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Author: William A. Stein
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\documentclass[11pt]{article}
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\include{macros}
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\title{A Gross-Zagier Style Conjecture and the \\Birch and Swinnerton-Dyer Conjecture}
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\author{William Stein}
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\begin{document}
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\maketitle
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\tableofcontents
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\section{Introduction}
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In this paper we describe a conjecture that has a similar style to the
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Gross-Zagier formula, and implies the Birch and Swinnerton-Dyer
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conjecture. We then discuss some relevant computations related to the
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conjecture for some curves of rank $>1$.
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\section{A Gross-Zagier Style Conjecture}\label{sec:thm}
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Let $E$ be an elliptic curve defined over $\Q$ with analytic rank at
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least $1$ and let $N$ be its conductor. Let $K$ be one of the
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infinitely many quadratic imaginary fields with discriminant $D<-4$
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coprime to $N$ such that each prime dividing $N$ splits in $K$, and
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such that
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$$\ord_{s=1} L(E^D,s)\leq 1.$$
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Fix any odd prime $\ell$ such that $\rhobar_{E,\ell}$ is
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surjective. Below we only consider primes $p\nmid N D$ that are inert
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in $K$. Set $N_p = \#E(\F_p)$, and let $\tilde{a}_p =
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\ell^{\ord_{\ell}(p+1-N_p)}$ be the $\ell$-part of $a_p=p+1-N_p$. Let
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$$
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b_p = \#(E(\F_p)/\tilde{a}_p E(\F_p)),
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$$
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Note that $b_p = \gcd(p+1,\tilde{a}_p)$,
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by \cite[Lemma~5.1]{ggz}.
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For any squarefree positive integer $n$, let
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$$
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b_n = \gcd(\{b_p : p \mid n\}).
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$$
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Let $P_n = J_n I_n y_n \in E(K_n)$ be the Kolyvagin point associated to $n$, where
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$K_n$ is the ring class field of $K$ of conductor $n$.
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The elements $J_n, I_n \in \Z[\Gal(K_n/K)]$ are constructed so that
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$$
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[P_n] \in (E(K_n)/ b_n E(K_n))^{\Gal(K_n/K)}.
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$$
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See \cite{ggz} for more details.
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Let $r_{\an}(E/\Q) = \ord_{s=1}L(E/\Q,s)$ and let $t<r_{\an}(E/\Q)$ be any nonnegative integer with
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$$
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t\equiv r_{\an}(E/\Q) - 1\pmod{2}.
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$$
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For any prime $p\nmid n$ with $\ell \mid b_p \mid b_n$, reducing modulo any
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choice of prime $\wp$ over $p\O_K$ yields a well defined point
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$$
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\overline{P}_n \in E(\F_{p})/\tilde{a}_p E(\F_p).
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$$
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The congruence condition on $t$ and our assumption that $\ell$ is odd implies
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that $\overline{P}_n\in E(\F_p)$ and not just in $E(\F_{p^2})$.
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Let $Y_p^t \subset E(\F_p)/\tilde{a}_p E(\F_p)$ be the subgroup
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generated by all points $\overline{P}_n$ as we vary over all $n$
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divisible by exactly~$t$ primes such that $b_p\mid b_n$.
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The Chebotarev density theorem implies that there are infinity many
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such integers $n$.
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Let $\pi_{p}:E(\Q) \to E(\F_p)/\tilde{a}_p E(\F_p)$ be the natural
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quotient map, and let
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$$
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W_p^t = \pi_{p}^{-1}(Y_p^t) \subset E(\Q).
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$$
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If $G$ is a subgroup of $E(\Q)$ and $n$ is a positive integer, let
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$$\langle G,G \rangle_n =\inf\{|\det\langle g_i, g_j\rangle| :
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\text{ independent points } g_1, \ldots g_n \in G \},$$ where
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$g_1,\ldots, g_n$ run through all choices of indepedent elements of
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$G/_{\tor}$. If $G$ has rank $<n$, then $\langle G,G \rangle_n=0$
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since it is the infimum of the empty set. Also not that one can prove
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using reduction theory for quadratic forms that there are only
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finitely many subgroups of $G$ of bounded height, so we can replace
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the infimum by a minimum. Also, if $G$ has rank $n>0$, then $\langle G,
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G\rangle_n = \Reg(G)$ is the regulator of $G$.
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Chose {\em any} maximal chain of subgroups
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$W_{p_1}^t \supsetneq W_{p_2}^t \supsetneq W_{p_3}^t \dots$ associated to primes
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$\cC=\{p_1, p_2, \ldots\}$, and let
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$$
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W^t_{\cC} = \bigcap_{p_i\in \cC} W_{p_i}^t.
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$$
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Note that $\cC$ could be either finite or infinite. The intersection
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$W^t_{\cC}$ may depend on $\cC$ and not just $t$, but we expect that
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for each $t$, there are only finitely many possibilities for $W^t$ and
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only one possibility for $[E(\Q):W^t]$. Also, since $Y_p$ is a
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subgroup of the cyclic group $E(\F_p)/\tilde{a}_p E(\F_p)$, if
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$W^t_{\cC}$ has finite index in $E(\Q)$, then the quotient
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$E(\Q)/W^t_{\cC}$ is cyclic.
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Finally, let $$v = t + 1 + r_{\an}(E^D/\Q),$$ and note that
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$v \leq r_{\an}(E/K)$ and $v \equiv r_{\an}(E/K)\pmod{2}$.
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\begin{conjecture}\label{conj:ggz}
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Fix a prime $\ell$, an integer $t$ and set of primes $\cC$ as
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above. Then we have the following generalization of the Gross-Zagier
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formula:
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$$
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\frac{L^{(v)}(E/K,1)}{v!} = \frac{b\cdot \|\omega\|^2}{c^2 \sqrt{|D|}}
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\cdot\langle W^t_{\cC}, W^t_{\cC} \rangle_{t+1} \cdot \Reg(E^D/\Q),
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$$
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where $b$ is a positive integer not divisible by $\ell$.
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\end{conjecture}
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Let $B$ be divisible by $2$ and the primes where $\rhobar_{E,\ell}$ is
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not surjective.
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\begin{conjecture}\label{conj:globalggz}
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Let $t$ be an integer as above. For prime $\ell\nmid B$, make a choice of
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$W(\ell) = W^t_{\cC}$ as above. Let $W=\cap_{\ell\nmid B} W(\ell)$. Then
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$$
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\frac{L^{(v)}(E/K,1)}{v!} = \frac{b\cdot \|\omega\|^2}{c^2 \sqrt{|D|}}
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\cdot\langle W, W \rangle_{t+1} \cdot \Reg(E^D/\Q),
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$$
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where $b$ is an integer divisible only by prime divisors of $B$.
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\end{conjecture}
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The classical Gross-Zagier formula is like the above formula, but $v=1$,
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we have $\Reg(E^D/\Q)=1$, and $\langle W, W \rangle_{t+1}$ is the height of the
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Heegner point $P_1\in E(K)$.
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All the definitions above make sense with no assumption on
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$\ell$, but we are not confident making the analogue of
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Conjecture~\ref{conj:globalggz} without further data.
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\section{The Birch and Swinnerton-Dyer Conjecture}
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\begin{theorem}
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Conjecture~\ref{conj:ggz} implies the BSD conjecture. More
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precisely, if Conjecture~\ref{conj:ggz} is true, then
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$$
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\rank(E(\Q)) = \ord_{s=1} L(E/\Q,s)
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$$
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and $\Sha(E/\Q)[\ell^\infty]$ is finite.
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\end{theorem}
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\begin{proof}
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First take $t=r_{\an}(E/\Q)-1$. If $Y_p^t=0$ for all $p$, then
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using the Chebotarev density theorem (see ggz paper), we can find a
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sequence of primes $p_i$ so that if $\cC=\{p_1, p_2, \ldots\}$, then
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$W_{\cC}^t = 0$. However, in Conjecture~\ref{conj:ggz} we have
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$v=r_{\an}(E/K)$, so $L^{(v)}(E/K,1)\neq 0$, hence $\langle
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W_{\cC}^t,W_{\cC}^t \rangle_t\neq 0$ so $W_{\cC}^t$ is infinite.
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Consequently, some $Y_p^t\neq 0$, hence some class $[P_n] \neq 0$
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for some $n$ divisible by $t$ primes. Thus Kolyvagin's ``Conjecture
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A'' is true with $f\leq r_{\an}(E/\Q)-1$. It follows by
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\cite[Theorem~4.2]{ggz} that for all $m\gg0$ we have
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\begin{equation}\label{eqn:kolysel}
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\Sel^{(\ell^m)}(E/\Q) = (\ZZ/\ell^{m}\ZZ)^{f+1} \oplus S
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\end{equation}
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where $S$ is a finite group independent of $m$ (note that
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conjecturally, $S$ is the $\ell$ part of $\Sha(E/\Q)$).
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Thus $\rank(E/\Q) \leq f+1$.
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By Conjecture~\ref{conj:ggz} above there are $t+1$ independent
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points in $W_{\cC}^t\subset E(\Q)$, so $t+1 \leq \rank(E/\Q)$ and $t+1 \leq
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f+1$. Thus $f = r_{\an}(E/\Q)-1$, and the BSD conjecture that $\rank(E/\Q)
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= r_{\an}(E/\Q)$ is true. Finiteness of $\Sha(E/\Q)[\ell^\infty]$ then follows
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from \eqref{eqn:kolysel}.
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\end{proof}
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%Let
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%$$
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% W_p = \pi_p^{-1}(Y_p)\subset E(\Q).
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%$$
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\section{Explicit Computations}
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For the rest of this section, we let $t = r_{\an}(E/\Q) - 1$, and set $Y_p =
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Y_p^t$.
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\begin{theorem}\label{thm:yp}
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Assume Conjecture~\ref{conj:ggz}, the BSD
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formula at $\ell$ for $E$ over $K$, and Kolyvagin's Conjecture
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$D_{\ell}$. Then for any good prime $p$, the group $Y_p$ is the
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image of $I \cdot E(\Q)$ in $E(\F_p)/\tilde{a}_p E(\F_p)$, where
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$$
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I = c \prod c_q \prod \sqrt{\#\Sha(E/K)}.
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$$
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[[worry -- there is a ``sufficiently large'' in Kolyvagin? If so,
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make this a conjecture, then give a theorem for sufficiently large as
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evidence.]]
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[[worry -- the above only gives $W_p$, not $Y_p$]]
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\end{theorem}
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\begin{proof}
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This is Proposition~7.3 of \cite{ggz}. (It might be that assuming
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Kolyvagin's Conjecture $D_{\ell}$ is redundant.)
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\end{proof}
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Assuming the conclusion of Theorem~\ref{thm:yp}, we can {\em in
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practice} compute the group $Y_p$ for any elliptic curve $E$. We
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can thus conditionally verify Conjecture~\ref{conj:ggz}. Just
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verifying the conjecture is not worth doing, since under the above
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hypothesis, Conjecture~\ref{conj:ggz} is implied by the BSD formula,
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since $\pi_P^{-1}(Y_p)$ has small enough index that it must contain a
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Gross-Zagier subgroup (see \cite[Prop.~2.4]{ggz} and
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\cite[Lemma.~7.4]{ggz}). There is, however, extra information
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contained in {\em which} subgroup $\pi_p^{-1}(Y_p)$ we find for a
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given $p$, since that does depend in a possiby subtle way on $p$.
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A deeper structure on $Y_p$ is that it has labeled generators
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$\overline{P}_n$, indexed by positive integers $n$. So far, it
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appears to be a highly nontrivial calculation to explicitly compute a
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specific $\overline{P}_n$ in any particular case.
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In the rest of this section, we compute as much as we reasonably can
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about the objects above in some specific examples.
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For the computations below, we assume BSD and Kolyvagin's conjecture so we
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can use Theorem~\ref{thm:yp} to compute $Y_p$.
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\begin{example}
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Let $E$ be the rank 2 elliptic curve 389a, and let $\ell=3$. We have
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$v=0$, since $c=c_{389}=1$ and $\#\Sha_{\an}=1.000\ldots$. The
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primes $p<100$ such that $E(\F_p)/\tilde{a}_p E({\F}_P) \neq 0$ are
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$P = \{5, 17, 29, 41, 53, 59, 83\}$, and in each case $E(\F_p)/\tilde{a}_p
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E(\F_P)$ is cyclic of order $3$.
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We have
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$$
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E(\Q) = \Z P_1 \oplus \Z P_2
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$$
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where $P_1 = (-1,1)$ and $P_2=(0,-1)$.
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\end{example}
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\section{Future Directions and Projects}
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\begin{enumerate}
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\item Assuming the hypothesis of Theorem~\ref{thm:yp}, compute
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groups $W$ for various choices of $W_{p_1}\supsetneq W_{p_2} \supsetneq \cdots$
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when $I\neq 1$.
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\item Formulate Conjecture~\ref{conj:ggz} on $J_0(N)$ over the Hilbert
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class field of $K$, and deduce Conjecture~\ref{conj:ggz} from this more
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general conjecture.
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\item Formulate Conjecture~\ref{conj:ggz} at {\em all} primes $\ell$ hence
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get an exact formula for $\langle W, W \rangle_{t+1}$ as almost
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in Conjecture~\ref{conj:globalggz}.
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\item Find an algorithm to compute $Y_p$ or $W_p$. This would be
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especially interesting when Theorem~\ref{thm:yp} does not apply.
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Give a conjectural description of $Y_p$ in {\em all} cases.
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\end{enumerate}
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\end{document}
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