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Author: William A. Stein
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% denominator.tex
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\documentclass[12pt]{article}
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\include{macros}
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\title{The Denominator of the Special Value $L(A_f,1)/\Omega(A_f)$}
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\begin{document}
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\author{Amod Agashe and William Stein\footnote{email: {\tt amod} and {\tt was} at {\tt math.berkeley.edu}.}}
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\maketitle
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\begin{abstract}
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Let $f$ be a newform and $A_f$ the quotient
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of $J_0(N)$ constructed by Shimura.
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We prove that, up to a Manin constant and a power of $2$,
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the denominator of the rational number $L(A_f,1)/\Omega(A_f)$
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divides the order of the image of $(0)-(\infty)$ in $A_f(\Q)$.
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This provides evidence for the Birch and Swinnerton-Dyer
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conjecture and raises questions about the structure of $A_f(\Q)$.
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\end{abstract}
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\section{Introduction}
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Fix a positive integer $N$.
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Let $f\in S_2(\Gamma_0(N))$ be a newform and let
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$A_f$ be the corresponding optimal quotient of $J_0(N)$.
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The $L$-function of $A_f$ is
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$$L(A_f,s)=\prod_{i=1}^d L(f_i,s)$$
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where $f_1,\ldots,f_d$ are the Galois conjugates of $f$.
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Let
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$$\Omega(A_f) = \int_{A_f(\R)} |\omega|$$
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where $\omega$ is a differential $d$-form on the Neron
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model of $A_f$. For $p|N$ the analogous local quantity
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is $c_p$ which is
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the number of $\Fp$-rational components of the
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special fiber of the Neron model of $A_f$ at $p$.
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Let
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$$\Sha(A_f) = \Ker [
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H^1(\Q,A_f)\ra \prod_v H^1(\Qv,A_f) ]$$
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where the product is over all primes $p$ and $\infty$.
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The Birch and Swinnerton-Dyer conjecture (BSD conjecture),
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as generalized by Tate, predicts that $\Sha(A_f)$ is finite and
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$$\frac{L(A_f,1)}{\Omega(A_f)}
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= \frac{\#\Sha(A_f)\cdot\prod_{p|N} c_p}
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{\#A_f(\Q) \cdot \# A_f^{\vee}(\Q)}.$$
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When $\#A_f(\Q)$ is infinite the right hand side is $0$.
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This conjecture is currently the subject of much
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intensive research. We do have
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\begin{theorem}[Kolyvagin, Logachev]
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If $L(A_f,1)\neq 0$ then both $A_f(\Q)$ and
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$\Sha(A_f)$ are finite.
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\end{theorem}
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We first express $L(A_f,1)$ in terms of modular symbols
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in order to show that the denominator divides the order
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of a certain subgroup of $\#A_f(\Q)$. Not only does our result
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provide evidence for the BSD conjecture, but assuming the BSD conjecture
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it also suggests that the natural map
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$$A_f(\Q) \ra \prod_{p|N} \Phi_{A_f,p}$$
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should be (very close to) injective.
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\section{Modular Symbols Expression for $L(A_f,1)$}
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Fix a newform $f$ as in the introduction and assume that
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$L(A_f,1)\neq 0$.
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Let $H_1(X_0(N),\Z)$ be the first integral homology of
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the modular curve $X_0(N)$. The Hecke algebra $\T$ and the
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involution $*$ both act on $H_1(X_0(N),\Z)$, and their
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actions commute. We have an exact sequence
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$$H_1(X_0(N),\Z)\xrightarrow{\Phi_f} \C^d \ra A_f(\C) \ra 0$$
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where
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$$\Phi_f(\gamma) = (\int_\gamma f_1, \ldots, \int_\gamma f_d).$$
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Let $\e=\{0,\infty\}\in H_1(X_0(N),\Q)$ correspond to integration
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along the vertical path from $0$ to $i\infty$.
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\begin{definition}[Module Index]
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Let $V$ be a $\Q$-vector space and let $L, M\subset V$ be
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lattices (of full rank). Denote by $[L:M]$ the absolute
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value of the determinant of an automorphism of $V$ sending
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$L$ to $M$.
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\end{definition}
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Define
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$$\L(f) = [\Phi_f(H_1(X_0(N),\Z)^+) : \Phi_f(\T\e)]$$
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where the images are lattices in $V = \Phi_f(H_1(X_0(N),\Q)^+)$.
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\begin{theorem}
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We have
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$$\L(f) = \frac{|L(A_f,1)|}{\Omega(A_f)}\cdot c_\infty \cdot c_M$$
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where $c_\infty$ is the number of real components and
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$c_M$ is the Manin constant.
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\end{theorem}
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\comment{The key idea in the proof involves relating the space $S_2(\Gamma_0(N),\Z)$
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of integral cusp forms to the space of global differentials on the Neron model,
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then using the perfect duality between $\T$ and $S_2(\Gamma_0(N),\Z)$. }
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\section{The Denominator of the Special Value}
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Fix a newform $f=\sum a_n q^n$ as in the introduction and assume
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$$L(A_f,1)\neq 0.$$ The cusps $0$ and $\infty$ on
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$X_0(N)$ give rise to a rational torsion point
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$$P_e=(0)-(\infty)\in J_0(N).$$
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Let $\overline{P}_e$ denote the image of $P_e$ in $A_f(\Q)$ and let
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$$C_e=\Z\overline{P}_e$$
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be the cyclic subgroup of $A_f(\Q)$ which it generates.
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Let
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$$\p_f = \{ t \in \T : t(f) = 0\}.$$
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The Hecke algebra $\T$ acts on $A_f$ through a quotient:
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$$\O_f = \T/\p_f \isom \Z[a_1,a_2,\ldots].$$
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Let $\pi :\T\ra \O_f$ denote the natural surjection.
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\begin{proposition}
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The group $C_e\subset A_f(\Q)$ is $\O_f$-invariant.
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\end{proposition}
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\begin{proof}
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It suffices to show that each $\pi(T_p)$ acts as a scalar on $\overline{P}_e$,
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since the $\pi(T_p)$ generate $\O_f$. Let $p$ be a prime.\\
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Case $p\nmid N$: Then, following the proof the Manin-Drinfeld theorem,
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$$T_p P_e = (p+1) P_e.$$
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Thus
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$$\pi(T_p)\overline{P}_e = (p+1) \overline{P}_e.$$\\
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Case $p\nmid N$: Then $\pi(T_p)=a_p\in \{0,\pm 1\}$
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(see bottom of page 64 of \cite{diamondim}).
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Thus
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$$\pi(T_p)\overline{P}_e \in \{ 0, \pm \overline{P}_e\}\subset C_e.$$
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\end{proof}
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\begin{theorem}\label{thmdenom}
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The denominator of $\L(f)$ divides the order of
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the cyclic subgroup $C_e\subset A_f(\Q)$.
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\end{theorem}
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\begin{proof}
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In the definition of $\L(f)$ we can identify $V=\Phi_f(H_1(X_0(N),\Q)^+)$
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with the $\Q$-vector space
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$H_1(X_0(N),\Q)^+/\Ker(\Phi_f)$. Thus we may replace $\Phi_f$ by any
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homomorphism $\Phi$ eminating from $H_1(X_0(N),\Z)$ and having the
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same kernel as $\Phi_f$. The resulting module index $\L(f)$
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remains unchanged.
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Choose some nonzero
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$$\Phi \in \Hom(H_1(X_0(N),\Z)^+,\O_f)$$
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satisfying the following additional requirement:
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$$\Phi(t\gamma) = \pi(t)\Phi(\gamma),\qquad\text{for all $t\in\T$}$$
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By ``multiplicity one'' and duality between homology and differentials,
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$\Phi$ exists and is uniquely determined up to
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a nonzero scalar in $\O_f$.
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Both
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$\Phi(H_1(X_0(N),\Z)^+)$ and $\Phi(\T\e)$ are
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contained in $K=\Frac(\O_f)$. They are fractional $\O_f$-ideals.
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Furthermore
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$$\L(f) = [\Phi(H_1(X_0(N),\Z)^+) : \Phi(\T\e)].$$
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Next define an ideal $I\subset \O_f$ by exactness of
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$$0\ra I \ra \O_f \xrightarrow{t\mapsto t.\overline{P}_e} C_e \ra 0.$$
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The map $\O_f\ra C_e$ is surjective because $\O_f$ contains $\Z$ and $C_f$ is
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cyclic as an abelian group. Thus $\O_f/I\isom C_e$ is finite cyclic.
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Furthermore, the Abel-Jacobi theorem implies that
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$$ I = \{ t \in \O : t \Phi(\e) \in \Phi(H_1(X_0(N),\Z)^+) \}.$$
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We are now in a position to bound the denominator of $\L(f)$. Writing
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$H=H_1(X_0(N),\Z)^+$ we have
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\begin{eqnarray*}\L(f) = [\Phi(H):\Phi(\T\e)] &=& [\Phi(H):\O_f\Phi(\e)] \\
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&=& [\Phi(H) : I\Phi(\e)]\cdot [I\Phi(\e) : \O_f\Phi(\e)]\\
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&=& \frac{[\Phi(H) : I\Phi(\e)]}
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{[\O_f\Phi(\e) : I\Phi(\e)]}.
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\end{eqnarray*}
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Next observe that:\\
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\indent 1) $I \Phi(\e)\subset \Phi(H)$ because of the construction of $I$, and\\
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\indent 2) $I\Phi(\e)\subset\O_f\Phi(e)$ because $I\subset\O_f$.\\
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It follows that
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$$[\Phi(H):I\Phi(\e)]\in\Z\text{ and } [\O_f\Phi(\e) : I\Phi(\e)]\in \Z.$$
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Thus the denominator of $\L(f)$ divides
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$$[\O_f\Phi(\e):I\Phi(\e)]=[\O_f:I]=\#(\O_f/I)=\#C_e.$$
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\end{proof}
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\begin{question}
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Note that the ideal class of the ideal
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$$\Phi(H_1(X_0(N),\Z)^+)\subset \O_f$$
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is independent of the choice of $\Phi$.
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What is the significance of this class in
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the ideal class group of $\O_f$?
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What is its order?
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\end{question}
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\begin{question}
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In the proof we expressed $\L(f)$ as a quotient
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$$\L(f) = \frac{[\Phi(H):I\Phi(\e)]}
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{[\O_f\Phi(e): I\Phi(e)]}.$$
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Both the numerator and the denominator are well-defined, irregardless
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of the choice of $\Phi$. How do they relate to the numerator
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and the denominator in the BSD conjecture?
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In particular, can we:\\
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\indent connect $[\Phi(H):I\Phi(\e)]$ with $\#\Sha(A_f) \cdot|\prod_{p|N} c_p|$, or\\
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\indent connect $[\O_f\Phi(e): I\Phi(e)]$ with $\#A_f(\Q) \cdot\#A^{\vee}_f(\Q)$?\\
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\end{question}
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\section{Idea to Bound $A_f(\Q)$}
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Fix a newform $f$ of level $N$ and assume that $L(A_f,1)\neq 0$.
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\begin{theorem}
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Suppose $p\nmid 2N$. Then
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$$A_f(\Q)\hookrightarrow A_f(\F_p).$$
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\end{theorem}
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Thus $\#A_f(\Q)$ divides
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$$G(f) = \gcd\{\#A_f(\F_p) : p\nmid 2N\}.$$
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Let the notation be as in the proof of Theorem~\ref{thmdenom}.
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Thus we have
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$$\O_f=\T/\p_f=\Z[a_1,a_2,\ldots]$$
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and a map
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$$\Phi:H_1(X_0(N),\Z)^+\ra \O_f$$
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such that
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$$\Phi(t\gamma)=\pi(t)\Phi(\gamma).$$
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We also have the ideal
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$$I = \{ t \in \O_f : t\Phi(\e)\in \Phi(H_1(X_0(N),\Z)^+) \}.$$
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When proving the theorem we observed that
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$$\#(\O/I)=\#C_e \mid \#A_f(\Q).$$
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Consider the ideal $L\subset \O_f$ generated by the ``obvious'' elements of $I$:
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$$L = ( (p+1) - \pi(T_p) : p\nmid 2N )\subset I$$
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We will use the following theorem to relate $\#(\O_f/L)$ to $G(f)$.
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\begin{theorem}
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Suppose $p\nmid N$.
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Let $F(x)$ be the characteristic polynomial of $\pi(T_p)$.
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Then
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$$\#A_f(\F_p) = F(p+1).$$
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\end{theorem}
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Define the norm of $x\in \O_f$ to be the determinant of the
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linear map $\ell_x = $ left multiplication by $x$. Observe that
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$$|\Norm(x)|=[\O_f:x\O_f]=\#(\O_f/x\O_f).$$
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\begin{corollary}
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Suppose $p\nmid N$. Then
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$$\Norm((p+1) - \pi(T_p)) = \#A_f(\F_p).$$
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\end{corollary}
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\begin{proof}
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If $F(x)$ is the characteristic polynomial of $\pi(T_p)$ then
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$F(p+1)$ is the determinant of left multiplication by
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$(p+1) - \pi(T_p)$.
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\end{proof}
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\begin{lemma}
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Let $\a$ be an ideal of $\O_f$.
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Then
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$$\#(\O_f/\a) \mid \gcd\{\Norm(x) : x \in \a\}.$$
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\end{lemma}
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\begin{proof}
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We have
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$$\#(\O_f/\a) \mid [\O_f : \a]\cdot [\a : x\O_f] = [\O_f:x\O_f] = |\Norm(x)|\in \Z.$$
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\end{proof}
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Thus
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$$\#(\O_f/L) \mid G(f) = \gcd\{\#A_f(\F_p) : p\nmid 2N\}.$$
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\begin{remark}
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When I first looked at this I thought that maybe
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there would be equality. (For example, if $\O_f=\Z$ there is indeed equality.)
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But, I see no reason for equality now.
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\end{remark}
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In the elliptic case
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the index of $L$ in $I$ measures the failure of
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$C_e$ to equal $E(\Q)$.
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\begin{proposition}
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Suppose $E=A_f$ has dimension one. Then
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$$\#(\O_f/I) \mid \#E(\Q) \mid \#(\O_f/L).$$
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In particular
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$$\#(E(\Q) / C_e) \mid [I:L].$$
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\end{proposition}
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\begin{proof}
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$L$ is the ideal in $\O_f=\Z$ generated by
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the elements $p+1-a_p=\#A_f(\F_p)$ for $p\nmid 2N$.
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\end{proof}
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\begin{question}
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To what extent does this observation carry over to higher dimensional $A_f$?
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What is the relationship between $A_f(\Q)$ and $C_e$?
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Is $\#(A_f(\Q)/C_e)$ a power of $2$?
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\end{question}
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\section{Numerical Data}
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Let $G'(f) = \gcd\{\#A_f(\F_p) \,:\, p\nmid 2N, \,\,p\leq 97\}$.
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\begin{center}
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\begin{tabular}{|l|c|c|c|c|}\hline
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$f$ & $\L(f)$ & $|C_e|$ & $|A_f(\Q)|$ & $G'(f)$ \\ \hline\hline
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{\bf 11A1} & $1/5$ & $5$ & $5$ & $5$ \\ \hline
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{\bf 35B2} & $1/8$ & $8$ & ? & $16$ \\ \hline
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\end{tabular}
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\end{center}
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[This table will be extended later.]
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\begin{thebibliography}{HHHHHHH}
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\bibitem[DI]{diamondim} F. Diamon, J. Im, {\em Modular forms
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and modular curves}, Seminar on Fermat's Last Theorem, CMS Conference
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Proceedings, Volume 17, (1994).
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\end{thebibliography} \normalsize\vspace*{1 cm}
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\end{document}
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