Sharedwww / Tables / modular_degree.texOpen in CoCalc
Author: William A. Stein
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% denominator.tex
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\documentclass[11pt]{article}
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\include{macros}
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\title{The Kernel of the Modular Polarization of a Quotient of $J_0(N)$}
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\begin{document}
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\maketitle
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\tableofcontents
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\section{The Modular Polarization}
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Suppose $A$ is a quotient of $J_0(N)$.
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Because $J_0(N)$ is a Jacobian it possesses a canonical
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principal polarization. This induces a polarization on $A$.
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\begin{definition}
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The {\bf modular polarization} is the map $\delta:A^{\vee}\ra A$
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arising from autoduality of $J_0(N)$.
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\begin{center}
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\begin{picture}(80,140)
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\put(0,110){$A^{\vee}$}
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\put(65,110){$J_0(N)$}
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\put(70,40){$A$}
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\put(75,100){\vector(0,-1){45}}
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\put(75,63){\vector(0,-1){3}}
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\put(13,105){\vector(1,-1){55}}
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%\qbezier(65,45)(10,60)(5,103)\put(5,103){\vector(0,1){1}}
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\put(18,113){\vector(1,0){43}}
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%\qbezier(18,113)(14,115)(18,117)
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\put(40,117){$\pi^{\vee}$}
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\put(40,80){$\delta$}
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%\put(20,50){$\hat{\delta}$}
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\put(80,80){$\pi$}
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\end{picture}
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\end{center}
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\end{definition}
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Let $f\in S_2(\Gamma_0(N))$ be a newform, $A=A_f$ the corresponding
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optimal quotient of $J_0(N)$, and $\p_f=\Ann_\T(f)\subset\T$ the
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annihilator of $f$ in the Hecke algebra.
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Let $H_1=H_1(X_0(N),\Z)$ be the first integral homology of
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the modular curve $X_0(N)$.
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\begin{theorem}
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Let $\Phi:H_1\ra A(\C)$ be the period map. Then
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there is an exact sequence
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$$ 0 \ra \Phi(H_1[\p_f]) \ra \Phi (H_1) \ra \Ker(\delta) \ra 0.$$
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\end{theorem}
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\begin{proof}
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\mbox{}\\
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{\noindent\bf Step 1: Pass to lattices.}
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Over the complex number we may write each of $A$ and $A^{\vee}$ as complex tori
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$T/\Lambda$ where $T\isom\C^d$ and $\Lambda$ is a lattice. The isogeny $\delta:A^{\vee}\ra A$
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induces maps $T(A^{\vee})\ra T(A)$ and $\Lambda(A^{\vee})\ra \Lambda(A)$. We
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thus obtain the following commuting diagram with exact rows and columns.
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$$\begin{matrix}
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& & & 0 & & 0 & & & \\
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& & & \da & & \da & & & \\
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0 \lra & 0 &\lra & \Lambda(A^{\vee})& \lra &\Lambda(A) & \lra & L & \lra 0 \\
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& \da & & \da & & \da & & \da & \\
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0 \lra & 0 & \lra&T(A^{\vee})& \lra & T(A) & \lra & 0 & \lra 0 \\
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& \da & & \da & & \da & & \da & \\
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0 \lra & \Ker(\delta) & \lra& A^{\vee}& \xrightarrow{\,\,\,\delta\,\,\,} & A & \lra & 0 & \lra 0 \\
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& & & \da & & \da & & & \\
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& & & 0 & & 0 & & & \\
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\end{matrix}$$
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Applying the snake lemma we see that
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$$\Ker(\delta) \isom L = \coker(\Lambda(A^{\vee})\ra \Lambda(A)).$$
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{\noindent\bf Step 2: Identify lattices.}
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Proposition 6 of \cite{shimura} allows us to identify $\Lambda(A)$ and $\Lambda(A^{\vee})$
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in terms of the integral homology $H_1=H_1(X_0(N),\Z)$.
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First for $J_0(N)$ we have
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$$T(J)=\Hom(S_2(\Gamma_0(N),\C))$$
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and an exact sequence
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$$0\ra H_1 \ra T(J) \ra J(\C) \ra 0.$$
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As for $A$, we have
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$$H_1\xrightarrow{\Phi} T(A) \ra A(\C) \ra 0$$
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where
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$$\Phi(\gamma) = (\int_\gamma f_1, \ldots, \int_\gamma f_d).$$
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(See below for the basis for $T(A)$.)
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We thus have
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\begin{eqnarray*}
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T(A)&=&\Hom(S[\p_f],\C)\\
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\Lambda(A) &=& \Phi(H_1)
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\end{eqnarray*}
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In defining $\Phi$ we have chosen the basis $f_1,\ldots, f_d$ for $S[\p_f]$, in
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order to obtain a basis for $T(A)$.
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For $A^{\vee}$ we have
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\begin{eqnarray*}
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T(A^{\vee}) &=& T(J)[\p_f] \\
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\Lambda(A^{\vee}) &=& H_1[\p_f]
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\end{eqnarray*}
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{\noindent\bf Step 3: Compute $L$.}
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The map $\Lambda(A^{\vee})\ra\Lambda(A)$ induced by $\delta$ is the restriction of $\Phi$
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to $H_1[\p_f]$. Thus
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$$L = \frac{\Phi(H_1)}{\Phi(H_1[\p_f])},$$
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which, combined with step 1, completes the proof.
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\end{proof}
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\section{The $\infty$-Component Group}
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Fix a newform $f\in S_2(\Gamma_0(N))$ as before and let
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$A_f$ be the corresponding optimal quotient of $J_0(N)$.
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Define groups
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\begin{eqnarray*}
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Y_A &=& H_1[\p_f]\\
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Y_{A^{\vee}} &=& \Hom(H_1,\Z)[\p_f]
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\end{eqnarray*}
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\begin{conjecture}
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There is an exact sequence of abelian groups
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$$0\ra Y_{A^{\vee}} \ra \Hom(Y_A,\Z) \ra \Ker(\delta) \ra 0.$$
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\end{conjecture}
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{\bf Evidence.} It's only a conjecture because I have not worked out
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all the details yet. The basic idea is that in computing
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$\Phi(H_1)/\Phi(H_1[\p_f])$ we can replace $\Phi$ by any
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homomorphism $\Psi$ eminating from $H_1$ and satisfying
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$$\Ker(\Psi) = \Ker(\Phi).$$
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Now let's make a few assumptions related to the structure of the $\T$-module
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$H_1=H_1(X_0(N),\C)$: Let
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$$V_f = \Hom(H_1,\Z)[\p_f].$$
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{\bf\noindent Assumption 1.} $\dim_{\Z} V_f = 2d,$
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where $d$ is the number of conjugates of $f$. Fix a basis
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$\vphi_1,\ldots,\vphi_{2d}$ for $V_f$ and define
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$$\Psi : H_1 \ra \Z,$$
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$$\Psi(x) = (\vphi_1(x),\ldots, \vphi_{2d}(x)).$$
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{\bf\noindent Assumption 2.} $\Ker(\Psi) = \Ker(\Phi)$ \\
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Given these two assumptions, computing $\Psi(H_1)/\Psi(H_1[\p_f])$
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and computing the cokernel $Y_{A^{\vee}} \ra \Hom(Y_A,\Z)$ are
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the same thing.
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I think both of these assumption can be shown by looking
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at characteristic polynomials of Hecke operators and using
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the Atkin-Lehner multiplicity one theory.
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\vspace{.8in}
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In analogy with the Grothendieck-Raynaud-Ribet description of the local
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component groups of Neron models we make the following definition.
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\begin{definition}
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The {\bf $\infty$-component group} is $$\Phi_{A,\infty} := \Ker(\delta).$$
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\end{definition}
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\begin{thebibliography}{HHHHHHH}
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\bibitem[S]{shimura} G. Shimura, {\em On the factors of the jacobian
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variety of a modular function field}, J. Math. Soc. Japan,
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{\bf 25}, No. 3, 523--544 (1973).
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\end{thebibliography} \normalsize\vspace*{1 cm}
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\end{document}
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