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\begin{document}

\title{Component Groups of Quotients of $J_0(N)$}
\titlerunning{Component Groups}
\author{David Kohel\inst{1} \and William A. Stein\inst{2}}
\authorrunning{Kohel \and Stein}
\tocauthor{David Kohel (University of Sydney),
William A. Stein (University of California at Berkeley)}
%
\institute{University of Sydney\\
\email{kohel@maths.usyd.edu.au}\\
\texttt{http://www.maths.usyd.edu.au:8000/u/kohel/}
\and
University of California at Berkeley,\\
\email{was@math.berkeley.edu}\\
\texttt{http://shimura.math.berkeley.edu/\~{}was}
}

\maketitle
\begin{abstract}
Let~$f$ be a newform of weight~$2$ on $\Gamma_0(N)$, and let~$A_f$ be
the corresponding optimal abelian variety quotient of $J_0(N)$.  We
describe an algorithm to compute the order of the component group of
$A_f$ at primes~$p$ that exactly divide~$N$.  We give a table of
orders of component groups for all~$f$ of level $N\leq 127$ and four
examples in which the component group is very large, as predicted by
the Birch and Swinnerton-Dyer conjecture.
\end{abstract}

\section{Introduction}

Let $X_0(N)$ be the Riemann surface obtained by compactifying the 
quotient of the upper half-plane by the action of $\Gamma_0(N)$.  
Then $X_0(N)$ has a canonical structure of algebraic curve 
over~$\Q$; denote its Jacobian by $J_0(N)$.  It is equipped with 
an action of a commutative ring $\T=\Z[\ldots T_n \ldots]$ of 
Hecke operators.  For more details on modular curves, Hecke operators,
and modular forms see, e.g.,~\cite{diamond-im}.

Now suppose that~$f=\sum_{n=1}^{\infty} a_n q^n$ is a modular newform 
of weight~$2$ for the congruence subgroup $\Gamma_0(N)$. 
The Hecke operators also act on~$f$ by $T_n(f) = a_n f$.  
The eigenvalues $a_n$ generate an order $R_f = \Z[\ldots a_n \ldots]$ 
in a number field $K_f$.  Let $I_f$ be the kernel ideal of the map 
$\T \rta R_f$ sending $T_n$ to~$a_n$.  
Following Shimura~\cite{shimura:factors}, we associate to~$f$ the 
quotient $A_f = J_0(N)/I_f J_0(N)$ of 
$J_0(N)$.  Then $A_f$ is an abelian variety over~$\Q$ of dimension 
$[K_f:\Q]$, with bad reduction exactly at the primes dividing~$N$. 

One dimensional quotients of $J_0(N)$ have been intensely studied 
in recent years, both computationally and theoretically.  
The original conjectures of Birch and 
Swinnerton-Dyer~\cite{birch-swd-I,birch-swd-II}, 
which concern elliptic curves,
were greatly influenced by computations.  
The scale of these computations was extended and systematized by 
Cremona in~\cite{cremona:algs}.  

In another direction, Wiles~\cite{wiles:fermat} and 
Taylor-Wiles~\cite{taylor-wiles} proved a special case of the
conjecture of Shimura-Taniyama, which asserts that every elliptic
curve over~$\Q$ is a quotient of some $J_0(N)$; this allowed them to establish
Fermat's Last Theorem.  The full Shimura-Taniyama 
conjecture was later proved by Breuil, Conrad, Diamond, 
and Taylor (see~\cite{breuil-conrad-diamond-taylor}). 
This illustrates the central roll played by quotients of $J_0(N)$.

\section{Component Groups of $A_f$}

The N\'eron model $\cA/\Z$ of an abelian variety $A/\Q$ is by definition
a smooth commutative group scheme over~$\Z$ with generic fiber~$A$ 
such that for any smooth scheme~$S$ over~$\Z$, the restriction map 
$$\Hom_\Z(S,\cA) \rta \Hom_\Q(S_\Q,A)$$ is a bijection.  
For more details, including a proof of existence, see, 
e.g.,~\cite{neronmodels}.

Suppose that $A_f$ is a quotient of $J_0(N)$ corresponding to a newform~$f$, 
and let $\cA_f$ be the N\'eron model of~$A_f$.  For any prime divisor~$p$ 
of~$N$, the closed fiber~${\cA_f}_{/\Fp}$ is a group scheme 
over~$\Fp$, which need not be connected.  Denote the connected 
component of the identity by~${\cA^{\circ}_f}_{/\Fp}$.  
There is an exact sequence
$$
0 \rightarrow {\cA^{\circ}_f}_{/\Fp}
  \rightarrow {\cA_f}_{/\Fp}\rightarrow\Phi_{A_f,p}
  \rightarrow 0
$$
with $\Phi_{A_f,p}$ a finite \'{e}tale group scheme over~$\Fp$
called the {\em component group} of $A_f$ at~$p$.  

We note that a finite \'etale group scheme is equivalent to a
finite abelian group together with an action of $\Gal(\Fpbar/\Fp)$.
In this paper we describe an algorithm for computing the order of 
this group. 

\section{The Algorithm}

Let~$J = J_0(N)$.  As the Jacobian of a curve, $J$ is canonically 
self-dual, so the projection $J \rta A_f$ induces a polarization 
$A_f^{\vee} \rta A_f$, where $A_f^{\vee}$ denotes the abelian variety
dual of $A_f$.    The {\em modular degree} $\delta_{A_f}$ 
is the positive square root of the degree of this polarization.

Let $T_{J,p}$ be the toric part of the closed fiber of the 
N\'eron model of $J$ at~$p$, and define $\cX_{J,p}$ to 
be the group $\Hom_{\Fpbar}(T_{J,p},\G_m)$ of characters of $T_{J,p}$.  
Then $\cX_{J,p}$ is a free abelian group equipped with an action 
of both~$\T$ and $\Gal(\Fpbar/\Fp)$ 
(see, e.g., Ribet~\cite{ribet:modreps}); there is a bilinear 
pairing 
$$
\langle\,,\,\rangle : \cX_{J,p} \times \cX_{J,p} \rightarrow \Z
$$
called the {\em monodromy pairing} such that 
$$
\Phi_{J,p} \isom \coker(\cX_{J,p} \rightarrow \Hom(\cX_{J,p},\Z)).
$$
Let $\cX_{J,p}[I_f]$ be the intersection of all kernels $\ker(t)$ for~$t$ 
in $I_f$, and let   
$$
\alpha_f: \cX_{J,p} \rightarrow \Hom(\cX_{J,p}[I_f],\Z)
$$
be the map induced by the monodromy pairing.  
The following theorem of Stein~\cite{stein:phd}, provides the 
basis for the computation of orders of component groups.

\begin{theorem}\label{thm:main}
$$
\#\Phi_{A_f,p} 
   = \frac{\#\coker(\alpha_f) \cdot \delta_{A_f}}
          {\# (\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))}\,.
$$
\end{theorem}

\subsection{Computing the modular degree $\delta_{A,f}$}

To compute the modular degree we require 
$H_1(X_0(N),\Z;\mbox{\rm cusps})$ together with the action 
of the Hecke ring $\T$.  
Using modular symbols (see, e.g., Cremona~\cite{cremona:algs}), we 
first compute $H_1(X_0(N),\Q;\mbox{\rm cusps})$.  
Using lattice reduction, we compute the $\Z$-submodule generated 
by all Manin symbols $(c,d)$; then $H_1(X_0(N),\Z)$ is the 
{\em integer} kernel of the boundary map.  We have a natural restriction
map
$$
\Hom(H_1(X_0(N),\Z),\Z)[I_f] \rightarrow 
   \Hom(H_1(X_0(N),\Z)[I_f],\Z)
$$
where $t \in \T$ acts on $\varphi \in \Hom(H_1(X_0(N),\Z),\Z)$ by
$(t.\varphi)(x) = \varphi(tx)$.  
By considering $J_0(N)$, $A_f$, and $A_f^{\vee}$
 as complex lattices, and applying the snake lemma,
we see that the order of the cokernel of this map is the square of the
modular degree $\delta_f$.

\subsection{Computing the character group $\cX_{J,p}$}

When $N=Mp$, with~$M$ small, the algorithm of Mestre and 
Oesterl\'e~\cite{mestre:graphs} can be used.  This algorithm 
constructs the graph of isogenies between $\Fpbar$-isomorphism
classes of pairs consisting of a supersingular elliptic curve
and an $M$-torsion subgroup.  In particular, the 
method is elementary to apply when $X_0(M)$ is of genus~$0$.   

In general, the above category of ``enhanced'' supersingular
elliptic curves can be replaced 
by one of left (or right) ideals of a quaternion order~$\cO$ of 
level~$M$ in the quaternion algebra over~$\Q$ ramified at~$p$. 
This gives an equivalent category, in which the computation of 
homomorphisms is efficient.  The character group $\cX_{J,p}$ is 
known by Deligne-Rapoport~\cite{deligne-rapoport} to be canonically 
equivalent to the degree zero subgroup $\cX(\cO)$ of the free 
abelian ``divisor group'' on the isomorphism classes of enhanced
supersingular  elliptic curves and of quaternion ideals.  Moreover, 
this isomorphism  is compatible with the operation of Hecke operators, 
which are  effectively computable in $\cX(\cO)$ in terms of ideal 
homomorphisms. 
 
The inner product of two classes in this setting is defined 
to be the number of isomorphisms between any two representatives. 
The linear extension to $\cX(\cO)$ gives an inner product which 
agrees, under the isomophism, with the monodromy pairing on 
$\cX_{J,p}$.  This gives, in particular, an isomorphism $\Phi_{J,p} 
\isom \coker(\cX(\cO) \rightarrow \Hom(\cX(\cO),\Z))$, and an 
effective means of computing $\#\coker(\alpha_f)$ and 
$\#(\alpha_f(\cX_{J,p})/\alpha_f(\cX_{J,p}[I_f]))$.

The ideal arithmetic of quaternions has been implemented in 
{\sc Magma} \cite{magma} by the first author.  Additional details and 
the application to Shimura curves, generalizing $X_0(N)$, can 
be found in Kohel~\cite{kohel}.

\subsection{The Galois action on $\Phi_{A_f,p}$}

To determine the Galois action on $\Phi_{A_f,p}$, we need only 
know the action of the Frobenius automorphism $\Frob_p$.  However 
$\Frob_p$ is known to equal $-W_p$ where $W_p$ is the $p$th 
Atkin-Lehner involution, which can be computed using modular symbols.
Since~$f$ is an eigenform, the involution $W_p$ acts as either 
$+1$ or $-1$ on $\Phi_{A_f,p}$.
Moreover, the operator $W_p$ is determined by an involution on 
the set of quaternion ideals, so can be determined explicitly 
on the character group. 

\section{Tables}

\subsection{Component groups at low level}

Table~\ref{tbl:lowlevel} gives the component groups of the quotients
$A_f$ of $J_0(N)$ for $N\leq 127$; it was 
computed using a {\sc Magma} implementation of an 
algorithm based on Theorem~\ref{thm:main}.  The column~$d$ is 
the dimension, and the column $\#\Phi_{A_f,p}$ contains a list 
of orders of component groups, one for each divisor~$p$ of~$N$. 
An order is starred if the $\Gal(\Fpbar/\Fp)$-action is nontrivial.
More data along these lines can be obtained from the second author.

\begin{table}
\begin{center}
\caption{Component groups at low level}
\end{center}
\label{tbl:lowlevel}
$$
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
11 & 1 & 5\\
14 & 1 & 6^*,3\\
15 & 1 & 4^*,4\\
17 & 1 & 4\\
19 & 1 & 3\\
20 & 1 & ?,2^*\\
21 & 1 & 4,2^*\\
23 & 2 & 11\\
24 & 1 & ?,2^*\\
26 & 1 & 3^*,3\\
 & 1 & 7,1^*\\
27 & 1 & ?\\
29 & 2 & 7\\
30 & 1 & 4^*,3,1^*\\
31 & 2 & 5\\
32 & 1 & ?\\
33 & 1 & 6^*,2\\
34 & 1 & 6,1^*\\
35 & 1 & 3^*,3\\
   & 2 & 8,4^*\\
36 & 1 & ?,?\\
37 & 1 & 1^*\\
   & 1 & 3\\
38 & 1 & 9^*,3\\
   & 1 & 5,1^*\\
39 & 1 & 2^*,2\\
   & 2 & 14,2^*\\
40 & 1 & ?,2\\
41 & 3 & 10\\
42 & 1 & 8,2^*,1^*\\
43 & 1 & 1^*\\
   & 2 & 7\\
44 & 1 & ?,1^*\\
45 & 1 & ?,1^*\\
46 & 1 & 10^*,1\\
47 & 4 & 23\\
48 & 1 & ?,2\\
49 & 1 & ?\\
50 & 1 & 1^*,?\\
   & 1 & 5,?\\
51 & 1 & 3,1^*\\
   & 2 & 16^*,4\\
52 & 1 & ?,2^*\\
53 & 1 & 1^*\\
\end{array}\quad
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
 & 3 & 13\\
54 & 1 & 3^*,?\\
 & 1 & 3,?\\
55 & 1 & 2,2^*\\
 & 2 & 14^*,2\\
56 & 1 & ?,1\\
 & 1 & ?,1^*\\
57 & 1 & 2^*,1^*\\
 & 1 & 2,2^*\\
 & 1 & 10,1^*\\
58 & 1 & 2^*,1^*\\
 & 1 & 10,1^*\\
59 & 5 & 29\\
61 & 1 & 1^*\\
 & 3 & 5\\
62 & 1 & 4,1^*\\
 & 2 & 66^*,3\\
63 & 1 & ?,1^*\\
 & 2 & ?,3\\
64 & 1 & ?\\
65 & 1 & 1^*,1^*\\
 & 2 & 3^*,3\\
 & 2 & 7,1^*\\
66 & 1 & 2^*,3,1^*\\
 & 1 & 4,1^*,1^*\\
 & 1 & 10,5,1\\
67 & 1 & 1\\
 & 2 & 1^*\\
 & 2 & 11\\
68 & 2 & ?,2^*\\
69 & 1 & 2,1^*\\
 & 2 & 22^*,2\\
70 & 1 & 4,2^*,1^*\\
71 & 3 & 5\\
 & 3 & 7\\
72 & 1 & ?,?\\
73 & 1 & 2\\
 & 2 & 1^*\\
 & 2 & 3\\
74 & 2 & 9^*,3\\
 & 2 & 95,1^*\\
75 & 1 & 1^*,?\\
 & 1 & 1,?\\
 & 1 & 5,?\\
\end{array}\quad
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
76 & 1 & ?,1^*\\
77 & 1 & 2^*,1^*\\
 & 1 & 3^*,2\\
 & 1 & 6,3^*\\
 & 2 & 2,2^*\\
78 & 1 & 16^*,5^*,1\\
79 & 1 & 1^*\\
 & 5 & 13\\
80 & 1 & ?,2\\
 & 1 & ?,2^*\\
81 & 2 & ?\\
82 & 1 & 2^*,1^*\\
 & 2 & 28,1^*\\
83 & 1 & 1^*\\
 & 6 & 41\\
84 & 1 & ?,1^*,2^*\\
 & 1 & ?,3,2\\
85 & 1 & 2^*,1\\
 & 2 & 2^*,1^*\\
 & 2 & 6,1^*\\
86 & 2 & 21^*,3\\
 & 2 & 55,1^*\\
87 & 2 & 5,1^*\\
 & 3 & 92^*,4\\
88 & 1 & ?,1^*\\
 & 2 & ?,2^*\\
89 & 1 & 1^*\\
 & 1 & 2\\
 & 5 & 11\\
90 & 1 & 2^*,?,3\\
 & 1 & 6,?,1^*\\
 & 1 & 4,?,1\\
91 & 1 & 1^*,1^*\\
 & 1 & 1,1\\
 & 2 & 7,1^*\\
 & 3 & 4^*,8\\
92 & 1 & ?,1^*\\
 & 1 & ?,1\\
93 & 2 & 4^*,1^*\\
 & 3 & 64,2^*\\
94 & 1 & 2,1^*\\
 & 2 & 94^*,1\\
95 & 3 & 10,2^*\\
 & 4 & 54^*,6\\
\end{array}\quad
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
96 & 1 & ?,2\\
 & 1 & ?,2^*\\
97 & 3 & 1^*\\
 & 4 & 8\\
98 & 1 & 2^*,?\\
 & 2 & 14,?\\
99 & 1 & ?,1^*\\
 & 1 & ?,1\\
 & 1 & ?,1^*\\
 & 1 & ?,1^*\\
100 & 1 & ?,?\\
101 & 1 & 1^*\\
 & 7 & 25\\
102 & 1 & 2^*,2^*,1^*\\
 & 1 & 6^*,6,1^*\\
 & 1 & 8,4,1\\
103 & 2 & 1^*\\
 & 6 & 17\\
104 & 1 & ?,1^*\\
 & 2 & ?,2\\
105 & 1 & 1,1,1\\
 & 2 & 10^*,2^*,2\\
106 & 1 & 4^*,1^*\\
 & 1 & 5^*,1\\
 & 1 & 24,1^*\\
 & 1 & 3,1^*\\
107 & 2 & 1^*\\
 & 7 & 53\\
108 & 1 & ?,?\\
109 & 1 & 1\\
 & 3 & 1^*\\
 & 4 & 9\\
110 & 1 & 7^*,1^*,3\\
 & 1 & 3,1^*,1^*\\
 & 1 & 5,5,1\\
 & 2 & 16^*,3,1^*\\
111 & 3 & 10^*,2\\
 & 4 & 266,2^*\\
112 & 1 & ?,1^*\\
 & 1 & ?,1\\
 & 1 & ?,1^*\\
113 & 1 & 2\\
 & 2 & 2\\
 & 3 & 1^*\\
\end{array}\quad
\begin{array}{lcl}
 N & \, d \, & \, \#\Phi_{A_f,p}\, \\
 & 3 & 7\\
114 & 1 & 2^*,5^*,1\\
 & 1 & 20,3^*,1^*\\
 & 1 & 6,3,1\\
115 & 1 & 5^*,1\\
 & 2 & 4^*,1^*\\
 & 4 & 32,4^*\\
116 & 1 & ?,1^*\\
 & 1 & ?,2^*\\
 & 1 & ?,1^*\\
117 & 1 & ?,1\\
 & 2 & ?,3\\
 & 2 & ?,1^*\\
118 & 1 & 2^*,1^*\\
 & 1 & 19^*,1\\
 & 1 & 10,1^*\\
 & 1 & 1,1^*\\
119 & 4 & 9,3^*\\
 & 5 & 48^*,8\\
120 & 1 & ?,1,1^*\\
 & 1 & ?,2,1\\
121 & 1 & ?\\
 & 1 & ?\\
 & 1 & ?\\
 & 1 & ?\\
122 & 1 & 4^*,1^*\\
 & 2 & 39^*,3\\
 & 3 & 248,1^*\\
123 & 1 & 1^*,1^*\\
 & 1 & 5,1\\
 & 2 & 7,1^*\\
 & 3 & 184^*,4\\
124 & 1 & ?,1^*\\
 & 1 & ?,1\\
125 & 2 & ?\\
 & 2 & ?\\
 & 4 & ?\\
126 & 1 & 8^*,?,1^*\\
 & 1 & 2,?,1\\
127 & 3 & 1^*\\
 & 7 & 21\\
&&\\
&&\\
&&\\
\end{array}$$
\end{table}



\subsection{Some large component groups}

Let $\Omega_{A_f}$ be the real period of $A_f$, as defined by
Tate~\cite{tate:bsd} in his extension of the Birch  and Swinnerton-Dyer 
conjecture to dimension greater than one.   The second author computed 
$L(A_f,1)/\Omega_{A_f}$ for every newform~$f$ of level $N\leq 1500$.  
The four ratios with the largest prime divisors in the numerator 
are given in Table~\ref{table:lratios}.  
The Birch and Swinnerton-Dyer conjecture predicts that the large 
prime divisors of the numerators of these special values must divide 
either the order of some $\Phi_{A_f,p}$ or of the Shafarevich-Tate 
group of~$A_f$.  In each instance this is the case.

\begin{table}
\label{table:lratios}
\begin{center}
\caption{Large $L(A_f,1)/\Omega_{A_f}$}
\end{center}
$$\begin{array}{ccll}
 \qquad N \qquad\quad &
 \quad \dim \quad & 
 \quad\quad L(A_f,1)/\Omega_{A_f} \qquad\qquad & 
 \qquad \#\Phi_{A_f,p} \\
       1154=2\tdot 577 & 20 & 2^?\tdot85495047371/17^2 
      & 2^?\tdot 17^2 \tdot 85495047371,  2^? \\
       1238=2\tdot 619  & 19 & 2^?\tdot 7553329019/5\tdot 31 
      & 2^?\tdot 5\tdot31\tdot7553329019 ,  2^?\\
       1322=2\tdot 661  & 21 & 2^?\tdot 57851840099/331 
      & 2^?\tdot 331 \tdot 57851840099, 2^?\\
       1382=2\tdot 691  & 20 & 2^?\tdot 37 \tdot 1864449649 /173
      & 2^?\tdot 37\tdot173\tdot 1864449649, 2^?\\
       1478=2\tdot 739  & 20 & 
            2^?\tdot 7\tdot 29\tdot 1183045463 / 5\tdot 37 
      & 2^?\tdot5\tdot 7\tdot29\tdot37\tdot1183045463, 2^?\\
\end{array}$$
\end{table}

\section{Further directions}

Further considerations are needed to compute the {\em group} 
structure of $\Phi_{A_f,p}$.
Once the group structure is determined, however, since the 
action of Frobenius is known, the structure as a group scheme 
is immediate. 

An equivalence with quaternion divisor groups is not known to 
hold for the character group at~$p$ when $p^2$ divides~$N$.  
Thus our methods say nothing about the component group at primes 
whose {\em square} divides the level.  The free abelian group
on classes of nonmaximal 
orders  of index~$p$ at a ramified prime give a well-defined 
divisor group.  Do the resulting Hecke modules determine the 
component groups for quotients of level $p^2M$? 

Is it possible to define quantities as in Theorem~\ref{thm:main}
even when the weight of~$f$ is {\em greater than~$2$}?
If so, how are the resulting quantities related to the Bloch-Kato 
Tamagawa  numbers (see~\cite{bloch-kato}) of the higher weight motive 
attached to~$f$?

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\end{thebibliography}

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