Author: William A. Stein
1\edit{{\bf Idea!} If I can give conditions under which $\Sha(J/\Q)[p]=\{0\}$
2(equivalently, $\Sha(E/K)[p]=\{0\}$) then I will know
3{\em automatically} that if there is something in
4$\Sha(A/\Q)[p]$, then that something is {\em visible}, hence
5$E(\Q)/p E(\Q)\neq 0$!!!!  That is, a strong form of finiteness
6of $\Sha$ of~$E$'', Conjecture~\ref{conj:nonvanishing}, and
7the rank-$0$ BSD lower bound on $\#\Sha$ together imply the
8following statement: $L(E,1)=0$ implies $E(\Q)$ is infinite.
9The strong finiteness'' is that there exists prime(s) $p$
10such that $\Sha(E/K)[p]=\{0\}$, where $K/\Q$ is appropriate
11cyclic of degree~$p$.}
12
13
14
15\comment{\section{Old Stuff}
16\noindent{\bf Outline of proof:}
17\begin{enumerate}
18
19\item There should be a twist $E'$ of~$E$ by a quadratic character
20unramified at~$p$ such that $E'(\Q)$ has rank exactly~$1$.  This
21should follow from Waldspurger, Kolyvagin, Gross-Zagier, and the other
22standard results.  Without loss, replace~$E$ by~$E'$ (need to show
23still that Tamagawa numbers don't change too much under a quadratic
24twist; should be easy?)
25
26
27
28\item (*) Use Conjecture~\ref{thm:nonvanishing} and surjective of
29$\rho_{E,p}$ to find an~$\ell$ such that $a_\ell\not\equiv \ell+1 30\pmod{p}$ (or $a_\ell\not\equiv 2\pmod{p}$)
31and $L(E,\chi_{p,\ell},1)\neq 0$.
32
33\item
34Let $K$ be the degree~$p$ and totally ramified at~$\ell$ extension
35of~$\Q$ corresponding to~$\chi_{p,\ell}$,
36let $J=\Res_{K/\Q}(E_K)$, and let~$B$ be the kernel of $\Tr : J \ra E$.
37
38\item The map $E\hookrightarrow J \ra E$ is multiplication by $[K:\Q]=p$,
39so $E[p]\subset B$.
40
41\item Use the irreducible assumption on $\rho_{E,p}$ to prove that
42$B(\Q)$ has no $p$-torsion.
43
44\item The bad reduction of~$B$ is $\ell\cdot N$.  For each prime~$q$
45dividing~$N$, the extension $K/\Q$ is unramified at~$q$ and
46$B_K \isom E^{\oplus (p-1)}$, so, because the formation of N\'eron models commutes
47with unramified base change, $c_{B,q}$ divides $\overline{c}_{E,q}^{(p-1)}$.
48In particular, the rigid hypothesis on~$p$ implies that
49$p\nmid c_{B,q}$.
50
51\item The only thing left to prove is that $p\nmid c_{B,\ell}$.
52This is Conjecture~\ref{conj:nonvanishing}.
53
54\item We conclude that $E(\Q)/p E(\Q)\hookrightarrow \Sha(A)$.
55Since $E(\Q)$ has rank~$1$, this concludes the proof.
56
57\end{enumerate}
58
59}
60
61
62
63
64
65
66
67\comment{
70If~$E$ is an elliptic curve over~$\Q$, then
71there is a (quadratic) twist $E'$ of~$E$ that has rank
72greater than~$0$.
73\end{proposition}
74\begin{proof}
75By the work of many people (see the introduction of
76Ono-Skinner \cite{}), there is a twist $E'$ of $E$ such
77that $L(E',1)\neq0$.  By, e.g., Murty-Murty \cite{}, there is
78a quadratic twist $E''$ of $E'$ such that
79$\ord_{s=1} L(E'',s) = 1$, then by Kolyvagin Theorem (see, e.g.,
80\cite{}), $E''$ has rank~$1$.
81\end{proof}
82\subsection{Twists of Prime Order}
83}
84