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\chapter{The Birch and Swinnerton-Dyer conjecture}
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\label{chap:bsd}\index{Birch and Swinnerton-Dyer conjecture|see{BSD conjecture}}%
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\index{Conjecture!Birch and Swinnerton-Dyer|see{BSD conjecture}}%
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Now that the Shimura-Taniyama%
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\index{Shimura-Taniyama conjecture}\index{Conjecture!Shimura and Taniyama}
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conjecture has been proved, many experts consider the Birch and
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Swinnerton-Dyer conjecture (BSD conjecture)
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to be one of the main outstanding problems in the field
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(see~\cite[pg.~549]{darmon-bsd} and \cite[Intro.]{cime-1997}).
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This conjecture ties together many of the arithmetic
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and analytic invariants of an elliptic curve. At present, there is no
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general class of elliptic curves for which the full BSD
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conjecture\index{BSD conjecture!is still unknown} is
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known, though a slightly weakened form is known for a fairly broad
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class of complex multiplication elliptic
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curves of analytic rank~$0$ (see~\cite{rubin:main-conjectures}), and
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several deep partial results have been obtained during
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the last twenty years (see, e.g.,~\cite{gross-zagier} and
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\cite{kolyvagin:mordellweil}).
96
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Approaches to the BSD conjecture\index{BSD conjecture}
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that rely on congruences between\index{Congruences!and BSD conjecture}
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modular forms\index{Modular forms!and BSD}
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are likely to require a deeper
101
understanding of the analogue of the BSD conjecture\index{BSD conjecture!in higher dimensions}
102
for higher-dimensional abelian varieties. As a first step, this chapter
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presents theorems and explicit computations of some of the arithmetic
104
invariants of modular abelian varieties.
105
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The reader is urged to also read A.~Agashe's 2000
107
Berkeley Ph.D.\ thesis which cover similar themes.\index{Agashe} The paper of
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Cremona and Mazur's~\cite{cremona-mazur}\index{Mazur} paints a detailed
109
experimental picture of the way in which congruences link
110
Mordell-Weil and Shafarevich-Tate groups of elliptic curves.
111
\index{Congruences!and BSD conjecture}
112
113
\section{The BSD conjecture}\index{BSD conjecture|textit}
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By~\cite{breuil-conrad-diamond-taylor} we now know
115
that every elliptic curve over~$\Q$ is
116
a quotient of the curve~$X_0(N)$, whose complex points
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are the isomorphism classes of pairs consisting of a
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(generalized) elliptic curve and a cyclic subgroup of order~$N$.
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Let~$J_0(N)$ denote the Jacobian\index{Jacobian} of $X_0(N)$; this is an abelian
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variety of dimension equal to the genus of~$X_0(N)$ whose points
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correspond to the degree~$0$ divisor classes on~$X_0(N)$.
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The survey article~\cite{diamond-im} is a good
123
guide to the facts and literature
124
about the family of abelian varieties $J_0(N)$.
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Following Mazur~\cite{mazur:rational}\index{Mazur}, we make the following definition.
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\begin{definition}[Optimal quotient]\index{Optimal quotient|textit}
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An {\em optimal quotient} of $J_0(N)$ is a quotient~$A$ of
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$J_0(N)$ by an abelian subvariety.
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\end{definition}
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Consider an optimal quotient~$A$ such that $L(A,1)\neq 0$.
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By~\cite{kolyvagin-logachev:totallyreal},~$A(\Q)$ and~$\Sha(A)$
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are both finite.
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The BSD conjecture\index{BSD conjecture!statement of}%
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asserts that
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$$\frac{L(A,1)}{\Omega_A} =
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\frac{\#\Sha(A)\cdot\prod_{p\mid N} c_p}
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{\# A(\Q)\cdot\#\Adual(\Q)}.$$
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Here the Shafarevich-Tate group\index{Shafarevich-Tate group}
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$$\Sha(A) := \ker\left(H^1(\Q,A) \ra \prod_{v} H^1(\Q_v,A)\right)$$
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is a measure of the failure
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of the local-to-global principle\index{Local-to-global principle};
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the Tamagawa numbers~$c_p$\index{Tamagawa numbers} are the
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orders of the groups of rational points of the
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component groups of~$A$ (see Chapter~\ref{chap:compgroups});
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the real number~$\Omega_A$ is
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the measure of~$A(\R)$ with respect to a basis of differentials having
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everywhere nonzero good reduction (see Section~\ref{sec:realmeasure});
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and~$\Adual$ is the abelian variety dual to~$A$ (see \cite[\S9]{milne:abvars}).
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This chapter makes a small contribution to the long-term goal
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of verifying the above conjecture for many specific abelian varieties
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on a case-by-case basis. In a large list of examples, we compute
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the conjectured order of $\Sha(A)$, up to a power of $2$, and then
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show that $\Sha(A)$ is at least as big as conjectured.
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We also discuss methods to obtain upper bounds on $\#\Sha(A)$, but do
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not carry out any computations in this direction.
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This is the first step in a program to verify the above
158
conjecture for an infinite family of quotients of~$J_0(N)$.
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\subsection{The ratio $L(A,1)/\Omega_A$}
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Extending classical work on elliptic curves,
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A.~Agashe\index{Agashe} and the author proved the following
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theorem.
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\begin{theorem}\label{thm:ratpart}
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Let~$m$ be the largest square dividing~$N$.
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The ratio $L(A,1)/\Omega_A$ is a rational number that can be
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explicitly computed, up to a unit (conjecturally $1$) in $\Z[1/(2m)]$.
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\end{theorem}
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\begin{proof}
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The proof uses modular symbols\index{Modular symbols}
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combined with an extension of the argument
172
used by Mazur\index{Mazur} in~\cite{mazur:rational} to bound
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the Manin constant\index{Manin constant}.
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The modular symbols part of the proof for $L$-functions attached
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to newforms of weight $k\geq 2$ is given in Section~\ref{sec:rationalvals};
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it involves expressing the
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ratio $L(A,1)/\Omega_A$ as the lattice
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index\index{Lattice index} of
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two modules over the Hecke algebra\index{Hecke algebra}.
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The bound on the Manin constant\index{Manin constant} is given in
181
Section~\ref{sec:maninconstant}.
182
\end{proof}
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The author has computed $L(A,1)/\Omega_A$ for all simple optimal
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quotients of level $N\leq 1500$; this table can be
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obtained from the author's web page.
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\begin{remark}
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The method of proof should also give similar results for special
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values of twists of $L(A,s)$, just as it does in the case $\dim A=1$
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(see~\cite[Prop.~2.11.2]{cremona:algs}).
192
\end{remark}
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\subsection{Torsion subgroup\index{Torsion subgroup}}
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We can compute upper and lower bounds on $\#A(\Q)_{\tor}$,
197
see Section~\ref{sec:torsionsubgroup};
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these frequently determine $\#A(\Q)_{\tor}$.
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These methods, combined with the method used
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to obtain Theorem~\ref{thm:ratpart},
202
yield the following corollary, which supports the expected
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cancellation between torsion and~$c_p$ coming from the reduction
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map sending rational points to their image in the component
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group of~$A$. The corollary also generalizes to higher weight forms,
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thus suggesting a geometric way to think about reducibility
207
of modular Galois representations.
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\begin{corollary}
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Let~$n$ be the order of the image of
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$(0)-(\infty)$ in $A(\Q)$, and
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let~$m$ be the largest square dividing~$N$.
212
Then
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$n\cdot L(A,1)/\Omega_A \in \Z[1/(2m)].$
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\end{corollary}
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For the proof, see Corollary~\ref{cor:denominator}
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in Chapter~\ref{chap:computing}.
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\subsection{Tamagawa numbers\index{Tamagawa numbers}}
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We prove the following theorem in Chapter~\ref{chap:compgroups}.
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\begin{theorem}\label{thm:tamagawa}
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When $p^2\nmid N$, the number~$c_p$ can be explicitly computed
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(up to a power of~$2$).
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\end{theorem}
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We can compute the order~$c_p$ of the group of rational
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points of the component group, but not
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its structure as a group.
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When $p^2 \mid N$ it may be possible
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to compute~$c_p$ using the
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Drinfeld-Katz-Mazur model
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of~$X_0(N)$, but we have not yet done this.
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There are also good bounds on the primes that can divide $c_p$ when
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$p^2\mid N$.
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Systematic computations (see Section~\ref{sec:compgroupconjectures})
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using this formula suggest the
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following conjectural refinement of a result
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of Mazur~\cite{mazur:eisenstein}\index{Mazur}.
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\begin{conjecture}
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Suppose~$N$ is prime and~$A$
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is an optimal quotient of $J_0(N)$ corresponding
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to a newform~$f$. Then $A(\Q)_{\tor}$ is
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generated by the image of $(0)-(\infty)$
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and $c_p = \#A(\Q)_{\tor}$. Furthermore,
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the product of the~$c_p$ over all simple optimal quotients
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corresponding to newforms equals the numerator of $(N-1)/12$.
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\end{conjecture}
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I have checked this conjecture for all $N\leq 997$ and,
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up to a power of~$2$, for all $N\leq 2113$.
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The first part is known when~$A$ is an elliptic
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curve (see~\cite{mestre-oesterle:crelle}).
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Upon hearing of this conjecture, Mazur\index{Mazur} reportedly
253
proved it when all ``$q$-Eisenstein quotients'' are simple.
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There are three promising approaches to finding
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a complete proof. One involves the explicit
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formula of Theorem~\ref{thm:tamagawa};
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another is based on Ribet's\index{Ribet} level lowering theorem
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(see~\cite{ribet:modreps}),
259
and a third makes use of a simplicity result of Merel\index{Merel}
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(see~\cite{merel:weil}).
261
262
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The formula that lies behind Theorem~\ref{thm:tamagawa} probably
264
has a natural analogue in weight greater than~$2$.
265
One could then guess that it produces Tamagawa numbers\index{Tamagawa numbers}
266
of motifs\index{Motifs} attached to eigenforms of higher weight; however,
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we have no idea if this is really the case. These numbers appear
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in the conjectures of Bloch and Kato,
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\index{Conjecture!Bloch and Kato}%
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\index{Bloch and Kato conjecture}%
271
which generalize the BSD conjecture\index{BSD conjecture!generalization of} to
272
motifs (see~\cite{bloch-kato}).
273
Anyone wishing to
274
try to compute them should be aware of Neil Dummigan's
275
paper~\cite{dummigan:cp}, which gives some information
276
about the Tamagawa numbers\index{Tamagawa numbers}
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of motifs\index{Motifs} attached by
278
Scholl in~\cite{scholl:motivesinvent}
279
to modular eigenforms.
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\subsection{Upper bounds on $\#\Sha(A)$}
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V.~Kolyvagin (see \cite{kolyvagin:structureofsha})
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and K.~Kato (see, e.g., \cite{scholl:kato})
284
constructed Euler systems\index{Euler system} that
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were used to prove that $\Sha(A)$ is {\em finite}
286
when $L(A,1)\neq 0$.
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To verify the full BSD conjecture\index{BSD conjecture!verification of}
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for certain abelian varieties, we must make the Kolyvagin-Kato
289
finiteness bound explicit.
290
Kolyvagin's bounds involve computations with Heegner
291
points\index{Heegner points},
292
and Kato's involve a study of the Galois representations
293
associated to~$A$.
294
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\subsubsection{Kolyvagin's bounds}%
296
\index{Bound of!Kolyvagin}
297
In~\cite{kolyvagin:mordellweil}, Kolyvagin obtains explicit upper
298
bounds for $\#\Sha(A)$ for a certain (finite) list of elliptic curves~$A$
299
by computing the index in $A(K)$ of the subgroup
300
generated by the Heegner point, where~$K$
301
is a suitable imaginary quadratic extension.
302
In~\cite{kolyvagin-logachev:totallyreal}, Kolyvagin and Logachev
303
generalize Kolyvagin's earlier results; in Section~1.6, ``Unsolved
304
problems'', they say that: ``If one were to compute the
305
height of a Heegner point~$y$ [...]
306
considered in the present paper, then one would have succeeded in
307
obtaining an upper bound for $\#\Sha$ for this curve.''
308
(By ``curve'' they mean abelian variety.)
309
This suggests that explicit computations should yield upper
310
bounds on the order of $\Sha(A)$, but that they had not yet
311
figured out how to carry out such computations.
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\subsubsection{Kato's bounds}%
314
\index{Bound of!Kato}
315
Kato has constructed Euler systems\index{Euler system} coming
316
from $K_2$-groups of modular
317
curves. These can be used to prove the following theorem (see, e.g.,
318
\cite[Cor.~3.5.19]{rubin:book}).
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\begin{theorem}[Kato]
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Suppose~$E$ is an elliptic curve over~$\Q$ without complex
321
multiplication that~$E$ has conductor~$N$,
322
that~$E$ has good reduction at~$p$, that~$p$ does not divide
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$2r_E\prod_{q\mid N} L_q(q^{-1})\#E(\Q_q)_{\tor}$, and
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the Galois representation $\rho_{E,p}:\GQ\ra\Aut(E[p])$
325
is surjective. Then
326
$$\#\Sha(E)_{p^{\infty}}\text{ divides }
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\frac{L(E,1)}{\Omega_E}.$$
328
\end{theorem}
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Here $L_q(x)$ is the local Euler factor at~$q$ and the constant
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$r_E$ arises in the construction of Kato's Euler system.
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Rubin suggests that computing $r_E$ is not very
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difficult (private communication).
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Appropriate variants of Kato's arguments
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give similar results for quotients of $J_0(N)$ of arbitrary
335
dimension, though these have not been written down.
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\comment{
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>How mysterious is the constant r_E in, for example, Corollary 3.5.19 of
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>your Euler Systems book?
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I think it's not too bad. Certainly nothing like Heegner points are
342
involved. When I wrote that part of my book, and the similar paper in
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the Durham proceedings, I did not really know what it was because Kato
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hadn't written anything. I don't have Scholl's paper here so I'm not
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certain, but I suspect that the only contribution to $r_E$ comes from
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passing from the modular curve to $E$, and perhaps some extra 2's and
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3's.
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Karl
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}
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\subsection{Lower bounds on $\#\Sha(A)$}
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One approach to showing that~$\Sha(A)$ is as {\em at least} as
355
large as predicted
356
by the BSD conjecture\index{BSD conjecture!and $\Sha$}
357
is suggested by Mazur's\index{Mazur} notion of
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the visible part $\Sha(A)^{\circ}$
359
of~$\Sha(A)$ (see~\cite{cremona-mazur, mazur:visthree}).
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Let~$\Adual\subset J_0(N)$ be the dual to~$A$.
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The \defn{visible part}\index{Visibility!of $\Sha$|textit}%
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\index{Shafarevich-Tate group!visible part of}
363
of $\Sha(\Adual)$ is the
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kernel of the natural map
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$\Sha(\Adual)\ra \Sha(J_0(N))$.
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Mazur\index{Mazur} observed that if an element of order~$p$
367
in~$\Sha(\Adual)$ is visible,
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then it is explained by a ``jump in the rank of Mordell-Weil''
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in the sense that there is another abelian subvariety $B\subset J_0(N)$
370
such that $p \mid \#(\Adual\intersect B)$ and the rank of~$B$ is positive.
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Mazur's\index{Mazur} observation can be turned around: if there is another abelian
373
variety~$B$ of positive rank such that $p\mid \#(\Adual\intersect B)$,
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then, under mild hypotheses (see Theorem~\ref{thm:shaexists}), there
375
is an element of~$\Sha(\Adual)$ of order~$p$. From a computational
376
point of view it is easy to understand the intersections
377
$\Adual\intersect B$; see Section~\ref{sec:intersection}.
378
From a theoretical point of view, nontrivial
379
intersections ``correspond'' to congruences between modular forms.
380
Thus the well-developed
381
theory of congruences between modular forms%
382
\index{Modular forms!congruences between}%
383
\index{Congruences!and lower bounds on $\Sha$}
384
can be used to obtain a lower bound on~$\#\Sha(\Adual)$.
385
386
\subsubsection{Invisible elements of $\#\Sha(\Adual)$}
387
\index{Shafarevich-Tate group!invisible elements of}
388
\index{Invisible elements of $\Sha$}
389
Numerical experiments suggest
390
that as $\Adual$ varies, $\Sha(\Adual)$ is
391
often {\em not} visible inside of~$J_0(N)$.
392
For example (see Table~\ref{table:primesha}), the
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BSD conjecture\index{BSD conjecture!predicts invisible elements}
394
predicts the existence of invisible elements of odd
395
order in~$\Sha(\Adual)$
396
for almost half of the~$37$ optimal quotients
397
of prime level $\leq 2113$.
398
399
\subsubsection{Visibility at higher level}
400
\index{Shafarevich-Tate group!visibility at higher level}
401
\index{Visibility!at higher level}
402
For every integer~$M$ (Ribet~\cite{ribet:raising}\index{Ribet}
403
tells us which~$M$
404
to choose), we can ask whether $\Sha(\Adual)$ maps to~$0$
405
under one of the natural maps $\Adual\ra J_0(NM)$; that is, we
406
can ask whether $\Sha(\Adual)$ ``becomes visible at
407
level $NM$.''
408
We have been unable to prove in any particular case that $\Sha(\Adual)$ is
409
not visible at level~$N$, but becomes visible at some level $NM$.
410
See Section~\ref{sec:higherlevel} for some computations which strongly
411
indicate that such examples exist.
412
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\subsubsection{Visibility in some Jacobian}%
414
\index{Visibility!in some Jacobian}%
415
\index{Jacobian!visibility in}%
416
Johan de Jong proved that if~$E$ is an elliptic curve
417
over a number field~$K$ and $c\in H^1(K,E)$ then there is a
418
Jacobian~$J$ and an imbedding $E\hookrightarrow J$ such that~$c$ maps
419
to~$0$ under the natural map $H^1(K,E)\ra H^1(K,J)$ (see Remark~3
420
in~\cite{cremona-mazur}); de Jong's proof appears to generalize
421
when~$E$ is replaced by an abelian variety, but time does not permit
422
going into the details here.
423
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\subsection{Motivation for considering abelian varieties}
425
If~$A$ is an elliptic curve, then explaining~$\Sha(A)$ using
426
only congruences between elliptic curves will probably fail.
427
\index{Congruences!between elliptic curves}
428
This is because pairs of non-isogenous elliptic curves with isomorphic
429
$p$-torsion for large~$p$ are, according to E.~Kani's\index{Kani}
430
conjecture, extremely rare.%
431
\index{Conjecture!Kani}
432
It is crucial to understand what happens in all dimensions.
433
434
Within the range accessible by computer, abelian varieties exhibit
435
more richly textured structure than elliptic curves. For example, there
436
is a visible element of prime order $83341$ in the
437
Shafarevich-Tate group\index{Shafarevich-Tate group} of an abelian
438
variety of prime conductor~$2333$; in contrast, over all optimal
439
elliptic curves of conductor up to $5500$, it appears that the largest
440
order of an element of a Shafarevich-Tate group is~$7$ (see the
441
discussion in~\cite{cremona-mazur}).
442
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\section{Existence of nontrivial visible elements of $\Sha(A)$}%
444
The reader who wants to see tables of Shafarevich-Tate groups can
445
safely skip to the next section.
446
447
Without relying on any unverified conjectures,
448
we will use the following theorem to exhibit abelian varieties~$A$
449
such that the visible part of $\Sha(A)$ is nonzero.
450
In the following theorem we do {\em not} assume that~$J$ is the
451
Jacobian\index{Jacobian} of a curve.
452
\begin{theorem}\label{thm:shaexists}\index{Visibility!existence theorem}
453
Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such
454
that $A\intersect B$ is finite and $A(\Q)$ is finite.
455
Assume that~$B$ has purely toric reduction
456
at each prime at which~$J$ has
457
bad reduction.
458
Let~$p$ be an odd prime at which~$J$ has good reduction, and
459
assume that~$p$ does not divide the orders of any of
460
the (geometric) component groups\index{Component group!geometric}
461
of~$A$ and~$B$,
462
or the orders of the torsion subgroups of $(J/B)(\Q)$ and $B(\Q)$.
463
Suppose further that $B[p] \subset A\intersect B$.
464
Then there exists an injection
465
$$B(\Q)/pB(\Q)\hookrightarrow \Sha(A)^{\circ}$$
466
of $B(\Q)/p B(\Q)$ into the visible part of $\Sha(A)$.
467
\end{theorem}
468
469
\begin{proof}
470
Let $C=J/A$.
471
The long exact sequence of Galois cohomology
472
associated to the short exact sequence
473
$$0 \ra A \ra J \ra C \ra 0$$
474
begins
475
$$0\ra A(\Q) \ra J(\Q) \ra C(\Q) \xrightarrow{\,\delta\,}
476
H^1(\Q,A) \ra \cdots.$$
477
Because $B[p]\subset A$, the map $B\ra C$, obtained by composing
478
the inclusion
479
$B\hookrightarrow J$ with $J\ra C$, factors through multiplication-by-$p$,
480
giving the following commutative diagram:
481
$$\xymatrix{
482
& B\ar[d] \ar[r]^{p}& B\ar[d]\\
483
A\ar[r]&J\ar[r]&C.}$$
484
Because $B(\Q)[p]=0$ and $B(\Q)\intersect A(\Q)=0$, we
485
deduce the following commutative diagram with exact
486
rows:
487
$$\xymatrix{
488
& 0\ar[d] & K_1\ar[d]& K_2\ar[d]\\
489
0 \ar[r] & B(\Q) \ar[r]^{p}\ar[d] & B(\Q)\ar[dr]^{\pi} \ar[r]\ar[d]
490
& B(\Q)/pB(\Q)\ar[r]\ar[d] & 0\\
491
0 \ar[r] & J(\Q)/A(\Q)\ar[r]\ar[d] & C(\Q) \ar[r] & \delta(C(\Q)) \ar[r] & 0\\
492
& K_3,
493
}$$
494
where $K_1$ and $K_2$ are the indicated kernels and $K_3$ is the cokernel.
495
We have the snake lemma exact sequence
496
$$0\ra K_1 \ra K_2 \ra K_3.$$
497
Because $B(\Q)[p]=0$ and $K_2$ is a $p$-torsion group,
498
we have $K_1=0$.
499
The quotient $J(\Q)/B(\Q)$ has no $p$-torsion because
500
it is a subgroup of $(J/B)(\Q)$; also, $A(\Q)$ is a finite group
501
of order coprime to~$p$,
502
so $K_3 = J(\Q)/(A(\Q)+B(\Q))$ has no $p$-torsion. Thus $K_2=0$.
503
504
The above argument shows that $B(\Q)/p B(\Q)$ is a subgroup of
505
$H^1(\Q,A)$. However, $H^1(\Q,A)$ contains infinitely many elements of
506
order~$p$ (see~\cite{shafarevich:exp}),
507
whereas $\Sha(A)[p]$ is a finite group, so we must work
508
harder to deduce that $B(\Q)/p B(\Q)$ lies in
509
$\Sha(A)[p]$. Fix $x\in B(\Q)$. We must show
510
that $\pi(x)$ lies in $\Sha(A)[p]$; equivalently, that
511
$\res_v(\pi(x))=0$ for all places~$v$ of~$\Q$.
512
513
At the archimedean place $v=\infty$, the restriction
514
$\res_v(\pi(x))$ is killed by~$2$ and the odd prime~$p$,
515
hence $\res_v(\pi(x))=0$.
516
517
Suppose that~$v$ is a place at which~$J$ has bad reduction.
518
By hypothesis, $B$ has purely toric reduction\index{Purely toric reduction},
519
so over the maximal unramified extension $\Q_v^{\ur}$
520
of $\Q_v$ there is an isomorphism $B\isom\Gm^d/\Gamma$
521
of $\Gal(\Qbar_v/\Q_v^{\ur})$-modules,
522
for some ``lattice'' $\Gamma$.
523
For example, when
524
$\dim B=1$, this is the Tate curve representation of~$B$.
525
Let~$n$ be the order of the component group of~$B$ at~$v$; thus~$n$
526
equals the order of the cokernel of the valuation
527
map $\Gamma\ra \Z^d$. Choose a representative $P=(x_1,\ldots,x_d)\in\Gm^d$
528
for the point~$x$. Then $nP$ can be adjusted by elements of~$\Gamma$
529
so that each of its components $x_i\in\Gm$ has valuation~$0$.
530
By assumption,~$p$ is a prime at which~$J$ has good reduction, so
531
$p\neq v$;
532
it follows that there is a point $Q\in\Gm^d(\Q_v^{\ur})$ such that
533
$pQ = nP$.
534
Thus the cohomology class $\res_v(\pi(nx))$ is unramified
535
at~$v$. By \cite[Prop.~I.3.8]{milne:duality},
536
$$H^1(\Q_v^{\ur}/\Q_v,A(\Q_v^{\ur}))
537
=H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}(\Fbar_v)),$$
538
where $\Phi_{A,v}$ is the component group of~$A$ at~$v$.
539
Since\item{There is a mistake here, but it is easy to fix.}
540
the component group $\Phi_{A,v}(\Fbar_v)$ has
541
order~$n$, it follows that $$\res_v(\pi(nx))=n\res_v(\pi(x))=0.$$
542
Since the order~$p$ of $\res_v(\pi(x))$ is coprime to~$n$,
543
we conclude that $\res_v(\pi(x))=0$.
544
545
Next suppose that~$J$ has good reduction at~$v$
546
and that~$v$ is {\em odd}, in the sense that the
547
residue characteristic of~$v$ is odd. To simplify notation in
548
this paragraph, since~$v$ is a non-archimedean place
549
of $\Q$, we will also let~$v$ denote the odd prime number
550
which is the residue characteristic of~$v$.
551
Let $\cA$, $\cJ$, $\cC$, be the N\'eron models
552
of~$A$,~$J$, and~$C$, respectively (for more on N\'eron
553
models, see Chapter~\ref{chap:compgroups}).
554
Let $A$, $J$, $C$, also denote the sheaves on
555
the \'etale-site over $\Spec(\Z_v)$ determined
556
by the group schemes $\cA$, $\cJ$, and $\cC$, respectively.
557
Since~$v$ is odd, $1=e<v-1$, so we may apply
558
\cite[Thm.~7.5.4]{neronmodels} to conclude that
559
the sequence of group schemes
560
$$0\ra \cA \ra \cJ\ra \cC \ra 0$$
561
is exact; in particular, it
562
is exact as a sequence of sheaves on the
563
\'etale site (see the proof of~\cite[Thm.~7.5.4]{neronmodels}).
564
Thus it is exact on the stalks, so by~\cite[2.9(d)]{milne:etale}
565
the sequence
566
$$0\ra \cA(\Z_v^{\ur})\ra \cJ(\Z_v^{\ur}) \ra \cC(\Z_v^{\ur})\ra 0$$
567
is exact; by the N\'eron mapping property the sequence
568
$$0\ra A(\Q_v^{\ur})\ra J(\Q_v^{\ur}) \ra C(\Q_v^{\ur})\ra 0$$
569
is also exact.
570
Thus $\res_v(\pi(x))$ in unramified,
571
so it arises by inflation from
572
an element of $H^1(\Q_v^{\ur}/\Q_v,A)$.
573
By \cite[Prop.~I.3.8]{milne:duality},
574
$$H^1(\Q_v^{\ur}/\Q_v,A) \isom H^1(\Q_v^{\ur}/\Q_v,\Phi_{A,v}),$$
575
where $\Phi_{A,v}$ is the component group of~$A$ at~$v$.
576
Since~$A$ has good reduction, $\Phi_{A,v}=0$, hence
577
$\res_v(\pi(x))=0$.
578
579
If~$J$ has bad reduction at~$v=2$, then we already dealt with~$2$ above.
580
Consider the case when~$J$ has good reduction at~$2$. Because the
581
absolute ramification index~$e$ of $\Z_2$ is~$1$, which is
582
{\em not} less than $v-1=1$, we can not apply \cite[Thm.~7.5.4]{neronmodels}.
583
However, we can modify our situation by an isogeny of degree a power
584
of~$2$, then apply a different theorem as follows.
585
The $2$-primary subgroup~$\Psi$ of $A\intersect B$ is
586
rational as a subgroup over~$\Q$, in the sense that the
587
conjugates of any point in $\Psi$ are
588
also contained in $\Psi$.
589
The abelian varieties
590
$\tilde{J}=J/\Psi$, $\tilde{A}=A/\Psi$, and
591
$\tilde{B}=B/\Psi$ also satisfy the hypothesis of
592
the theorem we are proving.
593
By \cite[Prop.~7.5.3(a)]{neronmodels}, the corresponding sequence of
594
N\'eron models
595
$$0\ra\tilde{\cA}\ra\tilde{\cJ}\ra\tilde{\cC}\ra 0$$
596
is exact, so the sequence
597
$$0\ra \tilde{A}(\Q_v^{\ur})\ra\tilde{J}(\Q_v^{\ur})
598
\ra\tilde{C}(\Q_v^{\ur})\ra 0$$
599
is exact. Thus the image of
600
$\res_v(\pi(x))$ in $H^1(\Q_v,\tilde{A})$ is unramified.
601
It equals~$0$, again by \cite[Prop.~3.8]{milne:duality},
602
since the component group of $\tilde{A}$ at~$v$ has order a power
603
of~$2$ (in fact it is trivial, since $\tilde{A}$ has
604
good reduction at~$2$), whereas $\pi(x)$ has odd prime order~$p$.
605
Thus $\res_v(\pi(x))=0$, since
606
the kernel of $H^1(\Q_v,A)\ra H^1(\Q_v,\tilde{A})$ is a
607
finite group of $2$-power order.
608
\end{proof}
609
610
611
612
613
\section{Description of tables}
614
In this section we describe our tables of optimal quotients of
615
$J_0(N)$, which have nontrivial Shafarevich-Tate group.
616
The tables, which can be found on
617
pages~\pageref{table:primesha}--\pageref{table:shacompgps},
618
were computed using a combination of
619
\hecke{}~\cite{stein:hecke}, {\sc LiDIA}, {\sc NTL}, {\sc Pari}, and
620
most successfully \magma{}~\cite{magma}. The component group
621
computations at non-prime level rely on Kohel's quaternion
622
algebra package, which was also written in \magma{}.
623
624
We compute the conjectural order of the Shafarevich-Tate group of an
625
abelian variety~$A$, and then make assertions about the
626
Shafarevich-Tate group of~$\Adual$. This is justified because the
627
order of $\Sha(\Adual)$ equal the order of~$\Sha(A)$, since both are
628
finite and the Cassells-Tate pairing sets up a nondegenerate duality
629
between them.
630
631
\subsection{Notation}\label{sec:optquo-notation}
632
Each optimal quotient~$A$ of $J_0(N)$ is denoted by a label, such as
633
{\bf 389E}, which consists of a level~$N$ and a letter indicating the
634
isogeny class. In the labeling,~$N$ is a positive integer and the
635
isogeny class is given by a letter: the first isogeny class is labeled
636
{\bf A}, the second is labeled {\bf B}, the third labeled {\bf C}, and
637
so on. Thus {\bf 389E} is the fifth isogeny class of optimal quotient
638
of $J_0(389)$, corresponding to a Galois-conjugacy class of newforms.
639
The isogeny classes that we consider are in bijection with the
640
Galois-conjugacy classes of newforms in $S_2(\Gamma_0(N))$. The
641
classes of newforms are ordered as described in Section~\ref{sec:sorting}.
642
643
{\bf WARNING:} The {\em odd part} of a rational number $x$ is $x/2^v$,
644
where $v=\ord_2(x)$. In the tables, only the {\bf odd parts} of the
645
arithmetic invariants of~$A$ are given.
646
647
\subsection{Table~\ref{table:primesha}: Shafarevich-Tate
648
groups at prime level}
649
650
Table~\ref{table:primesha} was constructed as follows. Using the
651
results of Section~\ref{sec:ratpartformula}, we computed the odd part
652
of the conjectural order $\Shaan(A)$ of the Shafarevich-Tate group of
653
every optimal quotient of $J_0(p)$ that corresponds to a single Galois
654
conjugacy-class of eigenforms and has analytic rank~$0$, for~$p$ a
655
prime with $p\leq 2161$. We also computed a few sporadic examples of
656
prime level~$p$ with $p>2161$. The sporadic examples appear at the
657
bottom of the table below a horizontal line.
658
659
\subsubsection{Notation}
660
The columns of the table contain the following information. The
661
abelian varieties~$A$ for which $\Shaan(A)$ is greater than~$1$ are
662
laid out in the first column of Table~\ref{table:primesha}. The
663
second column contains the dimensions of the abelian varieties in the
664
first column. The third column contains the {\em odd part} (i.e.,
665
largest odd divisor) of the order of the Shafarevich-Tate group, as
666
predicted by the BSD conjecture.\index{BSD conjecture!predicted order of $\Sha$}
667
Column four contains the odd parts
668
of the modular degrees of the abelian varieties in the first column.
669
670
The fifth column contains an optimal quotient~$B$ of $J_0(p)$ of
671
positive analytic rank, such that the $\ell$-torsion of $\Bdual$ is
672
contained in~$\Adual$, when one exists, where $\ell$ is a divisor
673
of $\Shaan(A)$. We computed this
674
intersection using the algorithm described in
675
Section~\ref{sec:intersection}. Such a~$B$ is called an
676
\defn{explanatory factor}.\index{Explanatory factor}
677
When no such abelian varieties exists, we write ``NONE'' in
678
the fifth column. The sixth column contains the dimensions of the
679
abelian varieties in the fifth column, and the seventh column contains
680
the odd parts of the modular degrees of the abelian varieties in the
681
fifth column.
682
683
\subsubsection{Ranks of the explanatory factors}
684
That the explanatory factors have positive analytic rank follows from
685
our modular symbols computation of $L(B,1)/\Omega_B$.
686
This is supported by the table in~\cite{brumer:rank}, except
687
in the case {\bf 2333A}, where there is a mistake in \cite{brumer:rank}
688
(see below).
689
690
The explanatory factor {\bf 389A} is the first elliptic curve of
691
rank~$2$. The table in \cite{brumer:rank} suggests that the
692
explanatory factor {\bf 1061B} is the first $2$-dimensional abelian
693
variety (attached to a newform) whose Mordell-Weil group when tensored
694
with the field of fractions~$F$ of the corresponding ring of Fourier
695
coefficients, is of dimension~$2$ over~$F$. Similarly
696
{\bf 1567B} appears to be the first $3$-dimensional one of rank~$2$, and
697
{\bf 2333A} is the first $4$-dimensional one of rank~$2$.
698
It thus seems very likely that the ranks of each explanatory factor
699
is exactly~$2$, though we have not proved this.
700
701
\subsubsection{Discussion of the data}
702
There are~$23$ examples in which~$\Sha(A)$ is
703
visible and~$18$ in which~$\Sha(A)$
704
is invisible. The largest visible
705
$\Sha(A)$ found occurs at level $2333$ and has order at least $83341^2$
706
($83341$ is prime).
707
The largest invisible\index{Invisible elements of $\Sha$} $\Sha(A)$
708
occurs in a $112$-dimensional abelian variety at level
709
$2111$ and has order at least $211^2$.
710
711
The example {\bf 1283C} demonstrates that $\Shaan(A)$ can divide the
712
modular degree, yet be {\em invisible}. In this case~$5$ divides
713
$\Shaan(A)$. Since~$5$ divides the
714
modular degree, it follows that there must be
715
other non-isogenous subvarieties of $J_0(1283)$ that
716
intersect {\bf 1283C} in a subgroup of order divisible
717
by~$5$. In this case, the only such subvariety is
718
{\bf 1283A}, which has dimension~$2$ and whose $5$-torsion is contained
719
in {\bf 1283C}. However {\bf 1283A} has analytic (hence algebraic) rank~$0$,
720
so $\Shaan(A)$ cannot be visible.
721
722
The cases {\bf 1483D}, {\bf 1567D}, {\bf 2029C}, and {\bf 2593B} are
723
interesting because {\em all} of~$\Sha$, even though it has two
724
nontrivial $p$-primary components in each of these cases, is made
725
visible in a single~$B$. In the case {\bf 1913E} only
726
the $5$-primary component of $\Sha$ is visible in {\bf 1913A}, but
727
still {\em both} the $5$-primary and
728
$61$-primary components of $\Sha$ are visible in {\bf 1913C}.
729
730
Examples {\bf 1091C} and {\bf 1429B} were first found in
731
\cite{agashe} and {\bf 1913B} in \cite{cremona-mazur}.
732
733
\subsubsection{Errata to Brumer's paper}
734
Contrary to our computations, \cite{brumer:rank} suggests that
735
{\bf 2333A} has rank~$0$. However, the author pointed the discrepancy out
736
to Brumer who replied:
737
\begin{quote}
738
I looked in vain for information about $\theta$-relations on~$2333$
739
and have concluded that I never ran the interval from~$2300$ to~$2500$
740
or else had lost all info by the time I wrote up the paper. The
741
punchline:~$4$ relations for~$2333$ and~$2$ relations for~$2381$ (also
742
missing from the table).
743
\end{quote}
744
745
\comment{
746
Date: Wed, 08 Sep 1999 18:24:10 -0400
747
From: armand brumer <brumer@murray.fordham.edu>
748
To: William Arthur Stein <was@math.berkeley.edu>
749
CC: ab <brumer@murray.fordham.edu>
750
Subject: Re: The rank of J_0(2333)
751
752
Dear William,
753
I just found your 3 emails (including one from the end of June)
754
sitting on a mail server I did not know existed until a few days ago (the
755
university did not tell us that the two addresses were on different
756
servers!!)
757
758
I then looked in vain for information about theta relations on 2333 and have
759
concluded that I never ran the interval from 2300 to 2500 or else had lost
760
all info by the time I wrote up the paper. The punchline:4 relations for 2333
761
and 2 relations for 2381 (also missing from the table). I may try to check as
762
much as possible in the background and would be grateful if you mention this
763
errata when you write up your stuff (I don't know any other way of
764
publicizing the correction).
765
766
Best regards and hope you did not think I was ``blowing you off" as my son
767
would say!
768
769
Armand
770
}
771
772
773
\subsection{Tables~\ref{table:newvis}--\ref{table:shacompgps}:
774
New visible Shafarevich-Tate groups}
775
776
Let~$n$ denote the largest odd square dividing the numerator of
777
$L(A,1)/\Omega_A$. Table~\ref{table:newvis} lists those~$A$ such that
778
for some $p\mid n$ there exists a quotient~$B$ of $J_0(N)$,
779
corresponding to a {\em newform} and having positive analytic rank,
780
such that $(\Adual\intersect B^{\vee})[p]\neq 0$. Our search was
781
systematic up to level $1001$, but there might be omitted examples
782
between levels $1001$ and $1028$.
783
Table~\ref{table:explain} contains
784
further arithmetic information about each explanatory factor.
785
Table~\ref{table:shacompgps} gives the quantities involved in the
786
formula of Chapter~\ref{chap:compgroups} for Tamagawa numbers, for
787
each of the abelian varieties~$A$ in Table~\ref{table:newvis}.
788
789
790
\subsubsection{Notation}
791
Most of the notation is the same as in Table~\ref{table:primesha}.
792
However the additional columns $w_q$ and $c_p$ contain the signs
793
of the Atkin-Lehner involutions and the Tamagawa numbers, respectively.
794
These are given in order, from smallest to largest prime divisor
795
of~$N$.
796
797
In each case~$B$ has dimension~$1$. When $4\mid N$, we write ``$a$''
798
for $c_2$ to remind us that we did not compute $c_2$ because the
799
reduction at~$2$ is additive. Again only
800
{\em odd parts} of the invariants are given.
801
Section~\ref{sec:compgrptables} contains a discussion of
802
the notation used in the
803
headings of Table~\ref{table:shacompgps}.
804
805
\subsubsection{Remarks on the data}
806
The explanatory factors~$B$ of level $\leq 1028$ are {\em exactly} the
807
set of rank~$2$ elliptic curves of level $\leq 1028$.
808
809
810
\section{Further visibility computations}
811
812
813
814
\subsection{Does $\Sha$ become visible at higher level?}
815
\label{sec:higherlevel}
816
817
This section is concerned with whether or not the examples of invisible
818
elements of Shafarevich-Tate groups of elliptic curves of level~$N$
819
that are given in \cite{cremona-mazur} become visible in abelian
820
surfaces inside appropriate $J_0(Np)$. We analyze each of the cases
821
in Table~1 of \cite{cremona-mazur}. For the reader's convenience, the
822
part of this table which concerns us is reproduced as
823
Table~\ref{table:cm}.
824
The most interesting examples we give
825
are {\bf 2849A} and {\bf 5389A}. As in
826
\cite{cremona-mazur}, the assertions below assume the
827
truth of the BSD conjecture.\index{BSD conjecture}
828
829
830
\begin{table}\index{Table of!odd invisible $\Sha_E$}
831
\ssp
832
\caption{Odd invisible $|\Sha_E|>1$, all $N\leq 5500$ (from Table~1 of~\cite{cremona-mazur})}
833
\label{table:cm}
834
$$
835
\begin{array}{lcclcl}
836
\mbox{\rm\bf E}&\sqrt{|\Sha_E|}& m_E & \mbox{\rm\bf F}
837
& m_F & \text{Remarks}\\
838
& & & & & \vspace{-3ex} \\
839
\mbox{\rm\bf 2849A}& 3 &2^5\cdot 5\cdot 61&\mbox{\rm\bf NONE}& - &\\
840
\mbox{\rm\bf 3364C}& 7 &2^6\cdot3^2\cdot5^2\cdot7 &\mbox{\rm\bf none}& - &\\
841
\mbox{\rm\bf 4229A}& 3 &2^3\cdot3\cdot7\cdot13 &\mbox{\rm\bf none}& - &\\
842
\mbox{\rm\bf 4343B}& 3 &2^4\cdot1583 &\mbox{\rm\bf NONE}& -&\\
843
\mbox{\rm\bf 4914N}& 3 &2^4\cdot 3^5 &\mbox{\rm\bf none}& - &\text{{\bf E} has rational $3$-torsion}\\
844
\mbox{\rm\bf 5054C}& 3 &2^3\cdot 3^3\cdot 11&\mbox{\rm\bf none}& - &\\
845
\mbox{\rm\bf 5073D}& 3 &2^5\cdot 3\cdot 5\cdot7\cdot23
846
&\mbox{\rm\bf none}& - & \\
847
\mbox{\rm\bf 5389A}& 3 &2^2\cdot 2333 &\mbox{\rm\bf NONE}& - &\\
848
\end{array}
849
$$
850
\end{table}
851
852
853
\subsubsection{How we found the explanatory curves}
854
We use a naive heuristic observation to find possible explanatory
855
curves of higher level, even though their conductors are out of the
856
range of Cremona's tables. Note that we have not proved that
857
these factors are actually explanatory in any cases, and expect
858
that in some cases they are not.
859
860
First we recall some of the notation from Table~1
861
of~\cite{cremona-mazur}, which is partially reproduced below.
862
The ``NONE'' label in the row corresponding
863
to an elliptic curve~$E$ indicates that there are elements in
864
$\Sha(E)$ whose order does not divide the modular degree of~$E$, and
865
hence they must be invisible. The label ``none'' indicates only that
866
no suitable explanatory elliptic curves could be found, so $\Sha(E)$ is
867
not visible in an {\em abelian surface} inside $J_0(N)$; it could,
868
however, be visible in the full abelian variety $J_0(N)$.
869
870
871
Studying the Weierstrass equations corresponding to the curves in
872
\cite{cremona-mazur} reveals that the elliptic curves labeled
873
``NONE'' have unusually large height, as compared to their conductors.
874
However, the explanatory factors often have unusually small height.
875
Motivated by this purely heuristic observation, we make a table of
876
all elliptic curves of the form
877
$$y^2 + a_1 xy + a_3 y = x^3 + a_2x^2 + a_4 x+ a_6,$$
878
with $a_1, a_2, a_3 \in \{-1,0,1\}$, $|a_4|, |a_6| < 1000$,
879
and conductor bounded by $50000$. The bound on the conductor
880
is required only so that the table will fit within
881
computer storage. This table took a few days to generate.
882
883
\subsubsection{2849A} Mazur\index{Mazur} and Adam Logan\index{Logan}
884
found the first known example of an
885
{\em invisible} Shafarevich-Tate group\index{Shafarevich-Tate group!first invisible example}. This
886
was $\Sha(E)$, where~$E$ is the elliptic curve {\bf 2849A},
887
which has minimal Weierstrass equation
888
$$E:\quad y^2 + xy + y = x^3 + x^2 - 53484x - 4843180.$$
889
Consulting our table of curves of small height,
890
we find an elliptic curve~$F$ of conductor
891
$8547=2839\cdot 3$ such that $f_E \con f_F \pmod{3}$, where $f_E$
892
and $f_F$ are the newforms attached to~$E$ and~$F$, respectively.
893
This is a congruence for {\em all} eigenvalues $a_p$ attached to~$E$ and~$F$.
894
The elliptic curve~$F$ is defined by the equation
895
$$F:\quad y^2 + xy + y = x^3 + x^2 - 154x - 478.$$
896
Cremona's program {\tt mwrank} reveals that
897
the Mordell-Weil group of~$F$ has rank~$2$.
898
Thus maybe $\Sha(E)$ becomes visible at level~$8547$.
899
Unfortunately, visibility of $\Sha(E)$ is not
900
implied by Theorem~\ref{thm:shaexists} because
901
the geometric component group of~$F$ at~$3$ has order
902
divisible by~$3$.
903
904
\subsubsection{4343B} Consider the elliptic curve $E$ labeled
905
{\bf 4343B}. According to Table~1 of \cite{cremona-mazur},
906
$\Sha(E)$ has order~$9$, but the modular degree prevents~$\Sha(E)$
907
from being visible in $J_0(4343)$.
908
At level $21715 = 5\cdot 4343$
909
there is an elliptic curve~$F$ of rank~$1$ that is
910
congruent to~$E$. Its equation is
911
$$F:\quad y^2 - xy - y = x^3 - x^2 + 78x - 256.$$
912
913
\subsubsection{5389A} The last curve labeled ``NONE'' in the table is curve
914
{\bf 5389A}, which has minimal Weierstrass equation
915
$$y^2+xy+y =x^3 - 35590x-2587197.$$
916
917
The main theorem of~\cite{ribet:raising} implies that there exists a
918
newform that is congruent modulo~$3$ to the newform corresponding to
919
{\bf 5389A} and of level $3\cdot 5389$. This is because $(-2)^2 =
920
(3+1)^2 \pmod{3}$. However, our table of curves of small height does
921
not contain any curve of conductor $3\cdot 5389$. Next we observe that
922
$(-2)^2 \con (7+1)^2 \pmod{3}$, so using Ribet's\index{Ribet} theorem we can
923
instead augment the level by~$7$. Our table of small-height curves
924
contains just one (up to isogeny) elliptic curve of
925
conductor~$37723$, and {\em luckily} the
926
corresponding newform is congruent modulo~$3$ to the newform
927
corresponding to {\bf 5389A} (away from primes dividing the level)!
928
The Weierstrass equation of this curve is
929
$$F:\quad y^2 - y = x^3 + x^2 + 34x - 248.$$
930
According to Cremona's program {\tt mwrank}, the rank of~$F$ is~$2$.
931
932
\subsubsection{3364C, 4229A, 5073D}
933
Perhaps $\Sha(E)$ is already visible in some of the cases in which the
934
curve is labeled ``none'', because the method fails in most
935
of these cases. Each of the curves {\bf 3364C}, {\bf
936
4229A}, and {\bf 5073D} is labeled ``none''.
937
In none of these 3 cases are we able to find
938
an explanatory factor at higher level, within the range of our table
939
of elliptic curves of small height.
940
941
\subsubsection{4194N, 5054C} The curve {\bf 4914N} is labeled ``none''
942
and we find the remark ``$E$ has rational $3$-torsion''.
943
There is a congruent curve~$F$ of conductor $24570$ given
944
by the equation
945
$$F: \quad y^2 - xy = x^3 - x^2 - 15x - 75,$$
946
and $F(\Q) = \{0\}$. The curve {\bf 5054C} is labeled ``none''
947
and its Shafarevich-Tate group contains invisible elements of
948
order~$3$. We find a congruent curve of level~$25270$ and rank~$1$.
949
The equation of the congruent curve is
950
$$F: y^2 - xy = x^3 + x^2 - 178x + 882.$$
951
952
953
\subsection{Positive rank example}
954
The abelian varieties with nontrivial $\Sha(A)$ that one
955
finds in both ours and Cremona's
956
tables all have rank~$0$. In this section we outline a computation
957
which sugggests, but does not prove, that there is a positive-rank abelian
958
subvariety $A$ of $J_0(5077)$ such that $\Sha(A)$
959
possesses a nontrivial visible element of order~$31$.
960
961
According to \cite{cremona:algs},
962
the smallest conductor elliptic curve~$E$ of rank~$3$ is found in
963
$J=J_0(5077)$. The number $5077$ is prime, and the isogeny
964
decomposition of~$J$ is\footnote{
965
This decomposition was found in about one minute
966
using the Mestre-Oesterl\'e\index{Mestre}
967
method of graphs (see~\cite{mestre:graphs}).}
968
$$J \sim A \cross B \cross E,$$
969
where each of~$A$, $B$, and~$E$ are abelian subvarieties of~$J$
970
associated to newforms, which have
971
dimensions $205$, $216$, and~$1$, respectively.
972
Using Remark~\ref{rem:moddegmestre} or
973
\cite{zagier:parametrizations},
974
we find that the modular degree of~$E$ is $1984=2^6\cdot 31$.
975
The sign of the Atkin-Lehner involution on~$E$ is the same
976
as its sign on~$A$, so $E[31]\subset A$.
977
We have $E(\Q)\isom \Z\cross\Z\cross\Z$, and the
978
component group of~$E$ is trivial.
979
The numerator of $(5077-1)/12$ is $3^2\cdot 47$, so \cite{mazur:eisenstein}
980
implies that none of the abelian varieties above have $31$-torsion.
981
It might be possible to find an analogue of Theorem~\ref{thm:shaexists}
982
that applies when~$A$ has positive rank, and deduce in this case
983
that $\Sha(A)$ contains visible elements of order~$31$.
984
985
%\section{Tables}
986
\begin{table}\index{Table of!$\Sha$ at prime level}
987
\ssp
988
\caption{Shafarevich-Tate groups at prime level.
989
(The entries in the columns
990
``mod deg'' and ``$\Shaan$'' are only really
991
the odd parts of ``mod deg'' and ``$\Shaan$''.)\label{table:primesha}}
992
\vspace{-.25in}$$
993
\begin{array}{lccclcc}
994
\mbox{\rm\bf A}& \mbox{\rm dim}& \Shaan(A) & \mbox{\rm mod deg}(A) & \mbox{\rm\bf B} & \mbox{\rm dim} & \mbox{\rm mod deg} (B)\\
995
& & & & & & \vspace{-3ex} \\
996
\mbox{\rm\bf 389E}& 20 &5^{2}&5&\mbox{\rm\bf 389A}& 1 &5\\
997
\mbox{\rm\bf 433D}& 16 &7^{2}&3\cdot7\cdot37&\mbox{\rm\bf 433A}& 1 &7\\
998
\mbox{\rm\bf 563E}& 31 &13^{2}&13&\mbox{\rm\bf 563A}& 1 &13\\
999
\mbox{\rm\bf 571D}& 2 &3^{2}&3^{2}\cdot127&\mbox{\rm\bf 571B}& 1 &3\\
1000
\mbox{\rm\bf 709C}& 30 &11^{2}&11&\mbox{\rm\bf 709A}& 1 &11\\
1001
\mbox{\rm\bf 997H}& 42 &3^{4}&3^{2}&\mbox{\rm\bf 997B}& 1 &3\\
1002
\mbox{\rm\bf 1061D}& 46 &151^{2}&61\cdot151\cdot179&\mbox{\rm\bf 1061B}& 2 &151\\
1003
\mbox{\rm\bf 1091C}& 62 &7^{2}&1&\mbox{\rm NONE} & & \\
1004
\mbox{\rm\bf 1171D}& 53 &11^{2}&3^{4}\cdot11&\mbox{\rm\bf 1171A}& 1 &11\\
1005
\mbox{\rm\bf 1283C}& 62 &5^{2}&5\cdot41\cdot59&\mbox{\rm NONE} & & \\
1006
\mbox{\rm\bf 1429B}& 64 &5^{2}&1&\mbox{\rm NONE} & & \\
1007
\mbox{\rm\bf 1481C}& 71 &13^{2}&5^{2}\cdot2833&\mbox{\rm NONE} & & \\
1008
\mbox{\rm\bf 1483D}& 67 &3^{2}\cdot5^{2}&3\cdot5&\mbox{\rm\bf 1483A}& 1 &3\cdot5\\
1009
\mbox{\rm\bf 1531D}& 73 &3^{2}&3&\mbox{\rm\bf 1531A}& 1 &3\\
1010
\mbox{\rm\bf 1559B}& 90 &11^{2}&1&\mbox{\rm NONE} & & \\
1011
\mbox{\rm\bf 1567D}& 69 &7^{2}\cdot41^{2}&7\cdot41&\mbox{\rm\bf 1567B}& 3 &7\cdot41\\
1012
\mbox{\rm\bf 1613D}& 75 &5^{2}&5\cdot19&\mbox{\rm\bf 1613A}& 1 &5\\
1013
\mbox{\rm\bf 1621C}& 70 &17^{2}&17&\mbox{\rm\bf 1621A}& 1 &17\\
1014
\mbox{\rm\bf 1627C}& 73 &3^{4}&3^{2}&\mbox{\rm\bf 1627A}& 1 &3^{2}\\
1015
\mbox{\rm\bf 1693C}& 72 &1301^{2}&1301&\mbox{\rm\bf 1693A}& 3 &1301\\
1016
\mbox{\rm\bf 1811D}& 98 &31^{2}&1&\mbox{\rm NONE} & & \\
1017
\mbox{\rm\bf 1847B}& 98 &3^{6}&1&\mbox{\rm NONE} & & \\
1018
\mbox{\rm\bf 1871C}& 98 &19^{2}&14699&\mbox{\rm NONE} & & \\
1019
\mbox{\rm\bf 1877B}& 86 &7^{2}&1&\mbox{\rm NONE} & & \\
1020
\mbox{\rm\bf 1907D}& 90 &7^{2}&3\cdot5\cdot7\cdot11&\mbox{\rm\bf 1907A}& 1 &7\\
1021
\mbox{\rm\bf 1913B}& 1 &3^{2}&3\cdot103&\mbox{\rm\bf 1913A}& 1 &3\cdot5^{2}\\
1022
\mbox{\rm\bf 1913E}& 84 &5^{4}\cdot61^{2}&5^{2}\cdot61\cdot103&\mbox{\rm\bf 1913A,C}& 1,2 &3\cdot5^{2}, 5^2\cdot 61\\
1023
\mbox{\rm\bf 1933C}& 83 &3^{2}\cdot7^{2}&3\cdot7&\mbox{\rm\bf 1933A}& 1 &3\cdot7\\
1024
\mbox{\rm\bf 1997C}& 93 &17^{2}&1&\mbox{\rm NONE} & & \\
1025
\mbox{\rm\bf 2027C}& 94 &29^{2}&29&\mbox{\rm\bf 2027A}& 1 &29\\
1026
\mbox{\rm\bf 2029C}& 90 &5^{2}\cdot269^{2}&5\cdot269&\mbox{\rm\bf 2029A}& 2 &5\cdot269\\
1027
\mbox{\rm\bf 2039F}& 99 &3^{4}\cdot5^{2}&19\cdot29\cdot7759\cdot3214201&\mbox{\rm NONE} & & \\
1028
\mbox{\rm\bf 2063C}& 106 &13^{2}&61\cdot139&\mbox{\rm NONE} & & \\
1029
\mbox{\rm\bf 2089J}& 91 &11^{2}&3\cdot5\cdot11\cdot19\cdot73\cdot139&\mbox{\rm\bf 2089B}& 1 &11\\
1030
\mbox{\rm\bf 2099B}& 106 &3^{2}&1&\mbox{\rm NONE} & & \\
1031
\mbox{\rm\bf 2111B}& 112 &211^{2}&1&\mbox{\rm NONE} & & \\
1032
\mbox{\rm\bf 2113B}& 91 &7^{2}&1&\mbox{\rm NONE} & & \\
1033
\mbox{\rm\bf 2161C}& 98 &23^{2}&1&\mbox{\rm NONE} & & \\\hline
1034
\mbox{\rm\bf 2333C}& 101 &83341^{2}&83341&\mbox{\rm\bf 2333A}& 4 &83341\\
1035
\mbox{\rm\bf 2339C}& 114 &3^{8}&6791&\mbox{\rm NONE} & & \\
1036
\mbox{\rm\bf 2411B}& 123 &11^{2}&1&\mbox{\rm NONE} & & \\
1037
\mbox{\rm\bf 2593B}& 109 &67^2\cdot 2213^2 & 67 \cdot 2213
1038
&\mbox {\bf 2593A}& 4
1039
& 67 \cdot 2213\\
1040
\end{array}
1041
$$
1042
\end{table}
1043
1044
\begin{table}\index{Table of!new visible $\Sha$}
1045
\ssp
1046
\caption{New visible Shafarevich-Tate groups\label{table:newvis}}\vspace{-2ex}
1047
$$\begin{array}{lcccccccl}
1048
\mathbf{A} &\text{dim} &\Shaan &w_q& c_p &\#A(\Q)
1049
&\frac{\#A(\Q)\cdot L(A,1)}{\Omega_A}
1050
&\mbox{\rm mod deg(A)} & \quad\mathbf{B}\\
1051
\vspace{-2ex} & & &&& & & \\
1052
\mathbf{389E} & 20&5^2&-&97&97&5^2&5&\mathbf{389A}\\
1053
\mathbf{433D} & 16&7^2&-&3^2&3^2&7^2&3\cdot 7\cdot 37&\mathbf{433A}\\
1054
\mathbf{446F}&8&11^2&+- &1,3&3&11^2&11\cdot359353&\mathbf{446B}\\
1055
\mathbf{563E}&31 & 13^2 & - & 281 &281 &13^2 &13 &\mathbf{563A}\\
1056
\mathbf{571D}&2 & 3^2 & - & 1 &1 & 3^2 & 3^2\cdot 127&\mathbf{571B}\\
1057
\mathbf{655D}&13 & 3^4 &+- & 1,1 & 1 & 3^4 & 3^2\cdot 19\cdot 515741&\mathbf{655A}\\
1058
1059
\mathbf{664F} & 8&5^2 &-+ &a,1&1& 5^2 &5 & \mathbf{664A}\\
1060
% Sha dim Wq c_p T TL/O delta B
1061
\mathbf{681B}&1 & 3^2 &+- & 1,1 & 1 & 3^2 & 3\cdot 5^3 & \mathbf{681C}\\
1062
\mathbf{707G}& 15& 13^2 &+- & 1,1 & 1 & 13^2 & 13\cdot 800077& \mathbf{707A}\\
1063
\mathbf{709C}&30& 11^2 &- & 59 &59 & 11^2 & 11 & \mathbf{709A}\\
1064
\mathbf{718F}&7& 7^2 &+- & 1,1 & 1 & 7^2 &7\cdot 151\cdot 35573 & \mathbf{718B}\\
1065
\mathbf{794G}&14& 11^2 &+- & 3,1 & 3 & 11^2 &3\cdot7\cdot11\cdot47\cdot35447& \mathbf{794A}\\
1066
\mathbf{817E}& 15& 7^2 &+- & 1,5 & 5 & 7^2 & 7\cdot 79 & \mathbf{817A}\\
1067
\mathbf{916G}&9& 11^2 &-+ & a,1 & 1 & 11^2 & 3^9\cdot 11\cdot 17\cdot 239 & \mathbf{916C}\\
1068
\mathbf{944O} &6& 7^2 &+- & a,1 & 1 & 7^2 & 7 & \mathbf{944E}\\
1069
\mathbf{997H}&42& 3^4 &- & 83 & 83 & 3^4 & 3^2 & \mathbf{997BC}\\
1070
\mathbf{1001L}&7& 7^2 &+-+& 1,1,1 & 1 & 7^2 & 7\cdot19\cdot47\cdot2273&\mathbf{1001C}\\
1071
\mathbf{1028E}&14& 11^2&-+ & a,1 & 3 & 3^4\cdot 11^2 & 3^{13}\cdot 11 & \mathbf{1028A}\\
1072
\end{array}$$
1073
\end{table}
1074
1075
\begin{table}\index{Table of!explanatory factors}
1076
\ssp
1077
\caption{Explanatory factors\label{table:explain}}\vspace{-2ex}
1078
$$\begin{array}{lcccccc}
1079
\mathbf{B}&\text{rank}&w_q&c_p&\#A(\Q)&\mbox{\rm mod deg(A)}&\text{Comments}\\
1080
\vspace{-2ex} & & && & & \\
1081
\mathbf{389A}& 2 &-&1 &1&5&\text{first curve of rank $2$}\\
1082
\mathbf{433A}&2 &-&1&1&7&\\
1083
\mathbf{446B}&2 &+-&1,1& 1 &11&\text{called $\mathbf{446D}$ in \cite{cremona:algs}}\\
1084
\mathbf{563A}&2 &- & 1 & 1 & 13 & \\
1085
\mathbf{571B}&2 &- & 1 & 1 & 3 & \\
1086
\mathbf{655A}&2 &+-& 1,1 & 1 & 3^2 & \\
1087
\mathbf{664A}&2 &-+& 1,1 & 1 & 5 & \\
1088
% RANK wq g_p Tor delta comments
1089
\mathbf{681C} & 2 & +- & 1,1 & 1 & 3 & \\
1090
\mathbf{707A} & 2 & +- & 1,1 & 1 & 13 & \\
1091
\mathbf{709A} & 2 & - & 1 & 1 & 11 & \\
1092
\mathbf{718B} & 2 & +- & 1,1 & 1 & 7 & \\
1093
\mathbf{794A} & 2 & +- & 1,1 & 1 & 11 & \\
1094
\mathbf{817A} & 2 & +- & 1,1 & 1 & 7 & \\
1095
\mathbf{916C} & 2 & -+ & 3,1 & 1 & 3\cdot 11 & \\
1096
\mathbf{944E} & 2 & +- & 1,1 & 1 & 7 & \\
1097
\mathbf{997B} & 2 & - & 1 & 1 & 3 & \\
1098
\mathbf{997C} & 2 & - & 1 & 1 & 3 & \\
1099
\mathbf{1001C} & 2 & +-+& 1,3,1& 1 & 3^2\cdot 7 & \\
1100
\mathbf{1028A} & 2 & -+ & 3,1 & 1 & 3\cdot 11& \text{intersects $\mathbf{1028E}$ mod $11$}\\
1101
\end{array}$$
1102
\end{table}
1103
1104
1105
\begin{table}\index{Table of!factorizations}
1106
\ssp
1107
\caption{Factorizations\label{table:factor}}\vspace{-1ex}
1108
$$\begin{array}{llll}
1109
\mathbf{446}=2\cdot 223&
1110
\mathbf{655}=5\cdot 131&
1111
\mathbf{664}=2^3\cdot 83&
1112
\mathbf{681}=3\cdot 227\\
1113
\mathbf{707}=7\cdot 101&
1114
\mathbf{718}=2\cdot 359&
1115
\mathbf{794}=2\cdot 397&
1116
\mathbf{817}=19\cdot 43\\
1117
\mathbf{916}=2^2\cdot 229&
1118
\mathbf{944}=2^4\cdot 59&
1119
\mathbf{1001}=7\cdot 11\cdot 13&
1120
\mathbf{1028}=2^2\cdot 257\\
1121
\end{array}$$
1122
\end{table}
1123
1124
\begin{table}\index{Table of!component groups of explanatory factors}
1125
\ssp
1126
\caption{Component groups\label{table:shacompgps}}\vspace{-2ex}
1127
$$\begin{array}{lcccccc}
1128
\vspace{-2ex}&&&&&&\\
1129
\mathbf{A} &\text{dim} & p & w_q &\#\Phi_{X,p} &m_{X,p}
1130
&\#\Phi_{A,p}(\Fpbar) \\
1131
\vspace{-2ex}& & & & & \\
1132
\mathbf{389E}&20& 389&-& 97 & 5\cdot 97 & 97 \\
1133
\mathbf{433D}&16& 433&-& 3^2& 3^3\cdot 7\cdot 37 & 3^2 \\
1134
\mathbf{446F}&8 & 223&-& 3 & 3\cdot 11\cdot 359353 & 3 \\
1135
&& 2 &+ & 3 & 3\cdot 11& 3\cdot 359353\\
1136
\mathbf{563E}&31& 563&-& 281& 13\cdot 281& 281 \\
1137
\mathbf{571D}&2 & 571&-& 1 & 3^2\cdot 127 &1 \\
1138
\mathbf{655D}&13& 131&-& 1 & 3^{2}\cdot19\cdot515741 & 1\\
1139
&& 5&+& 1 & 3^2 & 19\cdot 515741 \\
1140
\mathbf{664F}&8 & 83&+& 1 & 5 & 1 \\
1141
\mathbf{681B}&1 & 227&-& 1 & 3\cdot 5^3 & 1 \\
1142
& & 3&+& 1 & 3\cdot 5^2& 5 \\
1143
\mathbf{707G}&15& 101&-& 1 & 13\cdot800077 & 1 \\
1144
&& 7&+& 1 & 13& 800077 \\
1145
\mathbf{709C}&30& 709&-& 59& 11\cdot 59 & 59 \\
1146
\mathbf{718F}&7 & 359&-& 1 & 7\cdot 151\cdot 35573 &1 \\
1147
&& 2 &+& 1 & 7 & 151\cdot 35573 \\
1148
\mathbf{794G}&14& 397&-& 3 & 3^2\cdot7\cdot11\cdot47\cdot35447 & 3 \\
1149
&& 2&+& 3 & 3\cdot11& 3^2\cdot 7\cdot 47\cdot 35447 \\
1150
\mathbf{817E}&15& 43&- & 5 & 5\cdot7\cdot 79 & 5 \\
1151
&& 19&+ &1 & 7 & 79 \\
1152
\mathbf{916G}&9 & 229&+ &1 & 3^9\cdot 11\cdot 17\cdot 239 & 1 \\
1153
1154
\mathbf{944O}&6 & 59&-& 1 & 7 & 1\\
1155
\mathbf{997H}&42& 997&-& 83& 3^2\cdot 83 & 83 \\
1156
\mathbf{1001L}&7& 13&+& 1& 7\cdot 19\cdot 47\cdot 2273& 1\\
1157
&& 11&-& 1& 7\cdot19\cdot47\cdot2273 & 1 \\
1158
&& 7&+& 1& 7\cdot 19\cdot 47 & 2273 \\
1159
\mathbf{1028E}&14&257&+& 1 & 3^{13}\cdot 11 & 1 \\
1160
\end{array}$$
1161
\end{table}
1162
1163
1164
1165
\comment{
1166
1167
\begin{table}\index{Table of!odd square numerators}
1168
\caption{Square roots of odd square divisors of $L(A,1)/\Omega_A$\label{table:oddnumer}}\vspace{2ex}
1169
$$\begin{array}{lc}
1170
\mathbf{305D7}&3\\
1171
\mathbf{309D8}&5\\
1172
\mathbf{335E11}&3^2\\
1173
\mathbf{389E20}&5\\
1174
\mathbf{394A2}&5\\
1175
\mathbf{399G5}&3^4\\
1176
\mathbf{433D16}&7\\
1177
\mathbf{435G2}&3\\
1178
\mathbf{436C4}&3\\
1179
\mathbf{446E7}&3\\
1180
\mathbf{446F8}&11\\
1181
\mathbf{455D4}&3\\
1182
\mathbf{473F9}&3\\
1183
\mathbf{500C4}&3\\
1184
\mathbf{502E6}&11\\
1185
\mathbf{506I4}&5\\
1186
\mathbf{524D4}&3\\
1187
\mathbf{530G4}&7\\
1188
\mathbf{538E7}&3\\
1189
\mathbf{551H18}&3\\
1190
\mathbf{553D13}&3\\
1191
\mathbf{555E2}&3\\
1192
\mathbf{556C7}&3\\
1193
\mathbf{563E31}&13\\
1194
\mathbf{564C3}&3\\
1195
\mathbf{571D2}&3\\
1196
\end{array}\qquad
1197
\begin{array}{lc}
1198
\mathbf{579G13}&15\\
1199
\mathbf{597E14}&19\\
1200
\mathbf{602G3}&3\\
1201
\mathbf{604C6}&3 \\
1202
\mathbf{615F6}&5 \\
1203
\mathbf{615G8}&7 \\
1204
\mathbf{620D3}&3\\
1205
\mathbf{620E4}&3\\
1206
\mathbf{626F12}&5\\
1207
\mathbf{629G15}&3\\
1208
\mathbf{642D2}&3\\
1209
\mathbf{644C5}&3\\
1210
\mathbf{644D5}&3\\
1211
\mathbf{655D13}&3^2\\
1212
\mathbf{660F2}&3\\
1213
\mathbf{662E10}\!&\!\!43\\
1214
\mathbf{664F8}&5\\
1215
\mathbf{668B5}&3\\
1216
\mathbf{678I2}&3\\
1217
\mathbf{681B1}&3\\
1218
\mathbf{681I10}&3\\
1219
\mathbf{682I6}&11\\
1220
\mathbf{707G15}&13\\
1221
\mathbf{709C30}&11\\
1222
\mathbf{718F7}&7\\
1223
\mathbf{721F14}&3^2\\
1224
\end{array}\qquad
1225
\begin{array}{lc}
1226
\mathbf{724C8}&3\\
1227
\mathbf{756G2}&3\\
1228
\mathbf{764A8}&3\\
1229
\mathbf{765M4}&3\\
1230
\mathbf{766B4}&3\\
1231
\mathbf{772C9}&3\\
1232
\mathbf{790H6}&3\\
1233
\mathbf{794G12}\!&\!\!11\\
1234
\mathbf{794H14}&5^2\\
1235
\mathbf{796C8}&3\\
1236
\mathbf{817E15}&7\\
1237
\mathbf{820C4}&3\\
1238
\mathbf{825E2}&3\\
1239
\mathbf{844C10}\!&\!\!3^2\\
1240
\mathbf{855M4}&3\\
1241
\mathbf{860D4}&3\\
1242
\mathbf{868E5}&3\\
1243
\mathbf{876E5}&3\\
1244
\mathbf{878C2}&3\\
1245
\mathbf{884D6}&3\\
1246
\mathbf{885L9}&3^2\\
1247
\mathbf{894H2}&3\\
1248
\mathbf{902I5}&3\\
1249
\mathbf{913G17}&3\\
1250
\mathbf{916G9}&11\\
1251
\mathbf{918O2}&5\\
1252
\end{array}\qquad
1253
\begin{array}{lc}
1254
\mathbf{918P2}&3\\
1255
\mathbf{925K7}&3\\
1256
\mathbf{932B13}&3^2\\
1257
\mathbf{933E14}&19\\
1258
\mathbf{934I12}&7\\ %-+
1259
\mathbf{944O6}&7\\
1260
\mathbf{946K7}&3\\
1261
\mathbf{949B2}&3\\
1262
\mathbf{951D19}&3\\
1263
\mathbf{959D24}&3\\
1264
\mathbf{964C12}&3^2\\ % -+, same as EC 964A but that has rank=0.
1265
\mathbf{966J1}&3\\
1266
\mathbf{970I5}&3\\
1267
\mathbf{980F1}&3\\
1268
\mathbf{980J2}&3\\
1269
\mathbf{986J7}&5\\
1270
\mathbf{989E22}&5\\
1271
\mathbf{993B3}&3^2\\
1272
\mathbf{996E4}&3\\
1273
\mathbf{997H42}&3^2\\
1274
\mathbf{998A2}&3\\ % ++
1275
\mathbf{998H9}&3\\
1276
\mathbf{999J10}&3\\
1277
&\\
1278
&\\
1279
&\\
1280
\end{array}$$
1281
\end{table}
1282
1283
1284
}
1285
1286
1287
1288
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
1289
1290
1291
1292
1293
\comment{
1294
1295
\begin{lemma}
1296
Let~$A$ and~$B$ be abelian subvarieties of an abelian variety~$J$ such
1297
that $\Phi=A\intersect B$ is finite and and $A(\Q)$ is finite.
1298
Suppose that~$p$
1299
is a prime such that neither $(J/B)(\Q)$ nor $B(\Q)$ have any
1300
$p$-torsion and such that $B[p]\subset \Phi$.
1301
Then $B(\Q)/p B(\Q)$ is a subgroup of
1302
$(J/B)(\Q)/J(\Q)$.
1303
\end{lemma}
1304
\begin{proof}
1305
1306
\end{proof}
1307
1308
1309
1310
\section{Explanatory factors at higher level}
1311
Consider one of the items in Table~\ref{table:primesha} for which
1312
$\Sha$ is invisible. It is natural to ask whether these
1313
elements of~$\Sha$ ``become visible somewhere.''
1314
For example, Mazur~\cite{mazur:visthree}\index{Mazur} proved that if
1315
$E$ is an elliptic curve and $c\in\Sha(E)$ has order $3$ then
1316
there is some abelian surface $A$ and an
1317
injection $\iota: E\hookrightarrow A$ such that
1318
$\iota_*(c)=0\in H^1(\Q,A)$. T. Klenke has proved
1319
a partial statement in this direction for elements of
1320
order $2$ as part of his Harvard Ph.D. thesis.
1321
J.~de Jong (see \cite[Remark 3]{cremona-mazur})
1322
showed that every element of the Shafarevich-Tate
1323
group of an elliptic curve is visible in some Jacobian.\index{Jacobian}
1324
1325
Consider an abelian variety $A$, taken
1326
from Table~\ref{table:primesha}, for which
1327
$\Sha$ has an invisible element $c$. Thus
1328
$A$ sits inside $J_0(p)$ for some prime $p$,
1329
and we ask ``is there a prime $q$ such that $\delta(c)=0$
1330
for one of the degeneracy maps
1331
$\delta : J_0(p)\ra J_0(pq)$?''
1332
1333
The author has no idea\footnote{Lo\"\i{}c Merel\index{Merel} suspects
1334
the answer might be yes whereas Richard Taylor is more skeptical.}.
1335
To get a feeling for what might happen we consider in detail abelian
1336
variety $A=A_f$ at level $p=1091$ in which $\Shaan$ is divisible by
1337
$7$.
1338
1339
There is a prime $\lambda$ of the ring
1340
$\Z[f] = \Z[\ldots a_n\ldots]$ attached to $A$.
1341
The Fourier coefficients of $f$ modulo $\lambda$ are
1342
$$\begin{array}{rcccccccccccccccc}
1343
p= &2 &3 &5 &7 &11 &13 &17 &19 &23 &29 &31 &37 &41 &43 &47 &53\\
1344
a_p= &3 &0 &1 &5 &0 &2 &0 &5 &4 &6 &3 &3 &5 &5 &6 &5
1345
\end{array}$$
1346
These were computed by finding an eigenvector in $H_1(X_0(N);\F_7)$.
1347
[[SAY MORE ABOUT THE TRICK FOR FINDING ALL RIBET $q$'s.]]
1348
1349
According to Ribet's\index{Ribet} level raising theorem \cite{ribet:raising}
1350
there is a newform $g$ of level $1091\ell$ such that
1351
$f\con g$ modulo [[something]] if $a_\ell = \pm (\ell+1)\pmod{\lambda}$.
1352
Fortunately this criterion is already satisfied for $\ell=2$.
1353
Looking closely at level $2\cdot 1091$ (for example,
1354
in Cremona's online tables \cite{cremona:onlinetables})
1355
we find an elliptic curve $E$ whose corresponding newform
1356
$g=\sum b_n q^n$ has Fourier coefficients
1357
$$\begin{array}{rcccccccccccccccc}
1358
p = &2 & 3 & 5 & 7& 11& 13& 17& 19& 23& 29& 31& 37&41&43&47&53\\
1359
b_p=&1 & 0 & 1 &-2& 0 & -5& 0 & -2& 4& 6& -4& -4&-9&-9&-8&-2\\
1360
\end{array}$$
1361
1362
This is convincing evidence that one of the
1363
two images of~$A$ in $J_0(2\cdot 1091)$ shares some
1364
$7$-torsion with the elliptic curve \abvar{2182B}.
1365
This can be [[WILL BE!!, EASILY]] established by
1366
a direct computation with the period lattices.
1367
This is at first disconcerting because the rank of
1368
this elliptic curve is {\em not} $2$. However, the
1369
rank is still positive; it is $1$ with
1370
Mordell-Weil group $\E(\Q)=Z$.
1371
1372
I would not be at all surprised if your
1373
$7$-torsion in Sha does become visible in $J_0(2\cdot1091)$.
1374
The curve \abvar{2182B}, which shares 7-torsion with $A$ is
1375
1376
\begin{verbatim}
1377
e=ellinit([1,-1,1,-67,67]);
1378
The Tamagawa number c_2 is 14 (!!)
1379
? elllocalred(e,2)
1380
%2 = [1, 18, [1, 0, 0, 0], 14]
1381
The Tamagawa number c_1091 is 1.
1382
? elllocalred(e,1091)
1383
%3 = [1, 5, [1, 0, 0, 0], 1]
1384
1385
I have this feeling that the right statement about congruence
1386
and mordell-Weil is really something like
1387
congruence ==> "Selmer + Comp group"'s are identified.
1388
Anyway, the extra component group of order 7 may perhaps
1389
account for the other nontrivial element of Sha. This might
1390
just be wild speculation.
1391
1392
Good luck.
1393
1394
william
1395
Barry,
1396
1397
Amod asked me to investigate whether his element of order 7
1398
in the winding quotient J_e at level 1091 becomes visible at
1399
higher level. Luckily, Ribet's\index{Ribet} level raising theorem predicts the
1400
existence of a form at level 2*1091 congruent mod a prime over
1401
7 to the form corresponding to J_e. Even more luckily, one of the
1402
two rational newforms does the trick. Thus an image of J_e in
1403
J_0(2*1091) shares 7-torsion with an elliptic curve E (2182B
1404
in Cremona's tables). This elliptic curve has:
1405
1406
E(Q) = Z
1407
Sha(E/Q) = 0
1408
c_2 = 14, c_1091 = 1
1409
L^(1)(f,1)/1! = 4.27332686791516
1410
1411
1412
So there is reasonable hope that the elements of order 7 in
1413
Sha(J_e) are visible at this higher level, even though they
1414
are invisible even in J_1(1091).
1415
1416
Best,
1417
William
1418
1419
1420
Dear William,
1421
This is terrific. I assume that you will be showing that for J_e the
1422
winding
1423
(not quite quotient, but more conveniently sub-thing) in J_0(1091),
1424
the image of
1425
1426
Sha(J_e) ---> Sha(J_0(2*1091))
1427
1428
just dies? Since our working hope, I think, is that for any N there
1429
is an
1430
M so that
1431
Sha(J_0(N)) ---. Sha(J_0(N.M))
1432
1433
1434
dies, this suggests returning to the (mod 3) N=2849 example, where I
1435
"know" that there must exist such an M (because all three-torsion in Sha
1436
on elliptic curves is visible in some appropriate abelian surface which is
1437
isogenous to a product of two elliptic curves, and therefore, is abelian
1438
surface is probably "modular"). But I do not know a specific M.
1439
1440
Barry
1441
1442
\end{verbatim}
1443
1444
1445
\comment{
1446
\begin{remark}
1447
One reason we must assume~$p$ is odd, is because
1448
when~$B$ has good reduction at~$2$,
1449
in the proof we change~$J$ by an isogeny of $2$-power
1450
degree in order to apply~\cite[\S7.5, Prop.~3]{neronmodels} at $p=2$.
1451
When~$B$ has purely toric reduction,\index{Toric reduction}
1452
at~$2$ we use Tate
1453
uniformization to directly verify that points of $B(\Q)$ map into
1454
$\Sha(A)$, thereby avoiding exactness properties of
1455
N\'eron models\index{N\'eron model}.
1456
\end{remark}}
1457
1458
\subsubsection{Table~\ref{table:oddnumer}:
1459
Odd square divisors of $L(A,1)/\Omega_A$}
1460
In order to find candidate~$A$ with nontrivial visible
1461
$\Sha(A)$, we first enumerated those~$A$ for which the numerator
1462
of $L(A,1)/\Omega_A$ is divisible by an odd square~$n>1$.
1463
For $N<1000$, these are given in Table~\ref{table:oddnumer}.
1464
Any odd visible $\Sha(A)$ coprime to
1465
primes dividing torsion and $c_p$ must show up as a divisor
1466
of the numerator; it should show up as a square
1467
divisor because the Mordell-Weil rank of the explanatory factor
1468
should be even. It would be interesting to compute the conjectural
1469
order of $\Sha(A)$ for each abelian variety in this table, but
1470
not in table 1, and show (when possible) that the visible
1471
$\Sha(A)$ is old.
1472
}
1473
1474
1475
1476
1477
1478
1479