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% denominator.tex
\documentclass[12pt]{article}
\include{macros}
\title{The Denominator of the Special Value $L(A_f,1)/\Omega(A_f)$}
\begin{document}
\author{Amod Agashe and William Stein\footnote{email: {\tt amod} and {\tt was} at {\tt math.berkeley.edu}.}}
\maketitle
\begin{abstract}
Let $f$ be a newform and $A_f$ the quotient
of $J_0(N)$ constructed by Shimura.
We prove that, up to a Manin constant and a power of $2$,
the denominator of the rational number $L(A_f,1)/\Omega(A_f)$
divides the order of the image of $(0)-(\infty)$ in $A_f(\Q)$.
This provides evidence for the Birch and Swinnerton-Dyer
conjecture and raises questions about the structure of $A_f(\Q)$.
\end{abstract}

\section{Introduction}
Fix a positive integer $N$.
Let $f\in S_2(\Gamma_0(N))$ be a newform and let
$A_f$ be the corresponding optimal quotient of $J_0(N)$.
The $L$-function of $A_f$ is
$$L(A_f,s)=\prod_{i=1}^d L(f_i,s)$$
where $f_1,\ldots,f_d$ are the Galois conjugates of $f$.
Let
$$\Omega(A_f) = \int_{A_f(\R)} |\omega|$$
where $\omega$ is a differential $d$-form on the Neron
model of $A_f$. For $p|N$ the analogous local quantity
is $c_p$ which is
the number of $\Fp$-rational components of the
special fiber of the Neron model of $A_f$ at $p$.
Let
$$\Sha(A_f) = \Ker [ H^1(\Q,A_f)\ra \prod_v H^1(\Qv,A_f) ]$$
where the product is over all primes $p$ and $\infty$.

The Birch and Swinnerton-Dyer conjecture (BSD conjecture),
as generalized by Tate, predicts that $\Sha(A_f)$ is finite and
$$\frac{L(A_f,1)}{\Omega(A_f)} = \frac{\#\Sha(A_f)\cdot\prod_{p|N} c_p} {\#A_f(\Q) \cdot \# A_f^{\vee}(\Q)}.$$
When $\#A_f(\Q)$ is infinite the right hand side is $0$.
This conjecture is currently the subject of much
intensive research.  We do have
\begin{theorem}[Kolyvagin, Logachev]
If $L(A_f,1)\neq 0$ then both $A_f(\Q)$ and
$\Sha(A_f)$ are finite.
\end{theorem}

We first express $L(A_f,1)$ in terms of modular symbols
in order to show that the denominator divides the order
of a certain subgroup of $\#A_f(\Q)$.  Not only does our result
provide evidence for the BSD conjecture, but assuming the BSD conjecture
it also suggests that the natural map
$$A_f(\Q) \ra \prod_{p|N} \Phi_{A_f,p}$$
should be (very close to) injective.

\section{Modular Symbols Expression for $L(A_f,1)$}
Fix a newform $f$ as in the introduction and assume that
$L(A_f,1)\neq 0$.

Let $H_1(X_0(N),\Z)$ be the first integral homology of
the modular curve $X_0(N)$.  The Hecke algebra $\T$ and the
involution $*$ both act on $H_1(X_0(N),\Z)$, and their
actions commute.  We have an exact sequence
$$H_1(X_0(N),\Z)\xrightarrow{\Phi_f} \C^d \ra A_f(\C) \ra 0$$
where
$$\Phi_f(\gamma) = (\int_\gamma f_1, \ldots, \int_\gamma f_d).$$
Let $\e=\{0,\infty\}\in H_1(X_0(N),\Q)$ correspond to integration
along the vertical path from $0$ to $i\infty$.

\begin{definition}[Module Index]
Let $V$ be a $\Q$-vector space and let $L, M\subset V$ be
lattices (of full rank).  Denote by $[L:M]$ the absolute
value of the determinant of an automorphism of $V$ sending
$L$ to $M$.
\end{definition}

Define
$$\L(f) = [\Phi_f(H_1(X_0(N),\Z)^+) : \Phi_f(\T\e)]$$
where the images are lattices in $V = \Phi_f(H_1(X_0(N),\Q)^+)$.

\begin{theorem}
We have
$$\L(f) = \frac{|L(A_f,1)|}{\Omega(A_f)}\cdot c_\infty \cdot c_M$$
where $c_\infty$ is the number of real components and
$c_M$ is the Manin constant.
\end{theorem}
\comment{The key idea in the proof involves relating the space $S_2(\Gamma_0(N),\Z)$
of integral cusp forms to the space of global differentials on the Neron model,
then using the perfect duality between $\T$ and $S_2(\Gamma_0(N),\Z)$. }

\section{The Denominator of the Special Value}
Fix a newform $f=\sum a_n q^n$ as in the introduction and assume
$$L(A_f,1)\neq 0.$$  The cusps $0$ and $\infty$ on
$X_0(N)$ give rise to a rational torsion point
$$P_e=(0)-(\infty)\in J_0(N).$$
Let $\overline{P}_e$ denote the image of $P_e$ in $A_f(\Q)$ and let
$$C_e=\Z\overline{P}_e$$
be the cyclic subgroup of $A_f(\Q)$ which it generates.
Let
$$\p_f = \{ t \in \T : t(f) = 0\}.$$
The Hecke algebra $\T$ acts on $A_f$ through a quotient:
$$\O_f = \T/\p_f \isom \Z[a_1,a_2,\ldots].$$
Let $\pi :\T\ra \O_f$ denote the natural surjection.

\begin{proposition}
The group $C_e\subset A_f(\Q)$ is $\O_f$-invariant.
\end{proposition}
\begin{proof}
It suffices to show that each $\pi(T_p)$ acts as a scalar on $\overline{P}_e$,
since the $\pi(T_p)$ generate $\O_f$.  Let $p$ be a prime.\\
Case $p\nmid N$:  Then, following the proof the Manin-Drinfeld theorem,
$$T_p P_e = (p+1) P_e.$$
Thus
$$\pi(T_p)\overline{P}_e = (p+1) \overline{P}_e.$$\\
Case $p\nmid N$:  Then $\pi(T_p)=a_p\in \{0,\pm 1\}$
(see bottom of page 64 of \cite{diamondim}).
Thus
$$\pi(T_p)\overline{P}_e \in \{ 0, \pm \overline{P}_e\}\subset C_e.$$
\end{proof}

\begin{theorem}\label{thmdenom}
The denominator of $\L(f)$ divides the order of
the cyclic subgroup $C_e\subset A_f(\Q)$.
\end{theorem}
\begin{proof}
In the definition of $\L(f)$ we can identify $V=\Phi_f(H_1(X_0(N),\Q)^+)$
with the $\Q$-vector space
$H_1(X_0(N),\Q)^+/\Ker(\Phi_f)$.  Thus we may replace $\Phi_f$ by any
homomorphism $\Phi$ eminating from $H_1(X_0(N),\Z)$ and having the
same kernel as $\Phi_f$. The resulting  module index $\L(f)$
remains unchanged.
Choose some nonzero
$$\Phi \in \Hom(H_1(X_0(N),\Z)^+,\O_f)$$
satisfying the following additional requirement:
$$\Phi(t\gamma) = \pi(t)\Phi(\gamma),\qquad\text{for all t\in\T}$$
By multiplicity one'' and duality between homology and differentials,
$\Phi$ exists and is uniquely determined up to
a nonzero scalar in $\O_f$.

Both
$\Phi(H_1(X_0(N),\Z)^+)$ and $\Phi(\T\e)$ are
contained in $K=\Frac(\O_f)$.  They are fractional $\O_f$-ideals.
Furthermore
$$\L(f) = [\Phi(H_1(X_0(N),\Z)^+) : \Phi(\T\e)].$$

Next define an ideal $I\subset \O_f$ by exactness of
$$0\ra I \ra \O_f \xrightarrow{t\mapsto t.\overline{P}_e} C_e \ra 0.$$
The map $\O_f\ra C_e$ is surjective because $\O_f$ contains $\Z$ and $C_f$ is
cyclic as an abelian group.  Thus $\O_f/I\isom C_e$ is finite cyclic.
Furthermore, the Abel-Jacobi theorem implies that
$$I = \{ t \in \O : t \Phi(\e) \in \Phi(H_1(X_0(N),\Z)^+) \}.$$

We are now in a position to bound the denominator of $\L(f)$.  Writing
$H=H_1(X_0(N),\Z)^+$ we have
\begin{eqnarray*}\L(f) = [\Phi(H):\Phi(\T\e)] &=& [\Phi(H):\O_f\Phi(\e)] \\
&=& [\Phi(H) : I\Phi(\e)]\cdot [I\Phi(\e) : \O_f\Phi(\e)]\\
&=& \frac{[\Phi(H) : I\Phi(\e)]}
{[\O_f\Phi(\e) : I\Phi(\e)]}.
\end{eqnarray*}
Next observe that:\\
\indent 1) $I \Phi(\e)\subset \Phi(H)$ because of the construction of $I$, and\\
\indent 2) $I\Phi(\e)\subset\O_f\Phi(e)$ because $I\subset\O_f$.\\
It follows that
$$[\Phi(H):I\Phi(\e)]\in\Z\text{ and } [\O_f\Phi(\e) : I\Phi(\e)]\in \Z.$$
Thus the denominator of $\L(f)$ divides
$$[\O_f\Phi(\e):I\Phi(\e)]=[\O_f:I]=\#(\O_f/I)=\#C_e.$$
\end{proof}

\begin{question}
Note that the ideal class of the ideal
$$\Phi(H_1(X_0(N),\Z)^+)\subset \O_f$$
is independent of the choice of $\Phi$.
What is the significance of this class in
the ideal class group of $\O_f$?
What is its order?
\end{question}

\begin{question}
In the proof we expressed $\L(f)$ as a quotient
$$\L(f) = \frac{[\Phi(H):I\Phi(\e)]} {[\O_f\Phi(e): I\Phi(e)]}.$$
Both the numerator and the denominator are well-defined, irregardless
of the choice of $\Phi$.  How do they relate to the numerator
and the denominator in the BSD conjecture?
In particular, can we:\\
\indent connect $[\Phi(H):I\Phi(\e)]$ with $\#\Sha(A_f) \cdot|\prod_{p|N} c_p|$, or\\
\indent connect $[\O_f\Phi(e): I\Phi(e)]$ with $\#A_f(\Q) \cdot\#A^{\vee}_f(\Q)$?\\
\end{question}

\section{Idea to Bound $A_f(\Q)$}
Fix a newform $f$ of level $N$ and assume that $L(A_f,1)\neq 0$.
\begin{theorem}
Suppose $p\nmid 2N$.  Then
$$A_f(\Q)\hookrightarrow A_f(\F_p).$$
\end{theorem}

Thus $\#A_f(\Q)$ divides
$$G(f) = \gcd\{\#A_f(\F_p) : p\nmid 2N\}.$$

Let the notation be as in the proof of Theorem~\ref{thmdenom}.
Thus we have
$$\O_f=\T/\p_f=\Z[a_1,a_2,\ldots]$$
and a map
$$\Phi:H_1(X_0(N),\Z)^+\ra \O_f$$
such that
$$\Phi(t\gamma)=\pi(t)\Phi(\gamma).$$
We also have the ideal
$$I = \{ t \in \O_f : t\Phi(\e)\in \Phi(H_1(X_0(N),\Z)^+) \}.$$
When proving the theorem we observed that
$$\#(\O/I)=\#C_e \mid \#A_f(\Q).$$

Consider the ideal $L\subset \O_f$ generated by the obvious'' elements of $I$:
$$L = ( (p+1) - \pi(T_p) : p\nmid 2N )\subset I$$
We will use the following theorem to relate $\#(\O_f/L)$ to $G(f)$.
\begin{theorem}
Suppose $p\nmid N$.
Let $F(x)$ be the characteristic polynomial of $\pi(T_p)$.
Then
$$\#A_f(\F_p) = F(p+1).$$
\end{theorem}

Define the norm of $x\in \O_f$ to be the determinant of the
linear map $\ell_x =$ left multiplication by $x$.  Observe that
$$|\Norm(x)|=[\O_f:x\O_f]=\#(\O_f/x\O_f).$$

\begin{corollary}
Suppose $p\nmid N$.  Then
$$\Norm((p+1) - \pi(T_p)) = \#A_f(\F_p).$$
\end{corollary}
\begin{proof}
If $F(x)$ is the characteristic polynomial of $\pi(T_p)$ then
$F(p+1)$ is the determinant of left multiplication by
$(p+1) - \pi(T_p)$.
\end{proof}

\begin{lemma}
Let $\a$ be an ideal of $\O_f$.
Then
$$\#(\O_f/\a) \mid \gcd\{\Norm(x) : x \in \a\}.$$
\end{lemma}
\begin{proof}
We have
$$\#(\O_f/\a) \mid [\O_f : \a]\cdot [\a : x\O_f] = [\O_f:x\O_f] = |\Norm(x)|\in \Z.$$
\end{proof}

Thus
$$\#(\O_f/L) \mid G(f) = \gcd\{\#A_f(\F_p) : p\nmid 2N\}.$$
\begin{remark}
When I first looked at this I thought that maybe
there would be equality.  (For example, if $\O_f=\Z$ there is indeed equality.)
But, I see no reason for equality now.
\end{remark}

In the elliptic case
the index of $L$ in $I$ measures the failure of
$C_e$ to equal $E(\Q)$.
\begin{proposition}
Suppose $E=A_f$ has dimension one.  Then
$$\#(\O_f/I) \mid \#E(\Q) \mid \#(\O_f/L).$$
In particular
$$\#(E(\Q) / C_e) \mid [I:L].$$
\end{proposition}
\begin{proof}
$L$ is the ideal in $\O_f=\Z$ generated by
the elements $p+1-a_p=\#A_f(\F_p)$ for $p\nmid 2N$.
\end{proof}

\begin{question}
To what extent does this observation carry over to higher dimensional $A_f$?
What is the relationship between $A_f(\Q)$ and $C_e$?
Is $\#(A_f(\Q)/C_e)$ a power of $2$?
\end{question}

\section{Numerical Data}
Let $G'(f) = \gcd\{\#A_f(\F_p) \,:\, p\nmid 2N, \,\,p\leq 97\}$.
\begin{center}
\begin{tabular}{|l|c|c|c|c|}\hline
$f$  & $\L(f)$ & $|C_e|$ & $|A_f(\Q)|$ & $G'(f)$ \\ \hline\hline
{\bf 11A1} & $1/5$ & $5$ & $5$ & $5$ \\ \hline
{\bf 35B2} & $1/8$ & $8$ & ?   & $16$ \\ \hline
\end{tabular}
\end{center}
[This table will be extended later.]

\begin{thebibliography}{HHHHHHH}
\bibitem[DI]{diamondim} F. Diamon, J. Im, {\em Modular forms
and modular curves}, Seminar on Fermat's Last Theorem, CMS Conference
Proceedings, Volume 17, (1994).
\end{thebibliography} \normalsize\vspace*{1 cm}

\end{document}