CoCalc -- Collaborative Calculation in the Cloud
Sharedwww / Tables / modular_degree.texOpen in CoCalc
% denominator.tex
\title{The Kernel of the Modular Polarization of a Quotient of $J_0(N)$}
\section{The Modular Polarization}
Suppose $A$ is a quotient of $J_0(N)$.
Because $J_0(N)$ is a Jacobian it possesses a canonical
principal polarization.  This induces a polarization on $A$.

The {\bf modular polarization} is the map $\delta:A^{\vee}\ra A$
arising from autoduality of $J_0(N)$. 


Let $f\in S_2(\Gamma_0(N))$ be a newform, $A=A_f$ the corresponding
optimal quotient of $J_0(N)$, and $\p_f=\Ann_\T(f)\subset\T$ the
annihilator of $f$ in the Hecke algebra.  
Let $H_1=H_1(X_0(N),\Z)$ be the first integral homology of
the modular curve $X_0(N)$. 

Let $\Phi:H_1\ra A(\C)$ be the period map. Then
there is an exact sequence
     $$ 0 \ra  \Phi(H_1[\p_f]) \ra \Phi (H_1) \ra \Ker(\delta) \ra 0.$$
{\noindent\bf Step 1: Pass to lattices.}
Over the complex number we may write each of $A$ and $A^{\vee}$ as complex tori
$T/\Lambda$ where $T\isom\C^d$ and $\Lambda$ is a lattice.  The isogeny $\delta:A^{\vee}\ra A$
induces maps $T(A^{\vee})\ra T(A)$ and $\Lambda(A^{\vee})\ra \Lambda(A)$.  We
thus obtain the following commuting diagram with exact rows and columns.
        &         &     &   0     &           &     0     &          &      &          \\
        &         &     &  \da    &           &    \da    &          &      &          \\
0  \lra &     0   &\lra &  \Lambda(A^{\vee})&    \lra   &\Lambda(A) & \lra     &  L   &  \lra 0  \\
        &   \da   &     &   \da   &           &   \da     &          &  \da &          \\
0  \lra &  0      & \lra&T(A^{\vee})& \lra      & T(A)     & \lra     &   0  & \lra 0   \\
        & \da     &     &   \da   &           &  \da      &          &  \da &          \\
0  \lra & \Ker(\delta) & \lra&  A^{\vee}& \xrightarrow{\,\,\,\delta\,\,\,}    & A         & \lra     &   0  & \lra 0   \\
        &         &     &   \da   &           & \da       &          &      &          \\
        &         &     &    0    &           &     0     &          &      &          \\

Applying the snake lemma we see that 
    $$\Ker(\delta) \isom L = \coker(\Lambda(A^{\vee})\ra \Lambda(A)).$$

{\noindent\bf Step 2: Identify lattices.}
Proposition 6 of \cite{shimura} allows us to identify $\Lambda(A)$ and $\Lambda(A^{\vee})$
in terms of the integral homology $H_1=H_1(X_0(N),\Z)$.  

First for $J_0(N)$ we have
and an exact sequence
  $$0\ra H_1 \ra T(J) \ra J(\C) \ra 0.$$
As for $A$, we have
   $$H_1\xrightarrow{\Phi} T(A) \ra A(\C) \ra 0$$
   $$\Phi(\gamma) = (\int_\gamma f_1, \ldots, \int_\gamma f_d).$$
(See below for the basis for $T(A)$.)
We thus have
     \Lambda(A) &=& \Phi(H_1)
In defining $\Phi$ we have chosen the basis $f_1,\ldots, f_d$ for $S[\p_f]$, in
order to obtain a basis for $T(A)$. 
For $A^{\vee}$ we have
       T(A^{\vee}) &=& T(J)[\p_f] \\
       \Lambda(A^{\vee}) &=& H_1[\p_f]

{\noindent\bf Step 3: Compute $L$.}

The map $\Lambda(A^{\vee})\ra\Lambda(A)$ induced by $\delta$ is the restriction of $\Phi$
to $H_1[\p_f]$. Thus
$$L = \frac{\Phi(H_1)}{\Phi(H_1[\p_f])},$$
which, combined with step 1, completes the proof.

\section{The $\infty$-Component Group}
Fix a newform $f\in S_2(\Gamma_0(N))$ as before and let
$A_f$ be the corresponding optimal quotient of $J_0(N)$. 
Define groups
  Y_A &=& H_1[\p_f]\\
  Y_{A^{\vee}} &=& \Hom(H_1,\Z)[\p_f]

There is an exact sequence of abelian groups
$$0\ra Y_{A^{\vee}} \ra \Hom(Y_A,\Z) \ra \Ker(\delta) \ra 0.$$

{\bf Evidence.} It's only a conjecture because I have not worked out 
all the details yet.  The basic idea is that in computing
$\Phi(H_1)/\Phi(H_1[\p_f])$ we can replace $\Phi$ by any
homomorphism $\Psi$ eminating from $H_1$ and satisfying 
      $$\Ker(\Psi) = \Ker(\Phi).$$

Now let's make a few assumptions related to the structure of the $\T$-module
$H_1=H_1(X_0(N),\C)$: Let
       $$V_f =  \Hom(H_1,\Z)[\p_f].$$
{\bf\noindent Assumption 1.} $\dim_{\Z} V_f = 2d,$
where $d$ is the number of conjugates of $f$.  Fix a basis 
$\vphi_1,\ldots,\vphi_{2d}$ for $V_f$ and define 
      $$\Psi : H_1 \ra \Z,$$
      $$\Psi(x) = (\vphi_1(x),\ldots, \vphi_{2d}(x)).$$

{\bf\noindent Assumption 2.} $\Ker(\Psi) = \Ker(\Phi)$ \\

Given these two assumptions, computing $\Psi(H_1)/\Psi(H_1[\p_f])$
and computing the cokernel $Y_{A^{\vee}} \ra \Hom(Y_A,\Z)$ are
the same thing. 

I think both of these assumption can be shown by looking
at characteristic polynomials of Hecke operators and using
the Atkin-Lehner multiplicity one theory. 


In analogy with the Grothendieck-Raynaud-Ribet description of the local 
component groups of Neron models we make the following definition.
The {\bf $\infty$-component group} is $$\Phi_{A,\infty} := \Ker(\delta).$$

\bibitem[S]{shimura} G. Shimura, {\em On the factors of the jacobian
variety of a modular function field}, J. Math. Soc. Japan,
{\bf 25}, No. 3, 523--544 (1973).
\end{thebibliography} \normalsize\vspace*{1 cm}