\documentclass[11pt]{article}
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\title{A Gross-Zagier Style Conjecture and the \\Birch and Swinnerton-Dyer Conjecture}
\author{William Stein}
\begin{document}
\maketitle

\tableofcontents 

\section{Introduction}
In this paper we describe a conjecture that has a similar style to the
Gross-Zagier formula, and implies the Birch and Swinnerton-Dyer
conjecture.  We then discuss some relevant computations related to the
conjecture for some curves of rank $>1$.


\section{A Gross-Zagier Style Conjecture}\label{sec:thm}
Let $E$ be an elliptic curve defined over $\Q$ with analytic rank at
least $1$ and let $N$ be its conductor.  Let $K$ be one of the
infinitely many quadratic imaginary fields with discriminant $D<-4$
coprime to $N$ such that each prime dividing $N$ splits in $K$, and
such that
$$\ord_{s=1} L(E^D,s)\leq 1.$$


Fix any odd prime $\ell$ such that $\rhobar_{E,\ell}$ is
surjective. Below we only consider primes $p\nmid N D$ that are inert
in $K$.  Set $N_p = \#E(\F_p)$, and let $\tilde{a}_p =
\ell^{\ord_{\ell}(p+1-N_p)}$ be the $\ell$-part of $a_p=p+1-N_p$.  Let
$$
  b_p = \#(E(\F_p)/\tilde{a}_p E(\F_p)),
$$
Note that $b_p = \gcd(p+1,\tilde{a}_p)$, 
by \cite[Lemma~5.1]{ggz}.
For any squarefree positive integer $n$, let
$$
  b_n = \gcd(\{b_p : p \mid n\}).
$$
Let $P_n = J_n I_n y_n \in E(K_n)$ be the Kolyvagin point associated to $n$, where
$K_n$ is the ring class field of $K$ of conductor $n$.
The elements $J_n, I_n \in \Z[\Gal(K_n/K)]$ are constructed so that 
$$
  [P_n] \in (E(K_n)/ b_n E(K_n))^{\Gal(K_n/K)}.
$$
See \cite{ggz} for more details.

Let $r_{\an}(E/\Q) = \ord_{s=1}L(E/\Q,s)$ and let $t<r_{\an}(E/\Q)$ be any nonnegative integer with
$$
   t\equiv r_{\an}(E/\Q) - 1\pmod{2}.
$$
For any prime $p\nmid n$ with $\ell \mid b_p \mid b_n$, reducing modulo any
choice of prime $\wp$ over $p\O_K$ yields a well defined point
$$
   \overline{P}_n \in E(\F_{p})/\tilde{a}_p E(\F_p).
$$
The congruence condition on $t$ and our assumption that $\ell$ is odd implies 
that $\overline{P}_n\in E(\F_p)$ and not just in $E(\F_{p^2})$. 
Let $Y_p^t \subset E(\F_p)/\tilde{a}_p E(\F_p)$ be the subgroup
generated by all points $\overline{P}_n$ as we vary over all $n$
divisible by exactly~$t$ primes such that $b_p\mid b_n$.
The Chebotarev density theorem implies that there are infinity many
such integers $n$.

Let $\pi_{p}:E(\Q) \to E(\F_p)/\tilde{a}_p E(\F_p)$ be the natural
quotient map, and let
$$
   W_p^t = \pi_{p}^{-1}(Y_p^t) \subset E(\Q).
$$
If $G$ is a subgroup of $E(\Q)$ and $n$ is a positive integer, let 
$$\langle G,G \rangle_n =\inf\{|\det\langle g_i, g_j\rangle| : 
\text{ independent points } g_1, \ldots g_n \in G \},$$ where
$g_1,\ldots, g_n$ run through all choices of indepedent elements of
$G/_{\tor}$.  If $G$ has rank $<n$, then $\langle G,G \rangle_n=0$
since it is the infimum of the empty set.  Also not that one can prove
using reduction theory for quadratic forms that there are only
finitely many subgroups of $G$ of bounded height, so we can replace
the infimum by a minimum.  Also, if $G$ has rank $n>0$, then $\langle G,
G\rangle_n = \Reg(G)$ is the regulator of $G$.

Chose {\em any} maximal chain of subgroups
$W_{p_1}^t \supsetneq W_{p_2}^t \supsetneq W_{p_3}^t \dots$ associated to primes
$\cC=\{p_1, p_2, \ldots\}$, and let 
$$
  W^t_{\cC} = \bigcap_{p_i\in \cC} W_{p_i}^t.
$$

Note that $\cC$ could be either finite or infinite.  The intersection
$W^t_{\cC}$ may depend on $\cC$ and not just $t$, but we expect that
for each $t$, there are only finitely many possibilities for $W^t$ and
only one possibility for $[E(\Q):W^t]$.  Also, since $Y_p$ is a
subgroup of the cyclic group $E(\F_p)/\tilde{a}_p E(\F_p)$, if
$W^t_{\cC}$ has finite index in $E(\Q)$, then the quotient
$E(\Q)/W^t_{\cC}$ is cyclic.

Finally, let $$v = t + 1 + r_{\an}(E^D/\Q),$$ and note that
$v \leq r_{\an}(E/K)$ and $v \equiv r_{\an}(E/K)\pmod{2}$.
\begin{conjecture}\label{conj:ggz}
  Fix a prime $\ell$, an integer $t$ and set of primes $\cC$ as
  above. Then we have the following generalization of the Gross-Zagier
  formula:
$$
 \frac{L^{(v)}(E/K,1)}{v!} = \frac{b\cdot \|\omega\|^2}{c^2 \sqrt{|D|}} 
  \cdot\langle W^t_{\cC}, W^t_{\cC} \rangle_{t+1} \cdot \Reg(E^D/\Q),
$$
  where $b$ is a positive integer not divisible by $\ell$.
\end{conjecture}

Let $B$ be divisible by $2$ and the primes where $\rhobar_{E,\ell}$ is
not surjective.
\begin{conjecture}\label{conj:globalggz}
Let $t$ be an integer as above.  For prime $\ell\nmid B$, make a choice of 
$W(\ell) = W^t_{\cC}$ as above.  Let $W=\cap_{\ell\nmid B} W(\ell)$.  Then
$$
 \frac{L^{(v)}(E/K,1)}{v!} = \frac{b\cdot \|\omega\|^2}{c^2 \sqrt{|D|}} 
  \cdot\langle W, W \rangle_{t+1} \cdot \Reg(E^D/\Q),
$$
where $b$ is an integer divisible only by prime divisors of $B$.
\end{conjecture}
The classical Gross-Zagier formula is like the above formula, but $v=1$,
we have $\Reg(E^D/\Q)=1$, and $\langle W, W \rangle_{t+1}$ is the height of the
Heegner point $P_1\in E(K)$.

All the definitions above make sense with no assumption on
$\ell$, but we are not confident making the analogue of
Conjecture~\ref{conj:globalggz} without further data.

\section{The Birch and Swinnerton-Dyer Conjecture}

\begin{theorem}
  Conjecture~\ref{conj:ggz} implies the BSD conjecture. More
  precisely, if Conjecture~\ref{conj:ggz} is true, then
$$
 \rank(E(\Q)) = \ord_{s=1} L(E/\Q,s)
$$
and $\Sha(E/\Q)[\ell^\infty]$ is finite.
\end{theorem}
\begin{proof}
  First take $t=r_{\an}(E/\Q)-1$.  If $Y_p^t=0$ for all $p$, then
  using the Chebotarev density theorem (see ggz paper), we can find a
  sequence of primes $p_i$ so that if $\cC=\{p_1, p_2, \ldots\}$, then
  $W_{\cC}^t = 0$. However, in Conjecture~\ref{conj:ggz} we have
  $v=r_{\an}(E/K)$, so $L^{(v)}(E/K,1)\neq 0$, hence $\langle
  W_{\cC}^t,W_{\cC}^t \rangle_t\neq 0$ so $W_{\cC}^t$ is infinite.
  Consequently, some $Y_p^t\neq 0$, hence some class $[P_n] \neq 0$
  for some $n$ divisible by $t$ primes.  Thus Kolyvagin's ``Conjecture
  A'' is true with $f\leq r_{\an}(E/\Q)-1$.  It follows by
  \cite[Theorem~4.2]{ggz} that for all $m\gg0$ we have
\begin{equation}\label{eqn:kolysel}
\Sel^{(\ell^m)}(E/\Q) = (\ZZ/\ell^{m}\ZZ)^{f+1} \oplus S
\end{equation}
where $S$ is a finite group independent of $m$ (note that
conjecturally, $S$ is the $\ell$ part of $\Sha(E/\Q)$).
Thus $\rank(E/\Q) \leq f+1$.  

By Conjecture~\ref{conj:ggz} above there are $t+1$ independent
points in $W_{\cC}^t\subset E(\Q)$, so $t+1 \leq \rank(E/\Q)$ and $t+1 \leq
f+1$.  Thus $f = r_{\an}(E/\Q)-1$, and the BSD conjecture that $\rank(E/\Q)
= r_{\an}(E/\Q)$ is true.  Finiteness of $\Sha(E/\Q)[\ell^\infty]$ then follows
from \eqref{eqn:kolysel}.
\end{proof}

%Let
%$$
%  W_p = \pi_p^{-1}(Y_p)\subset E(\Q).
%$$
\section{Explicit Computations}
For the rest of this section, we let $t = r_{\an}(E/\Q) - 1$, and set $Y_p =
Y_p^t$.

\begin{theorem}\label{thm:yp}
   Assume Conjecture~\ref{conj:ggz}, the BSD
  formula at $\ell$ for $E$ over $K$, and Kolyvagin's Conjecture
  $D_{\ell}$.  Then for any good prime $p$, the group $Y_p$ is the
  image of $I \cdot E(\Q)$ in $E(\F_p)/\tilde{a}_p E(\F_p)$, where
$$
  I = c \prod c_q \prod \sqrt{\#\Sha(E/K)}.
$$

[[worry -- there is a ``sufficiently large'' in Kolyvagin?  If so,
make this a conjecture, then give a theorem for sufficiently large as
evidence.]]

[[worry -- the above only gives $W_p$, not $Y_p$]]
\end{theorem}
\begin{proof}
This is Proposition~7.3 of \cite{ggz}.     (It might be that assuming
Kolyvagin's Conjecture $D_{\ell}$ is redundant.)
\end{proof}

Assuming the conclusion of Theorem~\ref{thm:yp}, we can {\em in
  practice} compute the group $Y_p$ for any elliptic curve $E$.  We
can thus conditionally verify Conjecture~\ref{conj:ggz}.  Just
verifying the conjecture is not worth doing, since under the above
hypothesis, Conjecture~\ref{conj:ggz} is implied by the BSD formula,
since $\pi_P^{-1}(Y_p)$ has small enough index that it must contain a
Gross-Zagier subgroup (see \cite[Prop.~2.4]{ggz} and
\cite[Lemma.~7.4]{ggz}).  There is, however, extra information
contained in {\em which} subgroup $\pi_p^{-1}(Y_p)$ we find for a
given $p$, since that does depend in a possiby subtle way on $p$.

A deeper structure on $Y_p$ is that it has labeled generators
$\overline{P}_n$, indexed by positive integers $n$.  So far, it
appears to be a highly nontrivial calculation to explicitly compute a
specific $\overline{P}_n$ in any particular case.

In the rest of this section, we compute as much as we reasonably can
about the objects above in some specific examples.  

For the computations below, we assume BSD and Kolyvagin's conjecture so we
can use Theorem~\ref{thm:yp} to compute $Y_p$. 

\begin{example}
  Let $E$ be the rank 2 elliptic curve 389a, and let $\ell=3$.  We have
  $v=0$, since $c=c_{389}=1$ and $\#\Sha_{\an}=1.000\ldots$.  The
  primes $p<100$ such that $E(\F_p)/\tilde{a}_p E({\F}_P) \neq 0$ are
  $P = \{5, 17, 29, 41, 53, 59, 83\}$, and in each case $E(\F_p)/\tilde{a}_p
  E(\F_P)$ is cyclic of order $3$.
We have 
$$ 
  E(\Q) = \Z P_1 \oplus \Z P_2
$$
where $P_1 = (-1,1)$ and $P_2=(0,-1)$.



\end{example}


\section{Future Directions and Projects}

\begin{enumerate}
\item Assuming the hypothesis of Theorem~\ref{thm:yp}, compute
groups $W$ for various choices of $W_{p_1}\supsetneq W_{p_2} \supsetneq \cdots$
when $I\neq 1$.
\item Formulate Conjecture~\ref{conj:ggz} on $J_0(N)$ over the Hilbert
class field of $K$, and deduce Conjecture~\ref{conj:ggz} from this more
general conjecture. 
\item Formulate Conjecture~\ref{conj:ggz} at {\em all} primes $\ell$ hence
get an exact formula for $\langle W, W \rangle_{t+1}$ as almost
in Conjecture~\ref{conj:globalggz}.
\item Find an algorithm to compute $Y_p$ or $W_p$.  This would be
  especially interesting when Theorem~\ref{thm:yp} does not apply.
  Give a conjectural description of $Y_p$ in {\em all} cases.
\end{enumerate}

\end{document}