\documentclass[11pt]{article}1\include{macros}2\title{A Gross-Zagier Style Conjecture and the \\Birch and Swinnerton-Dyer Conjecture}3\author{William Stein}4\begin{document}5\maketitle67\tableofcontents89\section{Introduction}10In this paper we describe a conjecture that has a similar style to the11Gross-Zagier formula, and implies the Birch and Swinnerton-Dyer12conjecture. We then discuss some relevant computations related to the13conjecture for some curves of rank $>1$.141516\section{A Gross-Zagier Style Conjecture}\label{sec:thm}17Let $E$ be an elliptic curve defined over $\Q$ with analytic rank at18least $1$ and let $N$ be its conductor. Let $K$ be one of the19infinitely many quadratic imaginary fields with discriminant $D<-4$20coprime to $N$ such that each prime dividing $N$ splits in $K$, and21such that22$$\ord_{s=1} L(E^D,s)\leq 1.$$232425Fix any odd prime $\ell$ such that $\rhobar_{E,\ell}$ is26surjective. Below we only consider primes $p\nmid N D$ that are inert27in $K$. Set $N_p = \#E(\F_p)$, and let $\tilde{a}_p =28\ell^{\ord_{\ell}(p+1-N_p)}$ be the $\ell$-part of $a_p=p+1-N_p$. Let29$$30b_p = \#(E(\F_p)/\tilde{a}_p E(\F_p)),31$$32Note that $b_p = \gcd(p+1,\tilde{a}_p)$,33by \cite[Lemma~5.1]{ggz}.34For any squarefree positive integer $n$, let35$$36b_n = \gcd(\{b_p : p \mid n\}).37$$38Let $P_n = J_n I_n y_n \in E(K_n)$ be the Kolyvagin point associated to $n$, where39$K_n$ is the ring class field of $K$ of conductor $n$.40The elements $J_n, I_n \in \Z[\Gal(K_n/K)]$ are constructed so that41$$42[P_n] \in (E(K_n)/ b_n E(K_n))^{\Gal(K_n/K)}.43$$44See \cite{ggz} for more details.4546Let $r_{\an}(E/\Q) = \ord_{s=1}L(E/\Q,s)$ and let $t<r_{\an}(E/\Q)$ be any nonnegative integer with47$$48t\equiv r_{\an}(E/\Q) - 1\pmod{2}.49$$50For any prime $p\nmid n$ with $\ell \mid b_p \mid b_n$, reducing modulo any51choice of prime $\wp$ over $p\O_K$ yields a well defined point52$$53\overline{P}_n \in E(\F_{p})/\tilde{a}_p E(\F_p).54$$55The congruence condition on $t$ and our assumption that $\ell$ is odd implies56that $\overline{P}_n\in E(\F_p)$ and not just in $E(\F_{p^2})$.57Let $Y_p^t \subset E(\F_p)/\tilde{a}_p E(\F_p)$ be the subgroup58generated by all points $\overline{P}_n$ as we vary over all $n$59divisible by exactly~$t$ primes such that $b_p\mid b_n$.60The Chebotarev density theorem implies that there are infinity many61such integers $n$.6263Let $\pi_{p}:E(\Q) \to E(\F_p)/\tilde{a}_p E(\F_p)$ be the natural64quotient map, and let65$$66W_p^t = \pi_{p}^{-1}(Y_p^t) \subset E(\Q).67$$68If $G$ is a subgroup of $E(\Q)$ and $n$ is a positive integer, let69$$\langle G,G \rangle_n =\inf\{|\det\langle g_i, g_j\rangle| :70\text{ independent points } g_1, \ldots g_n \in G \},$$ where71$g_1,\ldots, g_n$ run through all choices of indepedent elements of72$G/_{\tor}$. If $G$ has rank $<n$, then $\langle G,G \rangle_n=0$73since it is the infimum of the empty set. Also not that one can prove74using reduction theory for quadratic forms that there are only75finitely many subgroups of $G$ of bounded height, so we can replace76the infimum by a minimum. Also, if $G$ has rank $n>0$, then $\langle G,77G\rangle_n = \Reg(G)$ is the regulator of $G$.7879Chose {\em any} maximal chain of subgroups80$W_{p_1}^t \supsetneq W_{p_2}^t \supsetneq W_{p_3}^t \dots$ associated to primes81$\cC=\{p_1, p_2, \ldots\}$, and let82$$83W^t_{\cC} = \bigcap_{p_i\in \cC} W_{p_i}^t.84$$8586Note that $\cC$ could be either finite or infinite. The intersection87$W^t_{\cC}$ may depend on $\cC$ and not just $t$, but we expect that88for each $t$, there are only finitely many possibilities for $W^t$ and89only one possibility for $[E(\Q):W^t]$. Also, since $Y_p$ is a90subgroup of the cyclic group $E(\F_p)/\tilde{a}_p E(\F_p)$, if91$W^t_{\cC}$ has finite index in $E(\Q)$, then the quotient92$E(\Q)/W^t_{\cC}$ is cyclic.9394Finally, let $$v = t + 1 + r_{\an}(E^D/\Q),$$ and note that95$v \leq r_{\an}(E/K)$ and $v \equiv r_{\an}(E/K)\pmod{2}$.96\begin{conjecture}\label{conj:ggz}97Fix a prime $\ell$, an integer $t$ and set of primes $\cC$ as98above. Then we have the following generalization of the Gross-Zagier99formula:100$$101\frac{L^{(v)}(E/K,1)}{v!} = \frac{b\cdot \|\omega\|^2}{c^2 \sqrt{|D|}}102\cdot\langle W^t_{\cC}, W^t_{\cC} \rangle_{t+1} \cdot \Reg(E^D/\Q),103$$104where $b$ is a positive integer not divisible by $\ell$.105\end{conjecture}106107Let $B$ be divisible by $2$ and the primes where $\rhobar_{E,\ell}$ is108not surjective.109\begin{conjecture}\label{conj:globalggz}110Let $t$ be an integer as above. For prime $\ell\nmid B$, make a choice of111$W(\ell) = W^t_{\cC}$ as above. Let $W=\cap_{\ell\nmid B} W(\ell)$. Then112$$113\frac{L^{(v)}(E/K,1)}{v!} = \frac{b\cdot \|\omega\|^2}{c^2 \sqrt{|D|}}114\cdot\langle W, W \rangle_{t+1} \cdot \Reg(E^D/\Q),115$$116where $b$ is an integer divisible only by prime divisors of $B$.117\end{conjecture}118The classical Gross-Zagier formula is like the above formula, but $v=1$,119we have $\Reg(E^D/\Q)=1$, and $\langle W, W \rangle_{t+1}$ is the height of the120Heegner point $P_1\in E(K)$.121122All the definitions above make sense with no assumption on123$\ell$, but we are not confident making the analogue of124Conjecture~\ref{conj:globalggz} without further data.125126\section{The Birch and Swinnerton-Dyer Conjecture}127128\begin{theorem}129Conjecture~\ref{conj:ggz} implies the BSD conjecture. More130precisely, if Conjecture~\ref{conj:ggz} is true, then131$$132\rank(E(\Q)) = \ord_{s=1} L(E/\Q,s)133$$134and $\Sha(E/\Q)[\ell^\infty]$ is finite.135\end{theorem}136\begin{proof}137First take $t=r_{\an}(E/\Q)-1$. If $Y_p^t=0$ for all $p$, then138using the Chebotarev density theorem (see ggz paper), we can find a139sequence of primes $p_i$ so that if $\cC=\{p_1, p_2, \ldots\}$, then140$W_{\cC}^t = 0$. However, in Conjecture~\ref{conj:ggz} we have141$v=r_{\an}(E/K)$, so $L^{(v)}(E/K,1)\neq 0$, hence $\langle142W_{\cC}^t,W_{\cC}^t \rangle_t\neq 0$ so $W_{\cC}^t$ is infinite.143Consequently, some $Y_p^t\neq 0$, hence some class $[P_n] \neq 0$144for some $n$ divisible by $t$ primes. Thus Kolyvagin's ``Conjecture145A'' is true with $f\leq r_{\an}(E/\Q)-1$. It follows by146\cite[Theorem~4.2]{ggz} that for all $m\gg0$ we have147\begin{equation}\label{eqn:kolysel}148\Sel^{(\ell^m)}(E/\Q) = (\ZZ/\ell^{m}\ZZ)^{f+1} \oplus S149\end{equation}150where $S$ is a finite group independent of $m$ (note that151conjecturally, $S$ is the $\ell$ part of $\Sha(E/\Q)$).152Thus $\rank(E/\Q) \leq f+1$.153154By Conjecture~\ref{conj:ggz} above there are $t+1$ independent155points in $W_{\cC}^t\subset E(\Q)$, so $t+1 \leq \rank(E/\Q)$ and $t+1 \leq156f+1$. Thus $f = r_{\an}(E/\Q)-1$, and the BSD conjecture that $\rank(E/\Q)157= r_{\an}(E/\Q)$ is true. Finiteness of $\Sha(E/\Q)[\ell^\infty]$ then follows158from \eqref{eqn:kolysel}.159\end{proof}160161%Let162%$$163% W_p = \pi_p^{-1}(Y_p)\subset E(\Q).164%$$165\section{Explicit Computations}166For the rest of this section, we let $t = r_{\an}(E/\Q) - 1$, and set $Y_p =167Y_p^t$.168169\begin{theorem}\label{thm:yp}170Assume Conjecture~\ref{conj:ggz}, the BSD171formula at $\ell$ for $E$ over $K$, and Kolyvagin's Conjecture172$D_{\ell}$. Then for any good prime $p$, the group $Y_p$ is the173image of $I \cdot E(\Q)$ in $E(\F_p)/\tilde{a}_p E(\F_p)$, where174$$175I = c \prod c_q \prod \sqrt{\#\Sha(E/K)}.176$$177178[[worry -- there is a ``sufficiently large'' in Kolyvagin? If so,179make this a conjecture, then give a theorem for sufficiently large as180evidence.]]181182[[worry -- the above only gives $W_p$, not $Y_p$]]183\end{theorem}184\begin{proof}185This is Proposition~7.3 of \cite{ggz}. (It might be that assuming186Kolyvagin's Conjecture $D_{\ell}$ is redundant.)187\end{proof}188189Assuming the conclusion of Theorem~\ref{thm:yp}, we can {\em in190practice} compute the group $Y_p$ for any elliptic curve $E$. We191can thus conditionally verify Conjecture~\ref{conj:ggz}. Just192verifying the conjecture is not worth doing, since under the above193hypothesis, Conjecture~\ref{conj:ggz} is implied by the BSD formula,194since $\pi_P^{-1}(Y_p)$ has small enough index that it must contain a195Gross-Zagier subgroup (see \cite[Prop.~2.4]{ggz} and196\cite[Lemma.~7.4]{ggz}). There is, however, extra information197contained in {\em which} subgroup $\pi_p^{-1}(Y_p)$ we find for a198given $p$, since that does depend in a possiby subtle way on $p$.199200A deeper structure on $Y_p$ is that it has labeled generators201$\overline{P}_n$, indexed by positive integers $n$. So far, it202appears to be a highly nontrivial calculation to explicitly compute a203specific $\overline{P}_n$ in any particular case.204205In the rest of this section, we compute as much as we reasonably can206about the objects above in some specific examples.207208For the computations below, we assume BSD and Kolyvagin's conjecture so we209can use Theorem~\ref{thm:yp} to compute $Y_p$.210211\begin{example}212Let $E$ be the rank 2 elliptic curve 389a, and let $\ell=3$. We have213$v=0$, since $c=c_{389}=1$ and $\#\Sha_{\an}=1.000\ldots$. The214primes $p<100$ such that $E(\F_p)/\tilde{a}_p E({\F}_P) \neq 0$ are215$P = \{5, 17, 29, 41, 53, 59, 83\}$, and in each case $E(\F_p)/\tilde{a}_p216E(\F_P)$ is cyclic of order $3$.217We have218$$219E(\Q) = \Z P_1 \oplus \Z P_2220$$221where $P_1 = (-1,1)$ and $P_2=(0,-1)$.222223224225\end{example}226227228\section{Future Directions and Projects}229230\begin{enumerate}231\item Assuming the hypothesis of Theorem~\ref{thm:yp}, compute232groups $W$ for various choices of $W_{p_1}\supsetneq W_{p_2} \supsetneq \cdots$233when $I\neq 1$.234\item Formulate Conjecture~\ref{conj:ggz} on $J_0(N)$ over the Hilbert235class field of $K$, and deduce Conjecture~\ref{conj:ggz} from this more236general conjecture.237\item Formulate Conjecture~\ref{conj:ggz} at {\em all} primes $\ell$ hence238get an exact formula for $\langle W, W \rangle_{t+1}$ as almost239in Conjecture~\ref{conj:globalggz}.240\item Find an algorithm to compute $Y_p$ or $W_p$. This would be241especially interesting when Theorem~\ref{thm:yp} does not apply.242Give a conjectural description of $Y_p$ in {\em all} cases.243\end{enumerate}244245\end{document}246247