CoCalc Shared Fileswww / papers / ars-congruence / current.tex
Author: William A. Stein
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309\begin{document}
310%%%%% ------------- fill in your data below this line  -------------------
311%%%%%    The following lines \Title ... \EndAddress must ALL be present
312%%%%%    and in the given order.
313\Title
314%%%%%    Put here the title. Line breaks will be recognized.
315The Modular Degree, Congruence Primes and Multiplicity One
316\ShortTitle
317The Modular Degree and Congruences
318%%%%%    Running title for odd numbered pages, ONE line, please.
319%%%%%    If none is given, \Title will be used instead.
320\SubTitle
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322\Author
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324Amod Agashe\\
325Kenneth A. Ribet\\
326William A. Stein
327\ShortAuthor
328Agashe, Ribet, Stein
329%%%%%%   Running title for even numbered pages, ONE line, please.
330%%%%%%   If none is given, \Author will be used instead.
331\EndTitle
332\Abstract
333%%%%%    Put here the abstract of your manuscript.
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336
337The modular degree and congruence number are two fundamental
338invariants of an elliptic curve over the rational field.  Frey and
339M{\"u}ller have asked whether these invariants coincide.  Although
340this question has a negative answer, we prove a theorem about the
341relation between the two invariants: one divides the other, and the
342ratio is divisible only by primes whose squares divide the conductor
343of the elliptic curve.  We discuss the ratio even in the case where
344the square of a prime does divide the conductor, and we study
345analogues of the two invariants for modular abelian varieties of
346arbitrary dimension.
347
348
349
350\EndAbstract
351\MSC
352%%%%%    2000 Mathematics Subject Classification:
353\EndMSC
354\KEY
355%%%%%    Keywords and Phrases:
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357%%%%%    All 4 \Address lines below must be present. To center the last
358%%%%%    entry, no empty lines must be between the following \Address
361%%%%%    Address of first Author here
363Amod Agashe
366Kenneth A. Ribet
369William A. Stein
370Department of Mathematics
371Harvard University
372Cambridge, MA  02138
373{\tt was@math.harvard.edu}
375%%
376%%       Make sure the last tex command in your manuscript
377%%       before the first \end{document} is the command  \Addresses
378%%
379%%---------------------Here the prologue ends---------------------------------
380
381
382
383
384%%--------------------Here the manuscript starts------------------------------
385
386
387\section{Introduction}
388Let~$E$ be an elliptic curve over~$\Q$.  By
389\cite{breuil-conrad-diamond-taylor}, we may view~$E$ as an abelian
390variety quotient over $\Q$ of the modular Jacobian $J_0(N)$, where $N$
391is the conductor of~$E$.  After possibly replacing $E$ by an isogenous
392curve, we may assume that the kernel of the map $J_0(N)\to E$ is
393connected, i.e., that~$E$ is an {\em optimal quotient} of $J_0(N)$.
394
395Let $f_E = \sum a_n q^n \in S_2(\Gamma_0(N))$ be the newform attached
396to $E$.  The {\em congruence number}~$\re$ of~$E$ is the largest
397integer such that there is an element $g =\sum b_n q^n \in 398S_2(\Gamma_0(N))$ with integer Fourier coefficients $b_n$ that is
399orthogonal to~$f_E$ with respect to the Peterson innner product, and
400congruent to~$f_E$ modulo~$\re$ (i.e., $a_n \equiv b_n\pmod{\re}$ for
401all~$n$).    The {\em modular
402  degree}~$\me$ is the degree of the composite map $X_0(N)\to 403J_0(N)\to E$, where we map $X_0(N)$ to $J_0(N)$ by sending $P\in 404X_0(N)$ to $[P]-[\infty] \in J_0(N)$.
405
406Section~\ref{congintro} is about relations between~$\re$ and~$\me$.
407For example, $\me \mid \re$.  In \cite[Q.~4.4]{frey-muller}, Frey and
408M{\"u}ller~ asked whether $\re = \me$.  We give examples in which $\re 409\neq \me$, then conjecture that for any prime $p$, $\ord_p(\re/\me) 410\leq \frac{1}{2}\ord_p(N)$.  We prove this conjecture when
411$\ord_p(N)\leq 1$.
412
413In Section~\ref{sec:quotients}, we consider analogues of congruence
414primes and the modular degree for optimal quotients that are not
415necessarily elliptic curves; these are quotients of~$J_0(N)$ and
416$J_1(N)$ of any dimension associated to ideals of the relevant Hecke
417algebras.  In Section~\ref{sec:main} we prove the main theorem of this
418paper, and in Section~\ref{sec:mult1} we give some new examples of
419failure of multiplicity one motivated by the arguments in
420Section~\ref{sec:main}.
421
422% For an introduction and the motivation for studying
423% the objects in the title of the paper, the reader
425% and~\ref{sec:quotients}, skipping the proofs.
426
427\smallskip
428
429{\bf \noindent Acknowledgment.} The authors are grateful to A.~Abbes,
430R.~Coleman, B.~Conrad, J.~Cremona, H.~Lenstra, E.~de Shalit,
431B.~Edixhoven, L.~Merel, and R.~Taylor for several discussions and
433
434\section{Congruence Primes and the Modular Degree}
435\label{sec:elliptic}
436
437
438Let~$N$ be a positive integer and let $X_0(N)$ be the modular curve
439over~$\Q$ that classifies isomorphism classes of elliptic curves with
440a cyclic subgroup of order~$N$.  The Hecke algebra~$\T$ of level~$N$
441is the subring of the ring of endomorphisms of $J_0(N)=\Jac(X_0(N))$
442generated by the Hecke operators $T_n$ for all $n\geq 1$.  Let~$f$ be
443a newform of weight~$2$ for~$\Gamma_0(N)$ with integer Fourier
444coefficients, and let $I_f$ be kernel of the
445homomorphism $\T\to \Z[\ldots, a_n(f), \ldots]$ that sends $T_n$ to
446$a_n$.  Then the quotient $E = J_0(N)/I_f J_0(N)$ is an elliptic curve
447over~$\Q$.  We call~$E$ the {\em optimal quotient} associated to~$f$.
448Composing the embedding $X_0(N)\hra J_0(N)$ that sends $\infty$ to~$0$
449with the quotient map $J_0(N) \ra E$, we obtain a surjective morphism
450of curves $\phie: X_0(N) \ra E$.
451\begin{defi}
452The {\em modular degree} $\me$ of~$E$ is the degree of~$\phie$.
453\end{defi}
454
455\label{congintro}
456
457%The congruence number $\re$ and the modular degree $\me$
458%are of great interest.
459Congruence primes have been studied by Doi, Hida, Ribet,
460Mazur and others (see, e.g.,~\cite[\S1]{ribet:modp}),
461and played an important role in Wiles's work~\cite{wiles}
462on Fermat's last theorem.  Frey and Mai-Murty have
463observed that an
464appropriate asymptotic bound on the modular degree is equivalent to
465the $abc$-conjecture (see~\cite[p.544]{frey:ternary}
466and~\cite[p.180]{murty:congruence}).
467Thus, results that relate congruence primes and the modular degree
468are of great interest.
469
470\begin{thm}\label{thm:ribet_au}
471\label{ddivsr}
472Let $E$ be an elliptic curve over $\Q$ of conductor~$N$, with modular
473degree $\me$ and congruence number $\re$.
474Then $\me \mid \re$ and if $\ord_p(N)\leq 1$ then $\ord_p(\re) = \ord_p(\me)$.
475\end{thm}
476We will prove a generalization of Theorem~\ref{thm:ribet_au}
477in Section~\ref{sec:main} below.
478
479The divisibility $\me\mid \re$ was first discussed
480in~\cite[Th.~3]{zagier}, where it is attributed to the second author
481(Ribet); however in \cite{zagier} the divisibility was mistakenly
482written in the opposite direction. For some other expositions of the
483proof, see~\cite[Lem~3.2]{abbull} and~\cite{cojo-kani}.  We generalize
484this divisibility in Proposition~\ref{ndivsm}.  The second part of
485Theorem~\ref{thm:ribet_au}, i.e., that if $\ord_p(N) \leq 1$ then
486$\ord_p(\re) = \ord_p(\me)$, follows from the more general
487Theorem~\ref{thm:ribet_gen} below.
489%in more generality in in Section~\ref{sec:proof_ribet} below.
490Note that \cite[Prop.~3.3--3.4]{abbull} implies the weaker
491statement that if $p\nmid N$ then $\ord_p(\re)=\ord_p(\me)$,
492since \cite[Prop.~3.3]{abbull}
493implies $$\ord_p(\re) - \ord_p(\me) = \ord_p(\#\mathcal{C}) 494- \ord_p(\ce) - \ord_p(\#\mathcal{D}),$$ and by \cite[Prop.~3.4]{abbull}
495$\ord_p(\#\mathcal{C}) =0$.  (Here $\ce$ is the Manin
496constant of $E$, which is an integer by results of
497Edixhoven and Katz-Mazur; see e.g., \cite{ars} for more details.)
498
499Frey 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
501Magma \cite{magma}, we quickly found that the answer is no. The
502counterexamples at conductor $N\leq 144$ are given in Table~\ref{table:moddeg},
503where the curve
504is given using the notation of \cite{cremona:alg}:
505\begin{table}\caption{Differing Modular Degree and
506Congruence Number\label{table:moddeg}}
507\begin{center}
508\begin{tabular}{|l|l|l|}\hline
509Curve & $\me$ & $\re$\\\hline
51054B1 & 2 & 6\\\hline
51164A1 & 2 & 4 \\\hline
51272A1 & 4 & 8 \\\hline
51380A1 & 4 & 8 \\\hline
51488A1 & 8 & 16 \\\hline
51592B1 & 6 & 12\\\hline
51696A1 & 4 & 8 \\\hline
51796B1 & 4 & 8 \\\hline
518\end{tabular}
519\begin{tabular}{|l|l|l|}\hline
520Curve & $\me$ & $\re$\\\hline
52199A1 & 4 & 12\\\hline
522108A1 & 6 & 18\\\hline
523112A1 & 8 & 16\\\hline
524112B1 & 4 & 8\\\hline
525112C1 & 8 & 16\\\hline
526120A1 & 8 & 16\\\hline
527124A1 & 6 & 12\\\hline
528126A1 & 8 & 24\\\hline
529\end{tabular}
530\begin{tabular}{|l|l|l|}\hline
531Curve & $\me$ & $\re$\\\hline
532128A1 & 4 & 32\\\hline
533128B1 & 8 & 32\\\hline
534128C1 & 4 & 32\\\hline
535128D1 & 8 & 32\\\hline
536135A1 & 12 & 36\\\hline
537144A1 & 4 & 8 \\\hline
538144B1 & 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
546For example, the elliptic curve 54B1, given by the equation $y^2 + xy + 547y = x^3 - x^2 + x - 1$, has $\re=6$ and $\me=2$.  To see explicitly
548that $3 \mid \re$, observe that the newform corresponding to~$E$ is
549$f=q + q^2 + q^4 - 3q^5 - q^7 + \cdots$ and the newform corresponding
550to $X_0(27)$ if $g=q - 2q^4 - q^7 + \cdots$, so $g(q) + g(q^2)$
551appears to be congruent to~$f$ modulo~$3$.  To prove this congruence,
552we checked it for $18$ Fourier coefficients, where the
553sufficiency of precision to degree $18$
554was 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
559In our computations, there appears to be no absolute bound on the~$p$
560that occur.  For example, for the curve 242B1 of conductor $N=2\cdot 11^2$
561we have\footnote{The curve 242a1 in modern notation.''}
562$$563\me = 2^4 \neq \re = 2^4\cdot 11. 564$$
565We 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}
575We verified Conjecture~\ref{conj:rm} using Magma for every optimal
576elliptic curve quotient of $J_0(N)$, with $N\leq 539$.
577
578If $p\geq 5$ then $\ord_p(N)\leq 2$, so a special case
579of 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}
617For $N\geq 4$, let~$\Gamma$ be a fixed choice of either~$\Gamma_0(N)$
618or~$\Gamma_1(N)$, let~$X$ be the modular curve over~$\Q$ associated
619to~$\Gamma$, 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 $\T/ I$
622is torsion free.  Then $A = A_I = J/IJ$ is an optimal quotient of~$J$
623since $IJ$ 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
633In this section, we generalize the notions of the congruence number
634and the modular degree to quotients~$A=A_I$, and state a theorem
635relating the two numbers, which we prove in
636Sections~\ref{sec:firstpart}--\ref{sec:secondpart}.
637
638Let $\phi_2$ denote the quotient map $J \ra A$.  By Poincare
639reducibility over $\Q$ there is a unique abelian subvariety $A^{\vee}$
640of $J$ that projects isogenously to the quotient $A$ (equivalently,
641which has finite intersection with $\ker(\phi_2)$), and so by Hecke
642equivariance of $J \to A$ it follows that $A^{\vee}$ is $\T$-stable.
643Let $\phi$ be the composite isogeny
644$$645 \phi: \Adual \stackrel{\po}{\lra} J \stackrel{\pt}{\lra} A. 646$$
647
648\begin{rmk}
649Note that $A^{\vee}$ is the dual abelian variety of $A$.  More
650generally, if~$C$ is any abelian variety, let $C^{\vee}$ denote the
651dual 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
654obtain a map $\po: \Adual \ra J$.  Note also that $\vphi$
655is 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
676The {\em exponent} of a finite group~$G$ is the smallest positive
677integer~$n$ such that every element of~$G$ has order dividing~$n$.
678
679\begin{defi}\label{defi:modular}
680The {\em modular exponent} of~$A$ is the exponent of the kernel
681of the isogeny~$\phi$, and the {\em modular number} of~$A$ is
682the degree of~$\phi$.
683\end{defi}
684
685We denote the modular exponent of~$A$ by~$\nAe$ and
686the modular number by~$\nA$.
687When~$A$ is an elliptic curve, the modular
688exponent is equal to the modular degree of~$A$,
689and 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
695If~$R$ is a subring of~$\C$,
696let $S_2(R)=S_2(\Gamma;R)$ denote the subgroup of~$S_2(\Gamma)$
697consisting of cups forms whose Fourier expansions at the cusp~$\infty$
698have coefficients in~$R$.  (Note that $\Gamma$ is fixed for this whole
699section.)
700Let $S_2(\Gamma;\Z)[I]^{\perp}$ denote the orthogonal complement of
701$S_2(\Gamma;\Z)[I]$ in $S_2(\Gamma;\Z)$ with respect to the Petersson inner
702product.
703
704The following is well known, but we had difficulty finding
705a good reference.
706\begin{prop}
707The 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}
720The 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}
724is the {\em congruence exponent} $\rAe$ of~$A$ and its
725order 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
737Our definition of~$\rA$ generalizes the definition in
738Section~\ref{congintro} when~$A$ is an elliptic curve (see
739\cite[p.~276]{abbull}), and the following generalizes
740Theorem~\ref{thm:ribet_au}:
741\begin{thm}\label{thm:ribet_gen}
742If $f \in S_2(\C)$ 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 $$\nAf = (2^{11}\cdot 6947)^2 \,\,\nmid\,\, r^2_{A_f} = (2^{10}\cdot 762 6947)^2.$$
763Note 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}
781In this section we prove Theorem~\ref{thm:ribet_gen}.
782We continue using the notation introduced so far.
783
784\subsection{Proof of Theorem~\ref{thm:ribet_gen} (a)}
785\label{sec:firstpart}
786
787We begin with a remark about compatibilities.  In general, the
788polarization of~$J$ induced by the theta divisor need not be Hecke
789equivariant, because if~$T$ is a Hecke operator on~$J$, then
790on~$\Jdual$ it acts as $W_N T W_N$, where $W_N$ is the Atkin-Lehner
791involution (see e.g.,~\cite[Rem.~10.2.2]{diamond-im}).  However,
792on~$J^{\rm new}$ the action of the Hecke operators commutes with that
793of~$W_N$, so if the quotient map $J \ra A$ factors through~$J^{\rm new}$,
794then the Hecke action on~$\Adual$ induced by the embedding $\Adual \to 795J^{\vee}$ and the action on $\Adual$ induced by $\phi_1:\Adual\to{}J$
796are the same.  Hence $\Adual$ is isomorphic to $\po(\Adual)$
797as a $\T$-module.
798
799Recall 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 $J/B\isom A$.
801The following lemma is well known, but we prove it here for the
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
838Let $\T_1$ be the image of~$\T$ in $\End(\Adual)$,
839and let $\T_2$ be the image of $\T$ in $\End(B)$.
840We 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] &
847}
848\end{equation}
849Let
850$$851e=(1,0)\in \T_1 \oplus \T_2, 852$$
853and let $e_1$ and $e_2$ denote the images of~$e$ in the groups $(\T_1 854\oplus \T_2)/\T$ and $(\End(\Adual) \oplus \End(B))/\End(J)$,
855respectively.  It follows from Lemma~\ref{lem:homzero} that the two
856quotient groups on the right hand side of (\ref{eqn:diagram}) are
857finite, so~$e_1$ and~$e_2$ have finite order.  Note that because $e_2$
858is 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
865The {\em denominator} of any $\vphi\in\End(J)\tensor\Q$ is the
866smallest 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
871Let $\piAd, \piB \in \End(J)\tensor\Q$ 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}
877The element $e_2\in (\End(\Adual) \oplus \End(B))/\End(J)$
878defined above has order $\nAe$.
879\end{lem}
880\begin{proof}
881Let $n$ be the order of $e_2$, so~$n$ is the denominator
882of $\piAd$, which, as mentioned above, is also the
883denominator of $\piB$. We want to show that $n$ is equal
884to~$\nAe$, the exponent of $\Adual\cap B$.
885
886Let $i_{\Adual}$ and $i_B$
887be the embeddings of $\Adual$ and $B$ into $J$, respectively.
888Then $$\vphi = (n\piAd,n\piB)\in\Hom(J,\Adual\times B)$$
889and $\vphi\circ (i_{\Adual} + i_B) = [n]_{\Adual\times B}.$
890We have an exact sequence
891$$8920\to \Adual\cap B\xra{x\mapsto (x,-x)}\Adual\times B \xra{i_{\Adual} + i_B} J \to 0. 893$$
894Let $\Delta$ be the image of $\Adual\cap B$.  Then by exactness,
895$$896 [n]\Delta = (\vphi\circ (i_{\Adual} + i_B))(\Delta) = 897\vphi\circ ((i_{\Adual} + i_B)(\Delta)) = \vphi(\{0\}) = \{0\}, 898$$
899so $n$ is a multiple of
900the exponent~$\nAe$ of $\Adual\cap B$.
901
902To show the opposite divisibility, consider the
903commutative diagram
904$$905\[email protected]=4em{ 9060 \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\\ 9090 \ar[r] & {\Adual \cap B} \ar[r]^{x\mapsto (x,-x)}& {\Adual \times B} 910 \ar[r]& J \ar[r] & 0, 911} 912$$
913where the middle vertical map is $(a,b)\mapsto (\nAe a,0)$
914and the map~$\psi$ exists because $[\nAe](\Adual\cap B)=0$.
915But $\psi = \nAe \piAd$ in $\End(J)\tensor\Q$.
916This shows that $\nAe \piAd \in \End(J)$, i.e.,
917that $\nAe$ is a multiple of the
918denominator~$n$ of $\piAd$.
919
920\end{proof}
921
922Let $\Ext^1 = \Ext^1_{\Z}$ denote the first $\Ext$ functor
923in the category of $\Z$-modules.
924
925\begin{lem}\label{lem:compare_with_dual}
926The group $(\T_1 \oplus \T_2)/\T$ is isomorphic to
927the quotient (\ref{eqn:congexp})
928 in Definition~\ref{def:congexp}, so
929 $\rA = \#((\T_1 \oplus \T_2)/\T)$ and $\rAe$ is the
930exponent of $(\T_1 \oplus \T_2)/\T$.
931More precisely,  $\Ext^1((\T_1 \oplus \T_2)/\T,\Z)$ is isomorphic as a
932$\T$-module to the quotient (\ref{eqn:congexp}).
933\end{lem}
934\begin{proof}
935Apply the $\Hom(-,\Z)$ functor to the first row of (\ref{eqn:diagram})
936to obtain a three-term exact sequence
937\begin{equation}\label{eqn:dualseq}
9380  \to \Hom(\T_1\oplus \T_2,\Z) \to \Hom(\T,\Z)
939\to \Ext^1((\T_1\oplus\T_2)/\T,\Z) \to 0.
940\end{equation}
941%The term $\Ext^1(\T_1\oplus \T_2,\Z)$ is $0$ is because
942%$\Ext^1(M,\Z)=0$ for any finitely generated free abelian group.  Also,
943%$\Hom((\T_1\oplus\T_2)/\T,\Z)=0$ since $(\T_1\oplus\T_2)/\T$ is
944%torsion.
945There is a $\T$-equivariant bilinear pairing $\T\times 946S_2(\Z)\to\Z$ given by $(t,g)\mapsto a_1(t(g))$, which is perfect by
948Using this pairing, we transform (\ref{eqn:dualseq}) into an exact
949sequence
950$$9510 \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$$
954of $\T$-modules.
955Here we use that $\Hom(\T_2,\Z)$ is the unique saturated
956Hecke-stable complement of $S_2(\Z)[I_f]$ in $S_2(\Z)$, hence
957must equal $S_2(\Z)[I_f]^{\perp}$.
958Finally 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}
963The 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}
984If $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
989vertical homomorphism in (\ref{eqn:diagram}), the order
990  of~$e_2$ divides that of~$e_1$.  Now
991apply Lemmas~\ref{lem:ord_e2} and \ref{lem:ord_e1}.
992\end{proof}
993
994This finishes the proof of the first statement in
995Theorem~\ref{thm:ribet_gen}.
996
997
998
999
1000\subsection{Proof of Theorem~\ref{thm:ribet_gen} (b)}
1001\label{sec:secondpart}
1002Let $\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$
1004some positive multiple of which lie in~$\T$.
1005%so
1006%$$1007% \T' = (\T\tensor\Q) \cap \End(J_0(N)), 1008%$$
1009%where the intersection is taken inside $\End(J_0(N))\tensor\Q$.
1010The
1011quotient $\T'/\T$ is a finitely generated abelian group because both
1012$\T$ and $\End(J_0(N))$ are finitely generated over~$\Z$. Since
1013$\T'/\T$ is also a torsion group, it is finite.
1014
1015In Section~\ref{sec:multone}, we will give some conditions under
1016which $\T$ and~$\T'$ agree locally at  maximal ideal of~$\T$.
1017In Section~\ref{sec:degrees}, we will explain how the ratio of
1018the congruence number to the modular degree is closely related
1019to the order of~$\T'/\T$, and finally deduce that this ratio is $1$
1020(for quotients associated to newforms) locally at a prime~$p$
1021such that $p^2 \nmid N$.
1022
1023\subsubsection{Multiplicity One} \label{sec:multone}
1024
1025%Suppose for the moment that $M=1$, so $p=pM$.
1026Fixt an integer $N$ and a prime $p\mid N$.
1027Suppose for a moment that $N$ is prime, so $p=N$.
1028In \cite{mazur:eisenstein},
1029Mazur proves that $\T=\T'$; he combines this result with
1030the equality
1031$$1032 \T\tensor\Q = \End(J_0(p)) \tensor\Q, 1033$$
1034to deduce that $\T=\End(J_0(p))$.
1035This result, combined with Ribet's result
1036\cite{ribet:endo} or \cite{ribet:endalg}
1037to the effect that $\T\tensor\Q = (\End_{\Qbar} J_0(N)) \tensor \Q$,
1038shows that~$\T$ is the full ring of endomorphisms of $J_0(N)$ over $\Qbar$.
1039When $N$ is no
1040longer necessarily prime,
1041the method of \cite{mazur:eisenstein} shows
1042that $\T$ and $\T'$ agree locally at a maximal ideal $\m$  of $\T$
1043that satisfies a simple condition involving
1044differentials form mod $\ell$, where $\ell$ is the residue
1045characteristic of $\m$.
1046%has dimension at most one.
1047\comment{
1048$\Supp_{\T}(\T'/\T)$ contains no maximal ideal $\m$ of~$\T$
1049for which his space $\H^0(X_0(pM)_{\Fell},\Omega)[\m]$ has
1050dimension $\leq 1$.  (Here $\ell$ is the residue characteristic
1051of $\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$.
1054Mazur's argument (see \cite[pg.~95]{mazur:eisenstein}) is quite
1055general; it relies on a multiplicity $1$ statement for spaces
1056of differentials in positive characteristic (see
1057\cite[Prop.~9.3, pg.~94]{mazur:eisenstein}).
1058}
1059
1060For the sake of completeness, we state and prove a lemma
1061that can be easily extracted from~\cite{mazur:eisenstein}.
1062Let $m$ be the largest square dividing $N$ and
1063let $R = \Z[\frac{1}{m}]$. Let $X_0(N)_{R}$ denote
1064the 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$.
1067Let $\Omega=\Omega_{X_0(N)/R}$ denote the sheaf of regular
1068differentials on $X_0(N)_{R}$, as in~\cite[\S2(e)]{mazur:rational}.
1069If~$\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
1071prime~$\ell$.
1072
1073\comment{
1074His method shows in
1075the general case (where~$M$ is no longer constrained to be~$1$)
1076that $\Supp_{\T}(\T'/\T)$ contains no maximal ideal $\m$ of~$\T$
1077for which his space $\H^0(X_0(pM)_{\Fell},\Omega)[\m]$ has
1078dimension $\leq 1$.  (Here $\ell$ is the residue characteristic
1079of $\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
1083}
1084
1085\begin{lem}[Mazur]\label{lem:m1}
1086Let $\m$ be a maximal ideal of $\T$ of residue characteristic~$\ell$
1087such that $\ell^2 \nmid N$.
1088Suppose that
1089$$1090 \dim_{\T/\m} \H^0(X_0(N)_{\Fell},\Omega)[\m] \leq 1. 1091$$
1092Then $\T$ and $\T'$ agree locally at~$\m$.
1093%$\m$ is not in the support of $\T'/\T$.
1094\end{lem}
1095
1096\begin{proof}
1097Let $M$ denote the group
1098$H^1(X_0(N)_R, \OO_{X_0(N)})$,
1099where $\OO_{X_0(N)}$ is the structure sheaf of $X_0(N)$.
1100As explained in~\cite[p.~95]{mazur:eisenstein},
1101we have an action of $\EJ$ on~$M$, and
1102the action of $\T$ on~$M$ via the inclusion $\T \subseteq \EJ$
1103is 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$.
1106Suppose~$\m$ is a maximal ideal of~$\T$ that satisfies the hypotheses
1107of the lemma.
1108To prove that $\T_\m=\T_\m'$ it suffices to
1109prove the following claim:\\
1110
1111\noindent {\em Claim:} The map~$\phi|_{\T}$ is surjective locally at~$\m$.
1112\begin{proof}
1113By Nakayama's lemma, to show that $M$ is generated
1114as 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$,
1118it 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$.
1121Since \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}).
1125Since 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{
1141Thus $M \tensor \T_\m$ is free of rank one over~$\T_\m$. The
1142composite $\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
1146that $\T_\m$ is saturated in $\EJ \tensor \T_\m$, i.e.,
1147that $\T$ and $\T'$ agree locally at~$\m$.
1148}
1149\end{proof}
1150
1151If $\m$ is a maximal ideal of the Hecke algebra~$\T$
1152of 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$$
1157By Lemma~\ref{lem:m1}, multiplicity one for
1158$\H^0(X_0(N)_{\Fell}, \Omega)[\m]$ implies that $\T$ and
1159$\T'$ agree at~$\m$.
1160
1161There is quite a bit of literature on the question of multiplicity~$1$
1162for $\H^0(X_0(N)_{\Fell},\Omega)[\m]$.
1163The easiest case is that~$\ell$ is prime to the level $N$:
1164
1165\begin{lem}[Mazur]\label{lem:m_ell}
1166If $\m$ is a maximal ideal of $\T$ of residue characteristic~$\ell$
1167such that $\ell \nmid N$, then
1168$$\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$$ for
1191%all $\m\mid \ell$.    Now apply Lemma~\ref{lem:m1}
1192%\edit{Is there a problem if $\ell=2$?  How do Lloyd Kilford's examples
1193%fit into this, where I guess $N=1$ and $\ell=2$ and multiplicity
1194%one in $J_0(p)$ fails.  Is it still OK in Mazur's
1195%differentials? -WAS}
1196
1197In the context of Mazur's paper, where the level~$N$ is prime, we see
1198from Lemma~\ref{lem:m_ell} that $\T$ and $\T'$ agree away from~$N$.
1199Locally at $N$, Mazur proved that $\T=\T'$ by an analogue of the
1200arguments that he used away from $N$; see Chapter II of
1201\cite{mazur:eisenstein} (and especially Prop.~9.4 and 9.5 of that
1202chapter) as well as \cite{mazur-ribet}, where these arguments are
1203taken up in a context where the level is no longer necessarily prime
1204(and where one works locally at a prime whose square does not divide
1205the level).
1206%At~$N$, we can still use the $q$-expansion principle
1207%because of the arguments in \cite[Ch.II~\S4]{mazur:eisenstein}.
1208Thus in the prime level case, $\T=\T'$, as we asserted above.
1209
1210
1211Now\label{NPnotation}
1212let $p$ be a prime such that $p \parallel N$, and let $M = N/p$.
1213The question of multiplicity $1$ at $p$ for $\H^0(X_0(pM)_{\Fp}, 1214\Omega)[\m]$ is discussed in \cite{mazur-ribet}, where the authors
1215establish multiplicity~$1$ for maximal ideals $\m\mid p$ for which the
1216associated mod~$p$ Galois representation is irreducible and {\em not}
1217$p$-old.  (A representation of level $pM$ is $p$-old if it arises from
1218$S_2(\Gamma_0(M))$.)
1219
1220If~$\m$ is a maximal ideal of~$\T$ of residue characteristic~$\ell$,
1221then we say that~$\m$ is ordinary
1222if $T_\ell \not\in \m$ (note that $T_\ell$ is often denoted $U_\ell$
1223if $\ell \mid N$). For our purposes, the following lemma is convenient:
1224
1225\begin{lem}[Wiles]\label{lem:wiles}
1226If $\m$ is an ordinary maximal ideal of $\T$ of characteristic~$p$, then
1227$$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
1233This is essentially Lemma~2.2 in~\cite[pg.~485]{wiles};
1234\comment{, which
1235proves, under a suitable hypothesis, that $\H^0(X_0(pM)_{\F_p},\Omega)[\m]$
1236is $1$-dimensional if $\m$ is a maximal ideal of~$\T$ that divides~$p$.
1237The 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.)
1239It follows from Wiles's lemma that $\T'=\T$ locally at~$\m$ whenever
1240$\m$ is an ordinary prime whose residue characteristic exactly
1241divides the level (which is $pM$ here).
1242}
1243we make a few comments about how it applies on our situation:
1244\begin{enumerate}
1245\item
1246Wiles considers $X_1(M,p)$ instead of $X_0(pM)$, which means that he is
1247using $\Gamma_1(M)$-structure instead of $\Gamma_0(M)$-structure.
1248This 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
1251this assumption is not relevant to our question.
1252
1253\item
1254The condition that $\m$ is ordinary does not appear
1255explicitly in the statement of Lemma~2.2 in~\cite{wiles};
1257assumption 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
1275We now summarize the conclusions we can make from the lemmas so far:
1276
1277\begin{prop} \label{prop:TT'}
1278The modules~$\T$
1279and $\T'$ agree locally at each maximal ideal~$\m$ that is either prime
1280to~$N$ or that satisfies the following supplemental hypothesis: the
1281residue characteristic of~$\m$ divides~$N$ only to the first power
1282and $\m$ is ordinary.
1283\end{prop}
1284\begin{proof}
1285This follows easily from Lemmas~\ref{lem:m1}, \ref{lem:m_ell},
1286and~\ref{lem:wiles}.
1287\end{proof}
1288\comment{
1289Wiles's lemma and the standard $q$-expansion argument
1290(Lemma~\ref{lem:m_ell} and Lemma~\ref{lem:wiles}) imply that~$\T$
1291and~$\T'$ agree locally at each rational prime that is prime to the
1292level $pM$, and also at each maximal ideal~$\m$ dividing~$p$ that is
1293ordinary, in the sense that $T_p \not\in \m$.  A more palatable
1294description of the situation involves considering the Hecke
1295algebra~$\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
1297to~$N$ or that satisfies the following supplemental hypothesis: the
1298residue characteristic of~$\m$ divides~$N$ only to the first power
1299and~$\m$ is ordinary.
1300}
1301
1302In Mazur's original context, where the level~$N$ is
1303prime, we have $T_N^2=1$ because there are no forms of
1304level~$1$. Accordingly, each~$\m$ dividing~$N$ is ordinary, and we
1305recover 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
1318Let $e \in \T\tensor\Q$ be as in Section~\ref{sec:firstpart},
1319and let $p,N,M$ be as before Lemma~\ref{lem:wiles}.
1320The image of $e$ in $J_0(pM)$ is the $\T$-stable abelian subvariety
1321denoted $\Adual$ in Section~\ref{sec:firstpart}, but since we shall
1322now exclusively work with this subvariety rather than the
1323corresponding optimal quotient of $J_0(pM)$ (which was denoted $A$
1324earlier), we will now write $A$ to denote the image of $e$ (without
1325risk of confusion).  We also write $B$ to denote the unique
1326$\T$-stable abelian subvariety of $J_0(pM)$ complementary to~$A$.
1327
1328For $t \in \T$, let $t_A$ be the restriction of~$t$ to $A$, and
1329let~$t_B$ be the image of~$t$ in $\End(B)$.  Let $\T_A$ be the
1330subgroup 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$
1335with finite cokernel.  Because~$j$ is an injection, we
1336refer to the maps $\pi_A:\T\to \T_A$ and $\pi_B : \T \to \T_B$,
1337given by $t \mapsto t_A$ and $t\mapsto t_B$, respectively,
1338as projections''.
1339
1340\begin{defi}
1341The {\em congruence ideal} associated with the projector~$e$ is
1342$I=\pi_A(\ker(\pi_B)) \subset \T_A.$
1343\end{defi}
1344
1345Viewing $\T_A$ as $\T_A\times \{0\}$, we may view $\T_A$ as a subgroup
1346of $\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}
1349We have $I=\T_A\cap \T$.
1350\end{lem}
1351
1352A larger ideal of $\T_A$ is
1353$1354 J = \Ann_{\T_A}(A \cap B); 1355$
1356it consists of restrictions to $A$ of Hecke operators that
1357vanish on $A\cap B$.
1358
1359\begin{lem}
1360We have $I\subset J$.
1361\end{lem}
1362\begin{proof}
1363The image in $\T_A$ of an operator that vanishes on $B$ also
1364vanishes on $A\cap B$.
1365\end{proof}
1366
1367\begin{lem}\label{lem:j_int}
1368We have
1369$J = \T_A \cap \End(J_0(pM)) = \T_A \cap \T'.$
1370\end{lem}
1371\begin{proof}
1372This is elementary; it is an analogue of Lemma~\ref{lem:i_int}.
1373\end{proof}
1374
1375\begin{prop}\label{prop:ji_inc}
1376There is a natural inclusion
1377$1378 J/I \hra \T'/\T 1379$
1380of $\T$-modules.
1381\end{prop}
1382\begin{proof}
1383Consider the map $\T\to \T\tensor\Q$ given by $t\mapsto te$.
1384This homomorphism factors through $\T_A$ and yields an injection
1385$\iota_A : \T_A \hra \T\tensor\Q$.  Symmetrically, we also
1386obtain $\iota_B : \T_B \hra \T\tensor\Q$.  The
1387map
1388$(t_A, t_B) \mapsto \iota_A(t_A) + \iota_B(t_B)$
1389is an injection
1390 $\T_A\times \T_B \hra \T\tensor\Q$.
1391The composite of this map with the inclusion $j:\T\hra \T_A\times \T_B$
1392defined above is the natural map $\T\hra \T\tensor\Q$.  We thus have
1393a 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$$
1398By Lemma~\ref{lem:i_int} and Lemma~\ref{lem:j_int},
1399we have $I=\T_A\cap \T$ and $J=\T_A\cap \T'$.
1400Thus $I=J\cap \T$, where the intersection is taken
1401inside $\T'$.  Thus
1402$$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}
1408If $\m$ is a maximal ideal not in $\Supp_{\T}(\T'/\T)$,
1409then $\m$ is not in the support of $J/I$, i.e.,
1410if $\T$ and $\T'$ agree locally at $\m$, then
1411$I$ and $J$ also agree locally at $\m$.
1412\end{cor}
1413
1414Note that the Hecke algebra $\T$ acts on $J/I$ through
1415its quotient $\T_A$,
1416since the action of~$\T$ on~$I$ and on~$J$ factors through
1417this quotient.
1418
1419Now we specialize to the case where $A$ is ordinary at $p$,
1420in the sense that the image of $T_p$ in $\T_A$, which we
1421denote $T_{p,A}$, is invertible modulo every maximal ideal
1422of $\T_A$ that divides~$p$.  (This case occurs when~$A$ is
1423a subvariety of the $p$-new subvariety of $J_0(pM)$, since
1424the square of $T_{p,A}$ is the identity.)
1425
1426
1427If $\m\mid p$
1428is a maximal ideal of $\T$ that arises by pullback from
1429a maximal ideal of $\T_A$, then~$\m$ is ordinary in the
1430sense used above.  When $A$ is ordinary at~$p$, it follows
1431from Proposition~\ref{prop:TT'} and Corollary~\ref{cor:ji_inc}
1432that $I=J$ locally at~$p$.  The reason is simple: regarding~$I$
1433and~$J$ as $\T_A$-modules, we realize that we need to test
1434that $I=J$ at maximal ideals of $\T_A$ that divide~$p$.
1435These ideals correspond to maximal ideals $\m\mid p$
1436of $\T$ that are automatically ordinary, so we have $I=J$
1437locally at $\m$ because of Proposition~\ref{prop:TT'}.
1438By Proposition~\ref{prop:TT'},
1439we have $\T=\T'$ locally at primes away from the
1440level $pM$.  Thus we conclude that $I=J$
1441locally at all primes $\ell\nmid pM$ and also at~$p$,
1442a prime that divides the level $pM$ exactly once.
1443
1444Suppose, finally, that $A$ is the abelian variety associated to a
1445newform~$f$ of level~$pM$.
1446%We then have $\T_A=\Z$.
1447The ideal $I\subset \T_A$ measures congruences between~$f$ and the space of forms
1448in $S_2(\Gamma_0(pM))$ that are orthogonal to the space generated
1449by~$f$.  Also, $A\cap B$ is the kernel in~$A$ of the map
1450multiplication by the modular element~$e$''.
1451In this case, the inclusion $I\subset J$ corresponds to the divisibility
1452$1453 \tilde{n}_A \mid \tilde{r}_A, 1454$
1455and we have equality at primes at which $I=J$ locally.
1456We conclude that the congruence exponent and the modular exponent
1457agree both at~$p$ and at primes not dividing $pM$, which completes our
1458proof of Theorem~\ref{thm:ribet_gen}(b).
1459
1460\begin{rmk}
1461The ring
1462$$1463 R = \End(J_0(pM)) \cap (\T_A \times \T_B) 1464$$
1465is often of interest, where the intersection is taken
1466in $\End(J_0(pM))\tensor \Q$.  We proved above that there
1467is a natural inclusion $J/I \hra \T'/\T$.  This
1468inclusion yields an isomorphism
1469$1470 J/I \xra{\sim} R/\T. 1471$
1472Indeed, if $(t_A, u_B)$ is an endomorphism of $J_0(pM)$,
1473where $t,u \in \T$, then
1474$(t_A, u_B) - u = (t_A, 0)$ is an element of~$J$.
1475The ideals~$I$ and~$J$ are equal to the extent that the
1476rings~$\T$ and $R$ coincide.  Even when $\T'$ is bigger than~$\T$,
1477its subring $R$ may be not far from~$\T$.
1478\end{rmk}
1479
1480\section{Failure of Multiplicity One}\label{sec:mult1}
1481In this section, we discuss examples of failure of multiplicity one
1482(in two different but related senses). The notion of multiplicity one,
1483originally due to Mazur~\cite{mazur:eisenstein}, has played an
1484important role in several places (e.g., in Wiles's proof of Fermat's
1485last theorem~\cite{wiles}). This notion is closely related to
1486Gorensteinness of certain Hecke algebras (e.g., see~\cite{tilouine:hecke}).
1487Kilford~\cite{kilford} found examples of failure of Gorensteinness
1488(and multiplicity one) at the prime~$2$ for certain prime levels.
1489Motivated by the arguments in Section~\ref{sec:main}, in this section
1490we give examples of failure of multiplicity one for primes (including
1491odd primes) whose square divides the level.
1492
1493\subsection{Multiplicity One for
1494  Differentials}\label{sec:dataind}
1495In connection with the arguments in Section~\ref{sec:main}, especially
1496Lemmas~\ref{lem:m1} and \ref{lem:wiles}, it is of interest
1497to compute the index $[\T':\T]$ for various $N$.
1498We can compute this index in Magma, e.g., the following
1499commands compute the index for $N=54$:
1500{\tt J := JZero(54); T := HeckeAlgebra(J); Index(Saturation(T), T);}''
1501We obtain Table~\ref{table:index}, where the first column
1502contains $N$ 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
150611 & 1 \\\hline
150712 & 1 \\\hline
150813 & 1 \\\hline
150914 & 1 \\\hline
151015 & 1 \\\hline
151116 & 1 \\\hline
151217 & 1 \\\hline
151318 & 1 \\\hline
151419 & 1 \\\hline
151520 & 1 \\\hline
151621 & 1 \\\hline
151722 & 1 \\\hline
151823 & 1 \\\hline
151924 & 1 \\\hline
152025 & 1 \\\hline
152126 & 1 \\\hline
152227 & 1 \\\hline
152328 & 1 \\\hline
152429 & 1 \\\hline
152530 & 1 \\\hline
152631 & 1 \\\hline
152732 & 1 \\\hline
152833 & 1 \\\hline
152934 & 1 \\\hline
153035 & 1 \\\hline
153136 & 1 \\\hline
153237 & 1 \\\hline
153338 & 1 \\\hline
153439 & 1 \\\hline
153540 & 1 \\\hline
153641 & 1 \\\hline
153742 & 1 \\\hline
153843 & 1 \\\hline
153944 & 2 \\\hline
154045 & 1 \\\hline
154146 & 2 \\\hline
154247 & 1 \\\hline
154348 & 1 \\\hline
154449 & 1 \\\hline
154550 & 1 \\\hline
1546\end{tabular}
1547\,\,\,\,\,\,
1548\begin{tabular}{|l|c|}\hline
154951 & 1 \\\hline
155052 & 1 \\\hline
155153 & 1 \\\hline
155254 & 3 \\\hline
155355 & 1 \\\hline
155456 & 2 \\\hline
155557 & 1 \\\hline
155658 & 1 \\\hline
155759 & 1 \\\hline
155860 & 2 \\\hline
155961 & 1 \\\hline
156062 & 2 \\\hline
156163 & 1 \\\hline
156264 & 2 \\\hline
156365 & 1 \\\hline
156466 & 1 \\\hline
156567 & 1 \\\hline
156668 & 2 \\\hline
156769 & 1 \\\hline
156870 & 1 \\\hline
156971 & 1 \\\hline
157072 & 2 \\\hline
157173 & 1 \\\hline
157274 & 1 \\\hline
157375 & 1 \\\hline
157476 & 2 \\\hline
157577 & 1 \\\hline
157678 & 2 \\\hline
157779 & 1 \\\hline
157880 & 4 \\\hline
157981 & 1 \\\hline
158082 & 1 \\\hline
158183 & 1 \\\hline
158284 & 2 \\\hline
158385 & 1 \\\hline
158486 & 1 \\\hline
158587 & 1 \\\hline
158688 & 8 \\\hline
158789 & 1 \\\hline
158890 & 1 \\\hline
1589\end{tabular}
1590\,\,\,\,\,\,
1591\begin{tabular}{|l|c|}\hline
159291 & 1 \\\hline
159392 & 16 \\\hline
159493 & 1 \\\hline
159594 & 4 \\\hline
159695 & 1 \\\hline
159796 & 8 \\\hline
159897 & 1 \\\hline
159998 & 1 \\\hline
160099 & 9 \\\hline
1601100 & 1 \\\hline
1602101 & 1 \\\hline
1603102 & 1 \\\hline
1604103 & 1 \\\hline
1605104 & 4 \\\hline
1606105 & 1 \\\hline
1607106 & 1 \\\hline
1608107 & 1 \\\hline
1609108 & 54 \\\hline
1610109 & 1 \\\hline
1611110 & 2 \\\hline
1612111 & 1 \\\hline
1613112 & 8 \\\hline
1614113 & 1 \\\hline
1615114 & 1 \\\hline
1616115 & 1 \\\hline
1617116 & 4 \\\hline
1618117 & 1 \\\hline
1619118 & 2 \\\hline
1620119 & 1 \\\hline
1621120 & 32 \\\hline
1622121 & 1 \\\hline
1623122 & 1 \\\hline
1624123 & 1 \\\hline
1625124 & 16 \\\hline
1626125 & 25 \\\hline
1627126 & 18 \\\hline
1628127 & 1 \\\hline
1629128 & 64 \\\hline
1630129 & 1 \\\hline
1631130 & 1 \\\hline
1632\end{tabular}
1633\,\,\,\,\,\,
1634\begin{tabular}{|l|c|}\hline
1635131 & 1 \\\hline
1636132 & 8 \\\hline
1637133 & 1 \\\hline
1638134 & 1 \\\hline
1639135 & 27 \\\hline
1640136 & 16 \\\hline
1641137 & 1 \\\hline
1642138 & 4 \\\hline
1643139 & 1 \\\hline
1644140 & 8 \\\hline
1645141 & 1 \\\hline
1646142 & 8 \\\hline
1647143 & 1 \\\hline
1648144 & 32 \\\hline
1649145 & 1 \\\hline
1650146 & 1 \\\hline
1651147 & 7 \\\hline
1652148 & 4 \\\hline
1653149 & 1 \\\hline
1654150 & 5 \\\hline
1655151 & 1 \\\hline
1656152 & 32 \\\hline
1657153 & 9 \\\hline
1658154 & 1 \\\hline
1659155 & 1 \\\hline
1660156 & 32 \\\hline
1661157 & 1 \\\hline
1662158 & 4 \\\hline
1663159 & 1 \\\hline
1664160 & 256 \\\hline
1665161 & 1 \\\hline
1666162 & 81 \\\hline
1667163 & 1 \\\hline
1668164 & 8 \\\hline
1669165 & 1 \\\hline
1670166 & 2 \\\hline
1671167 & 1 \\\hline
1672168 & 128 \\\hline
1673169 & 13 \\\hline
1674170 & 1 \\\hline
1675\end{tabular}
1676\,\,\,\,\,\,
1677\begin{tabular}{|l|c|}\hline
1678171 & 9 \\\hline
1679172 & 8 \\\hline
1680173 & 1 \\\hline
1681174 & 4 \\\hline
1682175 & 5 \\\hline
1683176 & 512 \\\hline
1684177 & 1 \\\hline
1685178 & 1 \\\hline
1686179 & 1 \\\hline
1687180 & 72 \\\hline
1688181 & 1 \\\hline
1689182 & 1 \\\hline
1690183 & 1 \\\hline
1691184 & 1024 \\\hline
1692185 & 1 \\\hline
1693186 & 4 \\\hline
1694187 & 1 \\\hline
1695188 & 256 \\\hline
1696189 & 243 \\\hline
1697190 & 8 \\\hline
1698191 & 1 \\\hline
1699192 & 4096 \\\hline
1700193 & 1 \\\hline
1701194 & 1 \\\hline
1702195 & 1 \\\hline
1703196 & 14 \\\hline
1704197 & 1 \\\hline
1705198 & 81 \\\hline
1706199 & 1 \\\hline
1707200 & 80 \\\hline
1708201 & 1 \\\hline
1709202 & 1 \\\hline
1710203 & 1 \\\hline
1711204 & 32 \\\hline
1712205 & 1 \\\hline
1713206 & 4 \\\hline
1714207 & 81 \\\hline
1715208 & 256 \\\hline
1716209 & 1 \\\hline
1717210 & 2 \\\hline
1718\end{tabular}
1719\end{center}
1720\end{table}
1721
1722Let $\m$ be a maximal ideal of the Hecke algebra
1723$\T\subset\End(J_0(N))$ of residue characteristic~$p$. Recall
1724that 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
1728In each case in which $[\T':\T]\neq 1$, Lemma~\ref{lem:m1} implies
1729that 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
1731of failure of multiplicity one for differentials.
1732
1733
1734In Table~\ref{table:index}, whenever $p\mid [\T':\T]$, then $p^2\mid 17352N$.  This is a consequence of Proposition~\ref{prop:TT'}, which
1736moreover asserts that when $2$ exactly divides $N$ and $2\mid[\T':\T]$
1737then there is a non-ordinary (old) maximal ideal of characteristic $2$
1738in 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
1745
1746Moreover, notice that
1747Theorem~\ref{thm:ribet_gen}(b) (whose proof is in
1748Section~\ref{sec:secondpart})
1749follows formally from two
1750key facts: that $A_f$ is new and that multiplicity one for differentials
1751holds for ordinary maximal ideals with residue characteristic
1752$p\mid\mid N$ and for all maximal ideals with residue
1753characteristic $p\nmid N$.  The conclusion of
1754Theorem~\ref{thm:ribet_gen}(b) does not hold for the counterexamples
1755in Section~\ref{sec:elliptic} (e.g., for~54B1), which are
1756all new elliptic curves, so multiplicity one for
1757differentials does not hold for certain
1758maximal ideals that arise from the new quotient of the Hecke algebra.
1759Note that in all examples we have $p\mid(r/m)$ with $p^2\mid N$,
1760which raises the question: are there non-ordinary counterexamples
1761with $p\mid\mid N$?
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$magma 1770% Magma V2.11-10 Sun Aug 28 2005 18:34:49 on modular [Seed = 1293693469] 1771% Type ? for help. Type <Ctrl>-D to quit. 1772% > J := JZero(46); 1773% > J; 1774% Modular abelian variety JZero(46) of dimension 5 and level 2*23 over Q 1775% > T := HeckeAlgebra(J); 1776% > S := Saturation(T); 1777% > m := S/T; 1778% > m; 1779% Abelian Group isomorphic to Z/2 1780% Defined on 1 generator 1781% Relations: 1782% 2*m.1 = 0 1783% > 1784% > S; 1785% Sat(HeckeAlg(JZero(46))): Group of homomorphisms from JZero(46) to JZero(46) 1786% > Basis(S); 1787% [ 1788% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix), 1789% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix), 1790% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix), 1791% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix), 1792% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix) 1793% ] 1794% > S.1; 1795% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix) 1796% > S.1 in T; 1797% true 1798% > S.2 in T; 1799% false 1800% > t2 := HeckeOperator(J,2); 1801% > t2 in T; 1802% true 1803% > t2*S.2; 1804% Homomorphism from JZero(46) to JZero(46) (not printing 10x10 matrix) 1805% > t2*S.2 in T; 1806% true 1807 1808\subsection{Multiplicity One for Jacobians} 1809 1810We say that a maximal ideal~$\m$of~$\T$satisfies {\it multiplicity one} 1811if$J_0(N)[\m]$is of dimension two over~$\T/\m$. We sometimes use the 1812phrase multiplicity one for~$J_0(N)$'' in order to distinguish this notion 1813from the notion of multiplicity one for differentials. 1814%\edit{I added these lines for clarity. --Amod} 1815 1816\begin{prop}\label{prop:mult1J} 1817 Suppose$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 1845Proposition~\ref{prop:mult1J} implies that any example in which 1846simultaneously$p\nmid m_E$and$\ord_p(r_E)\neq \ord_p(m_E)$produces 1847an example in which multiplicity one for$J_0(N)$fails. For example, 1848for 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 1862of the above two new sections.} 1863%\edit{[This is still very rough, and will need to be cleaned up. --Amod]} 1864 1865Let$\m$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 1874The proof of Theorem~\ref{thm:ribet_gen}(b) follows formally from two 1875key facts: that$A_f$is new and that strong multiplicity one holds 1876for ordinary maximal ideals if~$p^2 \nmid N$. The conclusion of 1877Theorem~\ref{thm:ribet_gen}(b) does not hold for the counterexamples 1878in Section~\ref{sec:elliptic} (at levels$54$,~$64$, etc.), which are 1879all new elliptic curves, which shows that strong multiplicity one does 1880not hold for certain ordinary\edit{William: I totally don't get this. 1881When$p^2\mid N$the ideals are {\em NOT} ordinary. We only got 1882ordinary in the proof above because$p\mid\mid N$. For example, for 1883the curve 54b, we have$a_3=0$, and the ideal is not ordinary.} 1884maximal ideals for the corresponding 1885levels (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 1890are not ordinary!} 1891\end{prop} 1892 1893There is another notion of multiplicity one: 1894suppose~$\rho_{\m}$, the representation attached to~$\m$, 1895is absolutely irreducible. 1896Then one 1897says that$\m$satisfies {\em weak multplicity one} if 1898$J_0(N)[\m]$is isomorphic to a single copy of~$\rho_{\m}$. 1899By standard arguments, strong multiplicity one implies 1900weak multiplicity one when$\overline{\rho}_{\m}$is absolutely irreducible. 1901\edit{William: I think this is wrong. 1902In connection with Remark~\ref{rem:24}, we find (again via a Magma 1903computation) that$\T'=\T$when$N=431$. This was also already known via 1904work of Mazur \cite{mazur:eisenstein}, and illustrates that 1905multiplicity 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}$1909%to be absolutely irreducible -- it might be worth checking 1910%if this holds in our couterexamples} 1911which in turn implies that the Hecke algebra~$\T$is Gorenstein.\edit{or 1912maybe weak multiplicity one is equivalent to Gorensteinness, and both 1913follow from strong multiplicity one; a good reference might be 1914Tilouine's article in the FLT conference.} Wiles proves that strong 1915multiplicity one holds for ordinary maximal ideals provided$p^2 \nmid
1916N$, which he used to show the Gorensteinness of certain Hecke 1917algebras. This Gorensteinness property was a key step in the proof of 1918Fermat's last theorem. Our finding above shows that the hypothesis 1919$p^2 \nmid N$is essential, and thus gives a limit to how far the 1920standard argument for proving Gorensteinness works. 1921 1922Note that while in our examples, we know that strong multiplicity one 1923fails, we do not know if weak multiplicity one or Gorensteinness 1924fails. 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