% 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}