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1\documentclass[draft,11pt]{llncs}
2\newcommand{\capt}{\caption}
3\newcommand{\cA}{\mathcal{A}}
4\newcommand{\cT}{\mathcal{T}}
5\newcommand{\cX}{\mathcal{X}}
6\newcommand{\cO}{\mathcal{O}}
7\newcommand{\coker}{\mbox{\rm coker}}
8\newcommand{\F}{\mathbf{F}}
9\newcommand{\G}{\mathbf{G}}
10\newcommand{\Fbar}{\overline{\F}}
11\newcommand{\Fp}{\F_p}
12\newcommand{\Fpbar}{\Fbar_p}
13\newcommand{\Frob}{\mbox{\rm Frob}}
14\newcommand{\Gal}{\mbox{\rm Gal}}
15\newcommand{\Hom}{\mbox{\rm Hom}}
16\newcommand{\Q}{\mathbf{Q}}
17\newcommand{\SL}{\mbox{\rm SL}}
18\newcommand{\T}{\mathbf{T}}
19\newcommand{\Z}{\mathbf{Z}}
20\newcommand{\isom}{\cong}
21\newcommand{\tdot}{\!\cdot\!}
22\newcommand{\rta}{\rightarrow}
23
24\begin{document}
25
26\title{Component Groups of Quotients of $J_0(N)$}
27\titlerunning{Component Groups}
28\author{David Kohel\inst{1} \and William A. Stein\inst{2}}
29\authorrunning{Kohel \and Stein}
30\tocauthor{David Kohel (University of Sydney),
31William A. Stein (University of California at Berkeley)}
32%
33\institute{University of Sydney\\
34\email{kohel@maths.usyd.edu.au}\\
35\texttt{http://www.maths.usyd.edu.au:8000/u/kohel/}
36\and
37University of California at Berkeley,\\
38\email{was@math.berkeley.edu}\\
39\texttt{http://shimura.math.berkeley.edu/\~{}was}
40}
41
42\maketitle
43\begin{abstract}
44Let~$f$ be a newform of weight~$2$ on $\Gamma_0(N)$, and let~$A_f$ be
45the corresponding optimal abelian variety quotient of $J_0(N)$.  We
46describe an algorithm to compute the order of the component group of
47$A_f$ at primes~$p$ that exactly divide~$N$.  We give a table of
48orders of component groups for all~$f$ of level $N\leq 127$ and four
49examples in which the component group is very large, as predicted by
50the Birch and Swinnerton-Dyer conjecture.
51\end{abstract}
52
53\section{Introduction}
54
55Let $X_0(N)$ be the Riemann surface obtained by compactifying the
56quotient of the upper half-plane by the action of $\Gamma_0(N)$.
57Then $X_0(N)$ has a canonical structure of algebraic curve
58over~$\Q$; denote its Jacobian by $J_0(N)$.  It is equipped with
59an action of a commutative ring $\T=\Z[\ldots T_n \ldots]$ of
60Hecke operators.  For more details on modular curves, Hecke operators,
61and modular forms see, e.g.,~\cite{diamond-im}.
62
63Now suppose that~$f=\sum_{n=1}^{\infty} a_n q^n$ is a modular newform
64of weight~$2$ for the congruence subgroup $\Gamma_0(N)$.
65The Hecke operators also act on~$f$ by $T_n(f) = a_n f$.
66The eigenvalues $a_n$ generate an order $R_f = \Z[\ldots a_n \ldots]$
67in a number field $K_f$.  Let $I_f$ be the kernel ideal of the map
68$\T \rta R_f$ sending $T_n$ to~$a_n$.
69Following Shimura~\cite{shimura:factors}, we associate to~$f$ the
70quotient $A_f = J_0(N)/I_f J_0(N)$ of
71$J_0(N)$.  Then $A_f$ is an abelian variety over~$\Q$ of dimension
72$[K_f:\Q]$, with bad reduction exactly at the primes dividing~$N$.
73
74One dimensional quotients of $J_0(N)$ have been intensely studied
75in recent years, both computationally and theoretically.
76The original conjectures of Birch and
77Swinnerton-Dyer~\cite{birch-swd-I,birch-swd-II},
78which concern elliptic curves,
79were greatly influenced by computations.
80The scale of these computations was extended and systematized by
81Cremona in~\cite{cremona:algs}.
82
83In another direction, Wiles~\cite{wiles:fermat} and
84Taylor-Wiles~\cite{taylor-wiles} proved a special case of the
85conjecture of Shimura-Taniyama, which asserts that every elliptic
86curve over~$\Q$ is a quotient of some $J_0(N)$; this allowed them to establish
87Fermat's Last Theorem.  The full Shimura-Taniyama
88conjecture was later proved by Breuil, Conrad, Diamond,
90This illustrates the central roll played by quotients of $J_0(N)$.
91
92\section{Component Groups of $A_f$}
93
94The N\'eron model $\cA/\Z$ of an abelian variety $A/\Q$ is by definition
95a smooth commutative group scheme over~$\Z$ with generic fiber~$A$
96such that for any smooth scheme~$S$ over~$\Z$, the restriction map
97$$\Hom_\Z(S,\cA) \rta \Hom_\Q(S_\Q,A)$$ is a bijection.
98For more details, including a proof of existence, see,
99e.g.,~\cite{neronmodels}.
100
101Suppose that $A_f$ is a quotient of $J_0(N)$ corresponding to a newform~$f$,
102and let $\cA_f$ be the N\'eron model of~$A_f$.  For any prime divisor~$p$
103of~$N$, the closed fiber~${\cA_f}_{/\Fp}$ is a group scheme
104over~$\Fp$, which need not be connected.  Denote the connected
105component of the identity by~${\cA^{\circ}_f}_{/\Fp}$.
106There is an exact sequence
107$$1080 \rightarrow {\cA^{\circ}_f}_{/\Fp} 109 \rightarrow {\cA_f}_{/\Fp}\rightarrow\Phi_{A_f,p} 110 \rightarrow 0 111$$
112with $\Phi_{A_f,p}$ a finite \'{e}tale group scheme over~$\Fp$
113called the {\em component group} of $A_f$ at~$p$.
114
115We note that a finite \'etale group scheme is equivalent to a
116finite abelian group together with an action of $\Gal(\Fpbar/\Fp)$.
117In this paper we describe an algorithm for computing the order of
118this group.
119
120\section{The Algorithm}
121
122Let~$J = J_0(N)$.  As the Jacobian of a curve, $J$ is canonically
123self-dual, so the projection $J \rta A_f$ induces a polarization
124$A_f^{\vee} \rta A_f$, where $A_f^{\vee}$ denotes the abelian variety
125dual of $A_f$.    The {\em modular degree} $\delta_{A_f}$
126is the positive square root of the degree of this polarization.
127
128Let $T_{J,p}$ be the toric part of the closed fiber of the
129N\'eron model of $J$ at~$p$, and define $\cX_{J,p}$ to
130be the group $\Hom_{\Fpbar}(T_{J,p},\G_m)$ of characters of $T_{J,p}$.
131Then $\cX_{J,p}$ is a free abelian group equipped with an action
132of both~$\T$ and $\Gal(\Fpbar/\Fp)$
133(see, e.g., Ribet~\cite{ribet:modreps}); there is a bilinear
134pairing
135$$136\langle\,,\,\rangle : \cX_{J,p} \times \cX_{J,p} \rightarrow \Z 137$$
138called the {\em monodromy pairing} such that
139$$140\Phi_{J,p} \isom \coker(\cX_{J,p} \rightarrow \Hom(\cX_{J,p},\Z)). 141$$
142Let $\cX_{J,p}[I_f]$ be the intersection of all kernels $\ker(t)$ for~$t$
143in $I_f$, and let
144$$145\alpha_f: \cX_{J,p} \rightarrow \Hom(\cX_{J,p}[I_f],\Z) 146$$
147be the map induced by the monodromy pairing.
148The following theorem of Stein~\cite{stein:phd}, provides the
149basis for the computation of orders of component groups.
150
151\begin{theorem}\label{thm:main}
152$$153\#\Phi_{A_f,p} 154 = \frac{\#\coker(\alpha_f) \cdot \delta_{A_f}} 155 {\# (\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))}\,. 156$$
157\end{theorem}
158
159\subsection{Computing the modular degree $\delta_{A,f}$}
160
161To compute the modular degree we require
162$H_1(X_0(N),\Z;\mbox{\rm cusps})$ together with the action
163of the Hecke ring $\T$.
164Using modular symbols (see, e.g., Cremona~\cite{cremona:algs}), we
165first compute $H_1(X_0(N),\Q;\mbox{\rm cusps})$.
166Using lattice reduction, we compute the $\Z$-submodule generated
167by all Manin symbols $(c,d)$; then $H_1(X_0(N),\Z)$ is the
168{\em integer} kernel of the boundary map.  We have a natural restriction
169map
170$$171\Hom(H_1(X_0(N),\Z),\Z)[I_f] \rightarrow 172 \Hom(H_1(X_0(N),\Z)[I_f],\Z) 173$$
174where $t \in \T$ acts on $\varphi \in \Hom(H_1(X_0(N),\Z),\Z)$ by
175$(t.\varphi)(x) = \varphi(tx)$.
176By considering $J_0(N)$, $A_f$, and $A_f^{\vee}$
177 as complex lattices, and applying the snake lemma,
178we see that the order of the cokernel of this map is the square of the
179modular degree $\delta_f$.
180
181\subsection{Computing the character group $\cX_{J,p}$}
182
183When $N=Mp$, with~$M$ small, the algorithm of Mestre and
184Oesterl\'e~\cite{mestre:graphs} can be used.  This algorithm
185constructs the graph of isogenies between $\Fpbar$-isomorphism
186classes of pairs consisting of a supersingular elliptic curve
187and an $M$-torsion subgroup.  In particular, the
188method is elementary to apply when $X_0(M)$ is of genus~$0$.
189
190In general, the above category of enhanced'' supersingular
191elliptic curves can be replaced
192by one of left (or right) ideals of a quaternion order~$\cO$ of
193level~$M$ in the quaternion algebra over~$\Q$ ramified at~$p$.
194This gives an equivalent category, in which the computation of
195homomorphisms is efficient.  The character group $\cX_{J,p}$ is
196known by Deligne-Rapoport~\cite{deligne-rapoport} to be canonically
197equivalent to the degree zero subgroup $\cX(\cO)$ of the free
198abelian divisor group'' on the isomorphism classes of enhanced
199supersingular  elliptic curves and of quaternion ideals.  Moreover,
200this isomorphism  is compatible with the operation of Hecke operators,
201which are  effectively computable in $\cX(\cO)$ in terms of ideal
202homomorphisms.
203
204The inner product of two classes in this setting is defined
205to be the number of isomorphisms between any two representatives.
206The linear extension to $\cX(\cO)$ gives an inner product which
207agrees, under the isomophism, with the monodromy pairing on
208$\cX_{J,p}$.  This gives, in particular, an isomorphism $\Phi_{J,p} 209\isom \coker(\cX(\cO) \rightarrow \Hom(\cX(\cO),\Z))$, and an
210effective means of computing $\#\coker(\alpha_f)$ and
211$\#(\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))$.
212
213The ideal arithmetic of quaternions has been implemented in
214{\sc Magma} \cite{magma} by the first author.  Additional details and
215the application to Shimura curves, generalizing $X_0(N)$, can
216be found in Kohel~\cite{kohel}.
217
218\subsection{The Galois action on $\Phi_{A_f,p}$}
219
220To determine the Galois action on $\Phi_{A_f,p}$, we need only
221know the action of the Frobenius automorphism $\Frob_p$.  However
222$\Frob_p$ is known to equal $-W_p$ where $W_p$ is the $p$th
223Atkin-Lehner involution, which can be computed using modular symbols.
224Since~$f$ is an eigenform, the involution $W_p$ acts as either
225$+1$ or $-1$ on $\Phi_{A_f,p}$.
226Moreover, the operator $W_p$ is determined by an involution on
227the set of quaternion ideals, so can be determined explicitly
228on the character group.
229
230\section{Tables}
231
232\subsection{Component groups at low level}
233
234Table~\ref{tbl:lowlevel} gives the component groups of the quotients
235$A_f$ of $J_0(N)$ for $N\leq 127$; it was
236computed using a {\sc Magma} implementation of an
237algorithm based on Theorem~\ref{thm:main}.  The column~$d$ is
238the dimension, and the column $\#\Phi_{A_f,p}$ contains a list
239of orders of component groups, one for each divisor~$p$ of~$N$.
240An order is starred if the $\Gal(\Fpbar/\Fp)$-action is nontrivial.
241More data along these lines can be obtained from the second author.
242
243\begin{table}
244\begin{center}
245\caption{Component groups at low level}
246\end{center}
247\label{tbl:lowlevel}
248$$249\begin{array}{lcl} 250 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 25111 & 1 & 5\\ 25214 & 1 & 6^*,3\\ 25315 & 1 & 4^*,4\\ 25417 & 1 & 4\\ 25519 & 1 & 3\\ 25620 & 1 & ?,2^*\\ 25721 & 1 & 4,2^*\\ 25823 & 2 & 11\\ 25924 & 1 & ?,2^*\\ 26026 & 1 & 3^*,3\\ 261 & 1 & 7,1^*\\ 26227 & 1 & ?\\ 26329 & 2 & 7\\ 26430 & 1 & 4^*,3,1^*\\ 26531 & 2 & 5\\ 26632 & 1 & ?\\ 26733 & 1 & 6^*,2\\ 26834 & 1 & 6,1^*\\ 26935 & 1 & 3^*,3\\ 270 & 2 & 8,4^*\\ 27136 & 1 & ?,?\\ 27237 & 1 & 1^*\\ 273 & 1 & 3\\ 27438 & 1 & 9^*,3\\ 275 & 1 & 5,1^*\\ 27639 & 1 & 2^*,2\\ 277 & 2 & 14,2^*\\ 27840 & 1 & ?,2\\ 27941 & 3 & 10\\ 28042 & 1 & 8,2^*,1^*\\ 28143 & 1 & 1^*\\ 282 & 2 & 7\\ 28344 & 1 & ?,1^*\\ 28445 & 1 & ?,1^*\\ 28546 & 1 & 10^*,1\\ 28647 & 4 & 23\\ 28748 & 1 & ?,2\\ 28849 & 1 & ?\\ 28950 & 1 & 1^*,?\\ 290 & 1 & 5,?\\ 29151 & 1 & 3,1^*\\ 292 & 2 & 16^*,4\\ 29352 & 1 & ?,2^*\\ 29453 & 1 & 1^*\\ 295\end{array}\quad 296\begin{array}{lcl} 297 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 298 & 3 & 13\\ 29954 & 1 & 3^*,?\\ 300 & 1 & 3,?\\ 30155 & 1 & 2,2^*\\ 302 & 2 & 14^*,2\\ 30356 & 1 & ?,1\\ 304 & 1 & ?,1^*\\ 30557 & 1 & 2^*,1^*\\ 306 & 1 & 2,2^*\\ 307 & 1 & 10,1^*\\ 30858 & 1 & 2^*,1^*\\ 309 & 1 & 10,1^*\\ 31059 & 5 & 29\\ 31161 & 1 & 1^*\\ 312 & 3 & 5\\ 31362 & 1 & 4,1^*\\ 314 & 2 & 66^*,3\\ 31563 & 1 & ?,1^*\\ 316 & 2 & ?,3\\ 31764 & 1 & ?\\ 31865 & 1 & 1^*,1^*\\ 319 & 2 & 3^*,3\\ 320 & 2 & 7,1^*\\ 32166 & 1 & 2^*,3,1^*\\ 322 & 1 & 4,1^*,1^*\\ 323 & 1 & 10,5,1\\ 32467 & 1 & 1\\ 325 & 2 & 1^*\\ 326 & 2 & 11\\ 32768 & 2 & ?,2^*\\ 32869 & 1 & 2,1^*\\ 329 & 2 & 22^*,2\\ 33070 & 1 & 4,2^*,1^*\\ 33171 & 3 & 5\\ 332 & 3 & 7\\ 33372 & 1 & ?,?\\ 33473 & 1 & 2\\ 335 & 2 & 1^*\\ 336 & 2 & 3\\ 33774 & 2 & 9^*,3\\ 338 & 2 & 95,1^*\\ 33975 & 1 & 1^*,?\\ 340 & 1 & 1,?\\ 341 & 1 & 5,?\\ 342\end{array}\quad 343\begin{array}{lcl} 344 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 34576 & 1 & ?,1^*\\ 34677 & 1 & 2^*,1^*\\ 347 & 1 & 3^*,2\\ 348 & 1 & 6,3^*\\ 349 & 2 & 2,2^*\\ 35078 & 1 & 16^*,5^*,1\\ 35179 & 1 & 1^*\\ 352 & 5 & 13\\ 35380 & 1 & ?,2\\ 354 & 1 & ?,2^*\\ 35581 & 2 & ?\\ 35682 & 1 & 2^*,1^*\\ 357 & 2 & 28,1^*\\ 35883 & 1 & 1^*\\ 359 & 6 & 41\\ 36084 & 1 & ?,1^*,2^*\\ 361 & 1 & ?,3,2\\ 36285 & 1 & 2^*,1\\ 363 & 2 & 2^*,1^*\\ 364 & 2 & 6,1^*\\ 36586 & 2 & 21^*,3\\ 366 & 2 & 55,1^*\\ 36787 & 2 & 5,1^*\\ 368 & 3 & 92^*,4\\ 36988 & 1 & ?,1^*\\ 370 & 2 & ?,2^*\\ 37189 & 1 & 1^*\\ 372 & 1 & 2\\ 373 & 5 & 11\\ 37490 & 1 & 2^*,?,3\\ 375 & 1 & 6,?,1^*\\ 376 & 1 & 4,?,1\\ 37791 & 1 & 1^*,1^*\\ 378 & 1 & 1,1\\ 379 & 2 & 7,1^*\\ 380 & 3 & 4^*,8\\ 38192 & 1 & ?,1^*\\ 382 & 1 & ?,1\\ 38393 & 2 & 4^*,1^*\\ 384 & 3 & 64,2^*\\ 38594 & 1 & 2,1^*\\ 386 & 2 & 94^*,1\\ 38795 & 3 & 10,2^*\\ 388 & 4 & 54^*,6\\ 389\end{array}\quad 390\begin{array}{lcl} 391 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 39296 & 1 & ?,2\\ 393 & 1 & ?,2^*\\ 39497 & 3 & 1^*\\ 395 & 4 & 8\\ 39698 & 1 & 2^*,?\\ 397 & 2 & 14,?\\ 39899 & 1 & ?,1^*\\ 399 & 1 & ?,1\\ 400 & 1 & ?,1^*\\ 401 & 1 & ?,1^*\\ 402100 & 1 & ?,?\\ 403101 & 1 & 1^*\\ 404 & 7 & 25\\ 405102 & 1 & 2^*,2^*,1^*\\ 406 & 1 & 6^*,6,1^*\\ 407 & 1 & 8,4,1\\ 408103 & 2 & 1^*\\ 409 & 6 & 17\\ 410104 & 1 & ?,1^*\\ 411 & 2 & ?,2\\ 412105 & 1 & 1,1,1\\ 413 & 2 & 10^*,2^*,2\\ 414106 & 1 & 4^*,1^*\\ 415 & 1 & 5^*,1\\ 416 & 1 & 24,1^*\\ 417 & 1 & 3,1^*\\ 418107 & 2 & 1^*\\ 419 & 7 & 53\\ 420108 & 1 & ?,?\\ 421109 & 1 & 1\\ 422 & 3 & 1^*\\ 423 & 4 & 9\\ 424110 & 1 & 7^*,1^*,3\\ 425 & 1 & 3,1^*,1^*\\ 426 & 1 & 5,5,1\\ 427 & 2 & 16^*,3,1^*\\ 428111 & 3 & 10^*,2\\ 429 & 4 & 266,2^*\\ 430112 & 1 & ?,1^*\\ 431 & 1 & ?,1\\ 432 & 1 & ?,1^*\\ 433113 & 1 & 2\\ 434 & 2 & 2\\ 435 & 3 & 1^*\\ 436\end{array}\quad 437\begin{array}{lcl} 438 N & \, d \, & \, \#\Phi_{A_f,p}\, \\ 439 & 3 & 7\\ 440114 & 1 & 2^*,5^*,1\\ 441 & 1 & 20,3^*,1^*\\ 442 & 1 & 6,3,1\\ 443115 & 1 & 5^*,1\\ 444 & 2 & 4^*,1^*\\ 445 & 4 & 32,4^*\\ 446116 & 1 & ?,1^*\\ 447 & 1 & ?,2^*\\ 448 & 1 & ?,1^*\\ 449117 & 1 & ?,1\\ 450 & 2 & ?,3\\ 451 & 2 & ?,1^*\\ 452118 & 1 & 2^*,1^*\\ 453 & 1 & 19^*,1\\ 454 & 1 & 10,1^*\\ 455 & 1 & 1,1^*\\ 456119 & 4 & 9,3^*\\ 457 & 5 & 48^*,8\\ 458120 & 1 & ?,1,1^*\\ 459 & 1 & ?,2,1\\ 460121 & 1 & ?\\ 461 & 1 & ?\\ 462 & 1 & ?\\ 463 & 1 & ?\\ 464122 & 1 & 4^*,1^*\\ 465 & 2 & 39^*,3\\ 466 & 3 & 248,1^*\\ 467123 & 1 & 1^*,1^*\\ 468 & 1 & 5,1\\ 469 & 2 & 7,1^*\\ 470 & 3 & 184^*,4\\ 471124 & 1 & ?,1^*\\ 472 & 1 & ?,1\\ 473125 & 2 & ?\\ 474 & 2 & ?\\ 475 & 4 & ?\\ 476126 & 1 & 8^*,?,1^*\\ 477 & 1 & 2,?,1\\ 478127 & 3 & 1^*\\ 479 & 7 & 21\\ 480&&\\ 481&&\\ 482&&\\ 483\end{array}$$
484\end{table}
485
486
487
488\subsection{Some large component groups}
489
490Let $\Omega_{A_f}$ be the real period of $A_f$, as defined by
491Tate~\cite{tate:bsd} in his extension of the Birch  and Swinnerton-Dyer
492conjecture to dimension greater than one.   The second author computed
493$L(A_f,1)/\Omega_{A_f}$ for every newform~$f$ of level $N\leq 1500$.
494The four ratios with the largest prime divisors in the numerator
495are given in Table~\ref{table:lratios}.
496The Birch and Swinnerton-Dyer conjecture predicts that the large
497prime divisors of the numerators of these special values must divide
498either the order of some $\Phi_{A_f,p}$ or of the Shafarevich-Tate
499group of~$A_f$.  In each instance this is the case.
500
501\begin{table}
502\label{table:lratios}
503\begin{center}
504\caption{Large $L(A_f,1)/\Omega_{A_f}$}
505\end{center}
506$$\begin{array}{ccll} 507 \qquad N \qquad\quad & 508 \quad \dim \quad & 509 \quad\quad L(A_f,1)/\Omega_{A_f} \qquad\qquad & 510 \qquad \#\Phi_{A_f,p} \\ 511 1154=2\tdot 577 & 20 & 2^?\tdot85495047371/17^2 512 & 2^?\tdot 17^2 \tdot 85495047371, 2^? \\ 513 1238=2\tdot 619 & 19 & 2^?\tdot 7553329019/5\tdot 31 514 & 2^?\tdot 5\tdot31\tdot7553329019 , 2^?\\ 515 1322=2\tdot 661 & 21 & 2^?\tdot 57851840099/331 516 & 2^?\tdot 331 \tdot 57851840099, 2^?\\ 517 1382=2\tdot 691 & 20 & 2^?\tdot 37 \tdot 1864449649 /173 518 & 2^?\tdot 37\tdot173\tdot 1864449649, 2^?\\ 519 1478=2\tdot 739 & 20 & 520 2^?\tdot 7\tdot 29\tdot 1183045463 / 5\tdot 37 521 & 2^?\tdot5\tdot 7\tdot29\tdot37\tdot1183045463, 2^?\\ 522\end{array}$$
523\end{table}
524
525\section{Further directions}
526
527Further considerations are needed to compute the {\em group}
528structure of $\Phi_{A_f,p}$.
529Once the group structure is determined, however, since the
530action of Frobenius is known, the structure as a group scheme
531is immediate.
532
533An equivalence with quaternion divisor groups is not known to
534hold for the character group at~$p$ when $p^2$ divides~$N$.
535Thus our methods say nothing about the component group at primes
536whose {\em square} divides the level.  The free abelian group
537on classes of nonmaximal
538orders  of index~$p$ at a ramified prime give a well-defined
539divisor group.  Do the resulting Hecke modules determine the
540component groups for quotients of level $p^2M$?
541
542Is it possible to define quantities as in Theorem~\ref{thm:main}
543even when the weight of~$f$ is {\em greater than~$2$}?
544If so, how are the resulting quantities related to the Bloch-Kato
545Tamagawa  numbers (see~\cite{bloch-kato}) of the higher weight motive
546attached to~$f$?
547
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630\end{thebibliography}
631
632\end{document}
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