%torsion.tex
\chapter{Quotients of~$J_0(N)$}
In this chapter $k=2$ and $\eps=1$; we
sometimes omit~$k$ and~$\eps$ from the notation.

\section{The torsion subgroup}
Modular symbols can be used to obtain both upper and lower bounds
on the rational torsion subgroup of $A_f$.
\subsection{Lower bounds: The cuspidal subgroup of $A_f$}
Let~$s$ be the projection onto $\sS_2(N;\Q)$ in
the  $\T$-module decomposition
$$\sM_2(N;\Q) 
   = \sS_2(N;\Q) \oplus \sE_2(N;\Q).$$
\begin{theorem}[Abel-Jacobi]
Let $[\alpha],\, [\beta]$ be cusps of $X_0(N)$ with
$\alpha, \beta\in\P^1(\Q)$.  Then the point 
$[\alpha]-[\beta]\in J_0(N)$ is~$0$ if and only if
$s(\{\alpha,\beta\})\in \sS_2(N;\Z)$.
\end{theorem}
\begin{definition}
The \defn{cuspidal subgroup}  of $J_0(N)$ is 
the group generated by the differences of all cusps.
The cuspdial subgroup of $A_f$ is the image of the
cuspidal subgroup of $J_0(N)$.
\end{definition}
\begin{proposition}
The cuspidal subgroup of $J_0(N)$ is a finite group isomorphic
to the subgroup~$C$ of $\sS_2(N;\Q)/\sS_2(N;\Z)$ generated by 
the images of  modular symbols $\{\alpha,\beta\}$ with 
$\alpha,\beta\in\P^1(\Q)$.
\end{proposition}

\begin{corollary}\label{cor:cuspidal}
The cuspidal subgroup of $A_f$ is isomorphic to 
the image of~$C$ in 
$$\Phi_f(\sS_2(N;\Q))/\Phi_f(\sS_2(N;\Z)).$$
\end{corollary}
Manin proved that $(0)-(\infty)$ defines an element
of $J_0(N)(\Q)_{\tor}$.  Thus the order of the
image of $(0)-(\infty)$ provides a  lower bounds on 
$\# A_f(\Q)_{\tor}$.

For the case $k>2$, see Section~\ref{sec:cuspdiff}.

\subsection{Upper bounds: Counting points mod~$p$}
Let~$f$ be a newform. The Hecke algebra~$\T$ acts through
a quotient $\overline{\T}$ on the subspace of $S_2(N)$ spanned
by the Galois conjugates of~$f$. Let $\chi_p(X)$ be the characteristic
polynomial of the image of $T_p$ in $\overline{\T}$.  
Suppose $p\nmid N$ and  let $N_p=\# A_f(\F_p)$ be the number of points 
on the mod~$p$ reduction of $A_f$.
\begin{proposition}
$$N_p = \chi_p(p+1)$$
\end{proposition}
\begin{proof}\footnote{Matt Baker told the author this proof.}
We have 
    $$T_p = \Frob + \Ver = \Frob + p/\Frob.$$  
If the characteristic polynomial of $\Frob$ on
an $\ell$-adic Tate module is $F(t)$, and the characteristic 
polynomial of $T_p$ 
on differentials is $f(t)$, then we have $f(t)=x^{-g}F(x)$, 
where $t=x+(p/x)$.  In
other words, Eichler-Shimura gives an easy conversion 
between~$f$ and~$F$.
Since it's a general fact that $\#J(\Fp)=F(1)$, we have
$$\#J(\Fp)=f(p+1)=\text{ constant term of }f(x+p+1).$$
\end{proof}

\begin{theorem}
Suppose $p\nmid N$.  Then the kernel of the reduction map
$A_f(\Q)_{\tor}\ra A_f(\F_p)$
is killed by~$p$.  If $p>2$ then 
the kernel is trivial.
\end{theorem}
Thus we obtain an upper bound on $\# A_f(\Q)_{\tor}$. 
This upper bound is not in general sharp, in fact it is unchanged 
if $A_f$ is replaced by any abelian variety isogeneous 
to $A_f$. 


% maninconst.tex
\section{The Manin constant}\label{sec:maninconstant}

Consider the optimal quotient~$A$ of~$J_0(N)$ corresponding
to a newform~$f$ on $\Gamma_0(N)$ of weight~$2$.  
Let~$I_A$ be the kernel of the natural map from the Hecke algebra
to $\End(A)$.
The \defn{Manin constant\label{defn:maninconstant}}~$c_A$ of~$A$ is the lattice
index 
 $$c_A := [S_2(\Gamma_0(N);\Z)[I_A]:H^0(\cA,\Omega_{\cA/\Z})]$$ 
taken inside of $S_2(\Gamma_0(N);\Q)$. 
Generalizing a theorem of Mazur, we prove that~$c_A$
is a unit in $\Z[\frac{1}{2m}]$, where~$m$ is the largest
square dividing~$N$.   We then conjecture that $c_A=1$,
and give supporting numerical evidence.


\subsection{The primes that might divide~$c_A$}
In the special case $\dim A=1$, the Manin constant is the classical Manin 
constant of~$A$, and in~\cite{mazur:rational} Mazur proved 
that~$c_A$ is a unit in $\Z[\frac{1}{2m}]$.
We generalize his proof to obtain the analogous result in
dimension greater than~$1$.
\begin{theorem}
Let $A$ be the new optimal quotient of~$J_0(N)$ corresponding
to a newform~$f$.  Then the Manin constant~$c_A$ is a unit
in $\Z[\frac{1}{2m}]$, where~$m$ is the largest square
dividing~$N$.  
\end{theorem}
\begin{proof}
Let~$\pi$ denote the map $J_0(N)\ra A$;
let~$\cA$ denote the Neron model 
of~$A$ over~$R:=\Z[\frac{1}{2m}]$, and~$\cJ$ the 
Neron model of $J_0(N)$ over~$R$.  
Let~$\cX$ be the smooth 
locus a minimal proper regular model for $X_0(N)$ over~$R$. 
Consider the diagram
\begin{equation}\label{eqn:qexp}
H^0(\cA,\Omega_{\cA}) \xrightarrow{\pi^*} H^0(\cJ,\Omega_{\cJ}) 
                      \isom H^0(\cX,\Omega_\cX) 
                          \xrightarrow{\text{ $q$-exp }} R[[q]]
\end{equation}

The map~$\pi^*$ must be an inclusion because 
$H^0(\cA,\Omega_{\cA})$ is torsion-free and~$\pi^*$ is an inclusion
after tensoring with~$\C$. 
To show that the Manin constant is a unit in~$R$, it
suffices to check that the image of $H^0(\cA,\Omega_{\cA})$
in $R[[q]]$ is {\em saturated}, in the sense that the cokernel
is torsion free; indeed, the image of 
$S_2(\Gamma_0(N);R)[I]$ is saturated and 
$S_2(\Gamma_0(N);R)[I]\tensor\Q = H^0(\cA,\Omega_{\cA})\tensor\Q$. 

For the image of $H^0(\cA,\Omega_{\cA})$ in $R[[q]]$ to be 
saturated means that the quotient~$D$
is torsion free.  Let~$\ell$ be a prime not dividing~$2m$;
tensoring
$$0\ra H^0(\cA,\Omega_{\cA})\xrightarrow{\text{$q$-exp}}
      R[[q]]\ra D\ra 0$$
with~$\Fl$ we obtain
$$0 \ra D[\ell] \ra H^0(\cA,\Omega_{\cA})\tensor\Fl\ra
     \Fl[[q]] \ra D\tensor\Fl \ra 0.$$
Here we have used that $\Tor^1(D,\Fl)$ is the $\ell$-torsion in~$D$,
and that $\Tor^1(-,\Fl)$ vanishes on the torsion free group $R[[q]]$.
To show that $D[\ell]=0$, it suffices to prove that the map
$\Theta:H^0(\cA,\Omega_{\cA})\tensor\Fl\ra \Fl[[q]]$
is injective. 

Since $\ell\neq 2$ and~$A$ is optimal, 
\cite[Cor 1.1]{mazur:rational} gives an exact sequence
  $$0 \ra H^0(\cA/\Zl,\Omega_{\cA/\Zl}) \ra 
          H^0(\cJ/\Zl,\Omega_{\cJ/\Zl}) \ra 
          H^0(\cB/\Zl,\Omega_{\cB/\Zl}) \ra 0$$
where $\cB=\ker(\cJ\ra\cA)$.  In particular, $H^0(\cB/\Zl,\Omega_{\cB/\Zl})$
is torsion free, so 
$$H^0(\cA/\Zl,\Omega_{\cA/\Zl})\tensor\Fl \ra 
          H^0(\cJ/\Zl,\Omega_{\cJ/\Zl})\tensor\Fl
         \isom H^0(\cX/\Fl,\Omega_{\cX/\Fl})$$
is injective. 
We also remark that 
$$H^0(\cA,\Omega_{\cA})\tensor\Fl\isom 
H^0(\cA/\Zl,\Omega_{\cA/\Zl})\tensor\Fl,$$
because~$\Zl$ is torsion-free, hence flat over~$R$.
Thus the map 
$$H^0(\cA,\Omega_{\cA})\tensor\Fl\ra
H^0(\cX/\Fl,\Omega_{\cX/\Fl})$$
is injective.

If $\ell\nmid N$, then injectivity of~$\Theta$ now follows from
the $q$-expansion principle, which asserts that the
$q$-expansion map $H^0(\cX/\Fl,\Omega_{\cX/\Fl})\ra \Fl[[q]]$
is injective. 

Suppose~$\ell$ does divide~$N$, and let 
$\omega\in \ker(\Theta)$.
Since $\ell \mid N$ and $\ell\nmid 2m$, we have that $\ell || N$;
thus $\cX/\Fl$ breaks up into a union of two irreducible
components, and the $q$-expansion principle implies only that~$\omega$
vanishes on the irreducible component containing the cusp~$\infty$. 
However, since~$A$ is {\em new} and corresponds to a {\em singe}
eigenform,~$\omega$ is an eigenvector for the involution~$W_N$
(since~$f$ and all of its conjugates are).  Since~$W_N$ permutes
the two components,~$\omega$ must be~$0$ on all $\cX/\Fl$.
Therefore $\omega=0$, and hence~$\Theta$ is injective.
\end{proof}

\subsection{Numerical evidence for a $c_A=1$ conjecture}
In the paper~\cite{empirical}, the authors 
show that $c_A=1$ for~$28$ two-dimensional optimal 
quotients of $J_0(N)$ (see Section~\ref{sec:analytic-empirical}).
The non square-free levels treated are:
 $$N=3^2\cdot 7,\quad 3^2\cdot 13,
     \quad 5^3,\quad 3^3\cdot 5,\quad 3\cdot 7^2,
      \quad 5^2\cdot 7,\quad 2^2\cdot 47,\quad 3^3\cdot 7.$$
In every case, $c_A = 1$.

\begin{conjecture}[Agash\'{e}, Stein]
Let~$A$ be an optimal quotient of $J_0(N)$, and let
$c_A$ be the corresponding Manin constant.  Then
$c_A=1$.
\end{conjecture}


