity one holds for ordinary maximal ideals provided $p^2 \nmid
1916
N$, which he used to show the Gorensteinness of certain Hecke
1917
algebras. This Gorensteinness property was a key step in the proof of
1918
Fermat's last theorem. Our finding above shows that the hypothesis
1919
$p^2 \nmid N$ is essential, and thus gives a limit to how far the
1920
standard argument for proving Gorensteinness works.
1921
1922
Note that while in our examples, we know that strong multiplicity one
1923
fails, we do not know if weak multiplicity one or Gorensteinness
1924
fails. It would be interesting to do calculations for the latter, like
1925
were done by Kilford.
1926
}
1927
1928
%\bibliographystyle{amsalpha}
1929
%\bibliography{biblio}
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%\end{document}
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\providecommand{\bysame}{\leavevmode\hbox to3em{\hrulefill}\thinspace}
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\href{http://www.ams.org/mathscinet-getitem?mr=#1}{#2}
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