Sharedwww / Tables / modular_degree.texOpen in CoCalc
Author: William A. Stein
1% denominator.tex
2\documentclass[11pt]{article}
3\include{macros}
4\title{The Kernel of the Modular Polarization of a Quotient of $J_0(N)$}
5\begin{document}
6\maketitle
7\tableofcontents
8\section{The Modular Polarization}
9Suppose $A$ is a quotient of $J_0(N)$.
10Because $J_0(N)$ is a Jacobian it possesses a canonical
11principal polarization.  This induces a polarization on $A$.
12
13\begin{definition}
14The {\bf modular polarization} is the map $\delta:A^{\vee}\ra A$
15arising from autoduality of $J_0(N)$.
16\begin{center}
17\begin{picture}(80,140)
18\put(0,110){$A^{\vee}$}
19\put(65,110){$J_0(N)$}
20\put(70,40){$A$}
21\put(75,100){\vector(0,-1){45}}
22\put(75,63){\vector(0,-1){3}}
23\put(13,105){\vector(1,-1){55}}
24%\qbezier(65,45)(10,60)(5,103)\put(5,103){\vector(0,1){1}}
25\put(18,113){\vector(1,0){43}}
26%\qbezier(18,113)(14,115)(18,117)
27
28\put(40,117){$\pi^{\vee}$}
29\put(40,80){$\delta$}
30%\put(20,50){$\hat{\delta}$}
31\put(80,80){$\pi$}
32\end{picture}
33\end{center}
34\end{definition}
35
36Let $f\in S_2(\Gamma_0(N))$ be a newform, $A=A_f$ the corresponding
37optimal quotient of $J_0(N)$, and $\p_f=\Ann_\T(f)\subset\T$ the
38annihilator of $f$ in the Hecke algebra.
39Let $H_1=H_1(X_0(N),\Z)$ be the first integral homology of
40the modular curve $X_0(N)$.
41
42\begin{theorem}
43Let $\Phi:H_1\ra A(\C)$ be the period map. Then
44there is an exact sequence
45     $$0 \ra \Phi(H_1[\p_f]) \ra \Phi (H_1) \ra \Ker(\delta) \ra 0.$$
46\end{theorem}
47\begin{proof}
48\mbox{}\\
49{\noindent\bf Step 1: Pass to lattices.}
50Over the complex number we may write each of $A$ and $A^{\vee}$ as complex tori
51$T/\Lambda$ where $T\isom\C^d$ and $\Lambda$ is a lattice.  The isogeny $\delta:A^{\vee}\ra A$
52induces maps $T(A^{\vee})\ra T(A)$ and $\Lambda(A^{\vee})\ra \Lambda(A)$.  We
53thus obtain the following commuting diagram with exact rows and columns.
54$$\begin{matrix} 55 & & & 0 & & 0 & & & \\ 56 & & & \da & & \da & & & \\ 570 \lra & 0 &\lra & \Lambda(A^{\vee})& \lra &\Lambda(A) & \lra & L & \lra 0 \\ 58 & \da & & \da & & \da & & \da & \\ 590 \lra & 0 & \lra&T(A^{\vee})& \lra & T(A) & \lra & 0 & \lra 0 \\ 60 & \da & & \da & & \da & & \da & \\ 610 \lra & \Ker(\delta) & \lra& A^{\vee}& \xrightarrow{\,\,\,\delta\,\,\,} & A & \lra & 0 & \lra 0 \\ 62 & & & \da & & \da & & & \\ 63 & & & 0 & & 0 & & & \\ 64 65\end{matrix}$$
66Applying the snake lemma we see that
67    $$\Ker(\delta) \isom L = \coker(\Lambda(A^{\vee})\ra \Lambda(A)).$$
68
69{\noindent\bf Step 2: Identify lattices.}
70Proposition 6 of \cite{shimura} allows us to identify $\Lambda(A)$ and $\Lambda(A^{\vee})$
71in terms of the integral homology $H_1=H_1(X_0(N),\Z)$.
72
73First for $J_0(N)$ we have
74   $$T(J)=\Hom(S_2(\Gamma_0(N),\C))$$
75and an exact sequence
76  $$0\ra H_1 \ra T(J) \ra J(\C) \ra 0.$$
77As for $A$, we have
78   $$H_1\xrightarrow{\Phi} T(A) \ra A(\C) \ra 0$$
79where
80   $$\Phi(\gamma) = (\int_\gamma f_1, \ldots, \int_\gamma f_d).$$
81(See below for the basis for $T(A)$.)
82We thus have
83   \begin{eqnarray*}
84     T(A)&=&\Hom(S[\p_f],\C)\\
85     \Lambda(A) &=& \Phi(H_1)
86    \end{eqnarray*}
87In defining $\Phi$ we have chosen the basis $f_1,\ldots, f_d$ for $S[\p_f]$, in
88order to obtain a basis for $T(A)$.
89For $A^{\vee}$ we have
90  \begin{eqnarray*}
91       T(A^{\vee}) &=& T(J)[\p_f] \\
92       \Lambda(A^{\vee}) &=& H_1[\p_f]
93  \end{eqnarray*}
94
95
96{\noindent\bf Step 3: Compute $L$.}
97
98The map $\Lambda(A^{\vee})\ra\Lambda(A)$ induced by $\delta$ is the restriction of $\Phi$
99to $H_1[\p_f]$. Thus
100$$L = \frac{\Phi(H_1)}{\Phi(H_1[\p_f])},$$
101which, combined with step 1, completes the proof.
102\end{proof}
103
104
105
106\section{The $\infty$-Component Group}
107Fix a newform $f\in S_2(\Gamma_0(N))$ as before and let
108$A_f$ be the corresponding optimal quotient of $J_0(N)$.
109Define groups
110\begin{eqnarray*}
111  Y_A &=& H_1[\p_f]\\
112  Y_{A^{\vee}} &=& \Hom(H_1,\Z)[\p_f]
113\end{eqnarray*}
114
115\begin{conjecture}
116There is an exact sequence of abelian groups
117$$0\ra Y_{A^{\vee}} \ra \Hom(Y_A,\Z) \ra \Ker(\delta) \ra 0.$$
118\end{conjecture}
119
120{\bf Evidence.} It's only a conjecture because I have not worked out
121all the details yet.  The basic idea is that in computing
122$\Phi(H_1)/\Phi(H_1[\p_f])$ we can replace $\Phi$ by any
123homomorphism $\Psi$ eminating from $H_1$ and satisfying
124      $$\Ker(\Psi) = \Ker(\Phi).$$
125
126Now let's make a few assumptions related to the structure of the $\T$-module
127$H_1=H_1(X_0(N),\C)$: Let
128       $$V_f = \Hom(H_1,\Z)[\p_f].$$
129{\bf\noindent Assumption 1.} $\dim_{\Z} V_f = 2d,$
130where $d$ is the number of conjugates of $f$.  Fix a basis
131$\vphi_1,\ldots,\vphi_{2d}$ for $V_f$ and define
132      $$\Psi : H_1 \ra \Z,$$
133      $$\Psi(x) = (\vphi_1(x),\ldots, \vphi_{2d}(x)).$$
134
135{\bf\noindent Assumption 2.} $\Ker(\Psi) = \Ker(\Phi)$ \\
136
137Given these two assumptions, computing $\Psi(H_1)/\Psi(H_1[\p_f])$
138and computing the cokernel $Y_{A^{\vee}} \ra \Hom(Y_A,\Z)$ are
139the same thing.
140
141I think both of these assumption can be shown by looking
142at characteristic polynomials of Hecke operators and using
143the Atkin-Lehner multiplicity one theory.
144
145\vspace{.8in}
146
147In analogy with the Grothendieck-Raynaud-Ribet description of the local
148component groups of Neron models we make the following definition.
149\begin{definition}
150The {\bf $\infty$-component group} is $$\Phi_{A,\infty} := \Ker(\delta).$$
151\end{definition}
152
153\begin{thebibliography}{HHHHHHH}
154\bibitem[S]{shimura} G. Shimura, {\em On the factors of the jacobian
155variety of a modular function field}, J. Math. Soc. Japan,
156{\bf 25}, No. 3, 523--544 (1973).
157\end{thebibliography} \normalsize\vspace*{1 cm}
158
159\end{document}
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