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Author: William A. Stein
1\documentclass[11pt]{article}
2\include{macros}
3\title{A Gross-Zagier Style Conjecture and the \\Birch and Swinnerton-Dyer Conjecture}
4\author{William Stein}
5\begin{document}
6\maketitle
7
8\tableofcontents
9
10\section{Introduction}
11In this paper we describe a conjecture that has a similar style to the
12Gross-Zagier formula, and implies the Birch and Swinnerton-Dyer
13conjecture.  We then discuss some relevant computations related to the
14conjecture for some curves of rank $>1$.
15
16
17\section{A Gross-Zagier Style Conjecture}\label{sec:thm}
18Let $E$ be an elliptic curve defined over $\Q$ with analytic rank at
19least $1$ and let $N$ be its conductor.  Let $K$ be one of the
20infinitely many quadratic imaginary fields with discriminant $D<-4$
21coprime to $N$ such that each prime dividing $N$ splits in $K$, and
22such that
23$$\ord_{s=1} L(E^D,s)\leq 1.$$
24
25
26Fix any odd prime $\ell$ such that $\rhobar_{E,\ell}$ is
27surjective. Below we only consider primes $p\nmid N D$ that are inert
28in $K$.  Set $N_p = \#E(\F_p)$, and let $\tilde{a}_p = 29\ell^{\ord_{\ell}(p+1-N_p)}$ be the $\ell$-part of $a_p=p+1-N_p$.  Let
30$$31 b_p = \#(E(\F_p)/\tilde{a}_p E(\F_p)), 32$$
33Note that $b_p = \gcd(p+1,\tilde{a}_p)$,
34by \cite[Lemma~5.1]{ggz}.
35For any squarefree positive integer $n$, let
36$$37 b_n = \gcd(\{b_p : p \mid n\}). 38$$
39Let $P_n = J_n I_n y_n \in E(K_n)$ be the Kolyvagin point associated to $n$, where
40$K_n$ is the ring class field of $K$ of conductor $n$.
41The elements $J_n, I_n \in \Z[\Gal(K_n/K)]$ are constructed so that
42$$43 [P_n] \in (E(K_n)/ b_n E(K_n))^{\Gal(K_n/K)}. 44$$
45See \cite{ggz} for more details.
46
47Let $r_{\an}(E/\Q) = \ord_{s=1}L(E/\Q,s)$ and let $t<r_{\an}(E/\Q)$ be any nonnegative integer with
48$$49 t\equiv r_{\an}(E/\Q) - 1\pmod{2}. 50$$
51For any prime $p\nmid n$ with $\ell \mid b_p \mid b_n$, reducing modulo any
52choice of prime $\wp$ over $p\O_K$ yields a well defined point
53$$54 \overline{P}_n \in E(\F_{p})/\tilde{a}_p E(\F_p). 55$$
56The congruence condition on $t$ and our assumption that $\ell$ is odd implies
57that $\overline{P}_n\in E(\F_p)$ and not just in $E(\F_{p^2})$.
58Let $Y_p^t \subset E(\F_p)/\tilde{a}_p E(\F_p)$ be the subgroup
59generated by all points $\overline{P}_n$ as we vary over all $n$
60divisible by exactly~$t$ primes such that $b_p\mid b_n$.
61The Chebotarev density theorem implies that there are infinity many
62such integers $n$.
63
64Let $\pi_{p}:E(\Q) \to E(\F_p)/\tilde{a}_p E(\F_p)$ be the natural
65quotient map, and let
66$$67 W_p^t = \pi_{p}^{-1}(Y_p^t) \subset E(\Q). 68$$
69If $G$ is a subgroup of $E(\Q)$ and $n$ is a positive integer, let
70$$\langle G,G \rangle_n =\inf\{|\det\langle g_i, g_j\rangle| : 71\text{ independent points } g_1, \ldots g_n \in G \},$$ where
72$g_1,\ldots, g_n$ run through all choices of indepedent elements of
73$G/_{\tor}$.  If $G$ has rank $<n$, then $\langle G,G \rangle_n=0$
74since it is the infimum of the empty set.  Also not that one can prove
75using reduction theory for quadratic forms that there are only
76finitely many subgroups of $G$ of bounded height, so we can replace
77the infimum by a minimum.  Also, if $G$ has rank $n>0$, then $\langle G, 78G\rangle_n = \Reg(G)$ is the regulator of $G$.
79
80Chose {\em any} maximal chain of subgroups
81$W_{p_1}^t \supsetneq W_{p_2}^t \supsetneq W_{p_3}^t \dots$ associated to primes
82$\cC=\{p_1, p_2, \ldots\}$, and let
83$$84 W^t_{\cC} = \bigcap_{p_i\in \cC} W_{p_i}^t. 85$$
86
87Note that $\cC$ could be either finite or infinite.  The intersection
88$W^t_{\cC}$ may depend on $\cC$ and not just $t$, but we expect that
89for each $t$, there are only finitely many possibilities for $W^t$ and
90only one possibility for $[E(\Q):W^t]$.  Also, since $Y_p$ is a
91subgroup of the cyclic group $E(\F_p)/\tilde{a}_p E(\F_p)$, if
92$W^t_{\cC}$ has finite index in $E(\Q)$, then the quotient
93$E(\Q)/W^t_{\cC}$ is cyclic.
94
95Finally, let $$v = t + 1 + r_{\an}(E^D/\Q),$$ and note that
96$v \leq r_{\an}(E/K)$ and $v \equiv r_{\an}(E/K)\pmod{2}$.
97\begin{conjecture}\label{conj:ggz}
98  Fix a prime $\ell$, an integer $t$ and set of primes $\cC$ as
99  above. Then we have the following generalization of the Gross-Zagier
100  formula:
101$$102 \frac{L^{(v)}(E/K,1)}{v!} = \frac{b\cdot \|\omega\|^2}{c^2 \sqrt{|D|}} 103 \cdot\langle W^t_{\cC}, W^t_{\cC} \rangle_{t+1} \cdot \Reg(E^D/\Q), 104$$
105  where $b$ is a positive integer not divisible by $\ell$.
106\end{conjecture}
107
108Let $B$ be divisible by $2$ and the primes where $\rhobar_{E,\ell}$ is
109not surjective.
110\begin{conjecture}\label{conj:globalggz}
111Let $t$ be an integer as above.  For prime $\ell\nmid B$, make a choice of
112$W(\ell) = W^t_{\cC}$ as above.  Let $W=\cap_{\ell\nmid B} W(\ell)$.  Then
113$$114 \frac{L^{(v)}(E/K,1)}{v!} = \frac{b\cdot \|\omega\|^2}{c^2 \sqrt{|D|}} 115 \cdot\langle W, W \rangle_{t+1} \cdot \Reg(E^D/\Q), 116$$
117where $b$ is an integer divisible only by prime divisors of $B$.
118\end{conjecture}
119The classical Gross-Zagier formula is like the above formula, but $v=1$,
120we have $\Reg(E^D/\Q)=1$, and $\langle W, W \rangle_{t+1}$ is the height of the
121Heegner point $P_1\in E(K)$.
122
123All the definitions above make sense with no assumption on
124$\ell$, but we are not confident making the analogue of
125Conjecture~\ref{conj:globalggz} without further data.
126
127\section{The Birch and Swinnerton-Dyer Conjecture}
128
129\begin{theorem}
130  Conjecture~\ref{conj:ggz} implies the BSD conjecture. More
131  precisely, if Conjecture~\ref{conj:ggz} is true, then
132$$133 \rank(E(\Q)) = \ord_{s=1} L(E/\Q,s) 134$$
135and $\Sha(E/\Q)[\ell^\infty]$ is finite.
136\end{theorem}
137\begin{proof}
138  First take $t=r_{\an}(E/\Q)-1$.  If $Y_p^t=0$ for all $p$, then
139  using the Chebotarev density theorem (see ggz paper), we can find a
140  sequence of primes $p_i$ so that if $\cC=\{p_1, p_2, \ldots\}$, then
141  $W_{\cC}^t = 0$. However, in Conjecture~\ref{conj:ggz} we have
142  $v=r_{\an}(E/K)$, so $L^{(v)}(E/K,1)\neq 0$, hence $\langle 143 W_{\cC}^t,W_{\cC}^t \rangle_t\neq 0$ so $W_{\cC}^t$ is infinite.
144  Consequently, some $Y_p^t\neq 0$, hence some class $[P_n] \neq 0$
145  for some $n$ divisible by $t$ primes.  Thus Kolyvagin's Conjecture
146  A'' is true with $f\leq r_{\an}(E/\Q)-1$.  It follows by
147  \cite[Theorem~4.2]{ggz} that for all $m\gg0$ we have
148\begin{equation}\label{eqn:kolysel}
149\Sel^{(\ell^m)}(E/\Q) = (\ZZ/\ell^{m}\ZZ)^{f+1} \oplus S
150\end{equation}
151where $S$ is a finite group independent of $m$ (note that
152conjecturally, $S$ is the $\ell$ part of $\Sha(E/\Q)$).
153Thus $\rank(E/\Q) \leq f+1$.
154
155By Conjecture~\ref{conj:ggz} above there are $t+1$ independent
156points in $W_{\cC}^t\subset E(\Q)$, so $t+1 \leq \rank(E/\Q)$ and $t+1 \leq 157f+1$.  Thus $f = r_{\an}(E/\Q)-1$, and the BSD conjecture that $\rank(E/\Q) 158= r_{\an}(E/\Q)$ is true.  Finiteness of $\Sha(E/\Q)[\ell^\infty]$ then follows
159from \eqref{eqn:kolysel}.
160\end{proof}
161
162%Let
163%$$164% W_p = \pi_p^{-1}(Y_p)\subset E(\Q). 165%$$
166\section{Explicit Computations}
167For the rest of this section, we let $t = r_{\an}(E/\Q) - 1$, and set $Y_p = 168Y_p^t$.
169
170\begin{theorem}\label{thm:yp}
171   Assume Conjecture~\ref{conj:ggz}, the BSD
172  formula at $\ell$ for $E$ over $K$, and Kolyvagin's Conjecture
173  $D_{\ell}$.  Then for any good prime $p$, the group $Y_p$ is the
174  image of $I \cdot E(\Q)$ in $E(\F_p)/\tilde{a}_p E(\F_p)$, where
175$$176 I = c \prod c_q \prod \sqrt{\#\Sha(E/K)}. 177$$
178
179[[worry -- there is a sufficiently large'' in Kolyvagin?  If so,
180make this a conjecture, then give a theorem for sufficiently large as
181evidence.]]
182
183[[worry -- the above only gives $W_p$, not $Y_p$]]
184\end{theorem}
185\begin{proof}
186This is Proposition~7.3 of \cite{ggz}.     (It might be that assuming
187Kolyvagin's Conjecture $D_{\ell}$ is redundant.)
188\end{proof}
189
190Assuming the conclusion of Theorem~\ref{thm:yp}, we can {\em in
191  practice} compute the group $Y_p$ for any elliptic curve $E$.  We
192can thus conditionally verify Conjecture~\ref{conj:ggz}.  Just
193verifying the conjecture is not worth doing, since under the above
194hypothesis, Conjecture~\ref{conj:ggz} is implied by the BSD formula,
195since $\pi_P^{-1}(Y_p)$ has small enough index that it must contain a
196Gross-Zagier subgroup (see \cite[Prop.~2.4]{ggz} and
197\cite[Lemma.~7.4]{ggz}).  There is, however, extra information
198contained in {\em which} subgroup $\pi_p^{-1}(Y_p)$ we find for a
199given $p$, since that does depend in a possiby subtle way on $p$.
200
201A deeper structure on $Y_p$ is that it has labeled generators
202$\overline{P}_n$, indexed by positive integers $n$.  So far, it
203appears to be a highly nontrivial calculation to explicitly compute a
204specific $\overline{P}_n$ in any particular case.
205
206In the rest of this section, we compute as much as we reasonably can
207about the objects above in some specific examples.
208
209For the computations below, we assume BSD and Kolyvagin's conjecture so we
210can use Theorem~\ref{thm:yp} to compute $Y_p$.
211
212\begin{example}
213  Let $E$ be the rank 2 elliptic curve 389a, and let $\ell=3$.  We have
214  $v=0$, since $c=c_{389}=1$ and $\#\Sha_{\an}=1.000\ldots$.  The
215  primes $p<100$ such that $E(\F_p)/\tilde{a}_p E({\F}_P) \neq 0$ are
216  $P = \{5, 17, 29, 41, 53, 59, 83\}$, and in each case $E(\F_p)/\tilde{a}_p 217 E(\F_P)$ is cyclic of order $3$.
218We have
219$$220 E(\Q) = \Z P_1 \oplus \Z P_2 221$$
222where $P_1 = (-1,1)$ and $P_2=(0,-1)$.
223
224
225
226\end{example}
227
228
229\section{Future Directions and Projects}
230
231\begin{enumerate}
232\item Assuming the hypothesis of Theorem~\ref{thm:yp}, compute
233groups $W$ for various choices of $W_{p_1}\supsetneq W_{p_2} \supsetneq \cdots$
234when $I\neq 1$.
235\item Formulate Conjecture~\ref{conj:ggz} on $J_0(N)$ over the Hilbert
236class field of $K$, and deduce Conjecture~\ref{conj:ggz} from this more
237general conjecture.
238\item Formulate Conjecture~\ref{conj:ggz} at {\em all} primes $\ell$ hence
239get an exact formula for $\langle W, W \rangle_{t+1}$ as almost
240in Conjecture~\ref{conj:globalggz}.
241\item Find an algorithm to compute $Y_p$ or $W_p$.  This would be
242  especially interesting when Theorem~\ref{thm:yp} does not apply.
243  Give a conjectural description of $Y_p$ in {\em all} cases.
244\end{enumerate}
245
246\end{document}
247