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\title{Visibility of Shafarevich-Tate Groups of\\ Abelian Varieties}
\author{Amod Agashe
\\University of Texas
\\Austin, TX
\\{\tt agashe@math.utexas.edu}
\and
William Stein
\\Harvard University
\\Cambridge, MA
\\{\tt was@math.harvard.edu}
}
\date{\today}
\begin{document}
\UseTips
\maketitle
\begin{abstract}
We investigate Mazur's notion of visibility of elements of
Shafarevich-Tate groups of abelian varieties. We give a proof that
every cohomology class is visible in a suitable abelian variety,
discuss the visibility dimension, and describe a construction of
visible elements of certain Shafarevich-Tate groups. This
construction can be used to give some of the first evidence for the
Birch and Swinnerton-Dyer Conjecture for abelian varieties of large
dimension. We then give examples of visible and invisible
Shafarevich-Tate groups.
\end{abstract}
\keywords{Visibility, Shafarevich-Tate Group, Birch and Swinnerton-Dyer Conjecture, Modular Abelian Variety}
\section*{Introduction}\label{sec:intro}
If a genus~$0$ curve~$X$ over~$\Q$ has a point in every local field
$\Q_p$ and in $\R$, then it has a global point over~$\Q$. For
genus~$1$ curves, this ``local-to-global principle'' frequently fails.
For example, the nonsingular projective curve defined by the equation
$3x^3+4y^3+5z^3=0$ has a point over each local field and
$\R$, but has no $\Q$-point.
The Shafarevich-Tate group of an elliptic curve~$E$, denoted
$\Sha(E)$, is a group that measures the extent to which a
local-to-global principle fails for the genus one curves
with Jacobian~$E$. More generally, if~$A$ is an abelian variety over
a number field~$K$, then the elements of
the Shafarevich-Tate group $\Sha(A)$ of~$A$ correspond to the
torsors for~$A$ that have a point everywhere locally, but not
globally. In this paper, we study a geometric way of realizing (or
``visualizing'') torsors corresponding to elements of~$\Sha(A)$.
Let~$A$ be an abelian variety over a
field~$K$. If $\iota: A\hookrightarrow J$ is
a closed immersion of abelian varieties,
then the subgroup of $H^1(K,A)$
{\em visible in~$J$} (with respect to~$\iota$) is
$\ker(H^1(K,A)\ra H^1(K,J))$.
We prove that every element of $H^1(K,A)$ is visible in
some abelian variety, and give bounds on the smallest size of an
abelian variety in which an element of $H^1(K,A)$ is visible.
Next assume that~$K$ is a number field. We give a construction of visible
elements of $\Sha(A)$, which we demonstrate by giving evidence for the
Birch and Swinnerton-Dyer conjecture for a certain $20$-dimensional
abelian variety. We also give an example of an elliptic curve~$E$
over~$\Q$ of conductor~$N$ whose Shafarevich-Tate group is not
visible in $J_0(N)$ but is visible in $J_0(N p)$ for some prime~$p$.
This paper is organized as follows. Section~\ref{sec:defs} contains
the definition of visibility for cohomology classes and elements of
Shafarevich-Tate groups. Then in
Section~\ref{sec:torsors}, we use a restriction of
scalars construction to prove that every cohomology class is visible
in some abelian variety. Next, in Section~\ref{sec:visdim}, we
investigate the visibility dimension of cohomology classes.
Section~\ref{sec:construction} contains a theorem that can be used to
construct visible elements of Shafarevich-Tate groups. The final
section, Section~\ref{sec:examples}, contains examples and applications
of our visibility results in the context of modular abelian varieties.
\begin{acknowledge}
We thank Barry Mazur for
his generous guidance, Brian Conrad for his extensive assistance,
Ralph Greenberg for suggesting the use of
restriction of scalars in Section~\ref{sec:torsors}, Fabrizio
Andreatta for suggesting that a semistability hypothesis was
unnecessary in Theorem~\ref{thm:shaexists},
and Loic Merel, Bjorn Poonen, and Ken Ribet for helpful conversations.
The first author would like to thank the Mathematical Sciences
Research Institute in Berkeley and the Institut des Hautes \'Etudes
Scientifiques in France, and the second author the Max Planck Institute
in Bonn, for their generous hospitality.
\end{acknowledge}
\section{Visibility}\label{sec:defs}
In Section~\ref{sec:visdef} we introduce visible cohomology classes,
then in Section~\ref{sec:visshadef} we discuss visible elements of
Shafarevich-Tate groups. In Section~\ref{sec:torsors}, we use
restriction of scalars to deduce that every cohomology class is
visible somewhere.
For a field~$K$ and a smooth commutative $K$-group scheme~$G$, we
write $H^i(K,G)$ to denote the group cohomology
$H^i(\Gal(K_s/K),G(K_s))$ where $K_s$ is a fixed separable closure
of~$K$; equivalently, $H^i(K,G)$ denotes the $i$th \'etale
cohomology of~$G$ viewed as an \'etale sheaf on
$\Spec(K)_{\mbox{\small\rm \'et}}$.
\subsection{Visible Elements of $H^1(K,A)$}
\label{sec:visdef}
In \cite{mazur:visthree}, Mazur introduced the following definition.
Let~$A$ be an abelian variety over an arbitrary field~$K$.
\begin{definition}
Let $\iota:A\hookrightarrow J$
be an embedding of~$A$ into an abelian variety~$J$ over~$K$.
Then the \defn{visible subgroup of $H^1(K,A)$ with respect
to the embedding~$\iota$} is
$$\Vis_J(H^1(K,A)) = \Ker(H^1(K,A)\ra{}H^1(K,J)).$$
\end{definition}
The visible subgroup $\Vis_J(H^1(K,A))$ depends on the choice of
embedding~$\iota$, but we do not include~$\iota$ in the notation, as
it is usually clear from context.
The Galois cohomology group $H^1(K,A)$ has a geometric interpretation
as the group of classes of torsors~$X$ for~$A$ (see~\cite{lang-tate}).
To a cohomology class $c\in H^1(K,A)$, there is a corresponding
variety~$X$ over~$K$ and a map $A\cross X \ra X$ that satisfies axioms
similar to those for a simply transitive group action. The set of
equivalence classes of such~$X$ forms a group, the Weil-Chatelet group
of~$A$, which is canonically isomorphic to $H^1(K,A)$.
There is a close relationship between visibility and the geometric
interpretation of Galois cohomology. Suppose $\iota: A\ra J$ is an
embedding and $c\in \Vis_J(H^1(K,A))$. We have an exact sequence of
abelian varieties $0\ra A\ra J\ra C\ra 0$, where $C=J/A$. A piece of
the associated long exact sequence of Galois cohomology is
$$0 \ra A(K) \ra J(K)\ra C(K) \ra H^1(K,A) \ra H^1(K,J) \ra \cdots,$$
so there is an exact sequence
\begin{equation}\label{eqn:vis}
0 \ra J(K)/A(K) \ra C(K) \ra \Vis_J(H^1(K,A)) \ra 0.
\end{equation}
Thus there is a point $x\in C(K)$ that maps to~$c$. The fiber~$X$
over~$x$ is a subvariety of~$J$, which, when equipped with its natural
action of~$A$, lies in the class of torsors corresponding to~$c$.
This is the origin of the terminology ``visible''. Also, we remark
that when~$K$ is a number field, $\Vis_J(H^1(K,A))$ is finite
because it is torsion
and is the surjective image of the finitely generated group $C(K)$.
\subsection{Visible Elements of $\Sha(A)$}
\label{sec:visshadef}
Let~$A$ be an abelian variety over a number field~$K$.
The Shafarevich-Tate group of~$A$, which is defined below,
measures the failure of the local-to-global principle for
certain torsors.
The \defn{Shafarevich-Tate group} of $A$ is
$$\Sha(A) := \Ker\left(H^1(K,A) \ra \prod_{v} H^1(K_v,A)\right),$$
where the product is over all places of~$K$.
\begin{definition}
If $\iota:A\hookrightarrow J$ is an embedding, then the
{\em visible subgroup of $\Sha(A)$ with respect to~$\iota$}
is
$$\Vis_J(\Sha(A)) := \Sha(A) \intersect \Vis_J(H^1(K,A))
= \Ker(\Sha(A)\ra \Sha(J)).$$
\end{definition}
\subsection{Every Element is Visible Somewhere}\label{sec:torsors}
\begin{proposition}\label{prop:allvisible}
Every element of $H^1(K,A)$ is visible in some abelian variety~$J$.
\end{proposition}
\begin{proof}
Fix $c\in H^1(K,A)$. There is a finite separable extension~$L$ of~$K$ such
that
$\res_L(c) = 0\in H^1(L,A)$. Let $J=\Res_{L/K}(A_L)$ be the
Weil restriction of scalars from~$L$ to~$K$ of the abelian variety~$A_L$
(see \cite[\S7.6]{neronmodels}).
Thus~$J$ is an abelian variety over~$K$ of dimension $[L:K]\cdot \dim(A)$,
and for any scheme~$S$ over~$K$, we have a natural (functorial)
group isomorphism $A_L(S_L)\isom{}J(S)$.
The functorial injection $A(S) \hookrightarrow A_L(S_L) \isom{}J(S)$
corresponds via Yoneda's Lemma to a natural $K$-group scheme
map $\iota:A \rightarrow J$, and by construction~$\iota$
is a monomorphism.
But~$\iota$ is proper and thus
is a closed immersion (see \cite[\S8.11.5]{ega4_3}).
Using the Shapiro lemma one finds, after a tedious computation, that
there is a canonical isomorphism
$H^1(K,J)\isom H^1(L,A)$
which identifies $\iota_*(c)$ with $\res_L(c)=0$.
\end{proof}
\begin{remark}\mbox{}\vspace{-1ex}
\begin{enumerate}
\item
In \cite{cremona-mazur}, de Jong gave a totally different proof
of the above proposition in the case when~$A$ is an elliptic
curve over a number field. His argument actually displays~$A$
as visible inside the Jacobian of a curve.
\item
L.~Clozel has remarked that the method of proof above is a
standard technique in the theory of algebraic groups.
\end{enumerate}
\end{remark}
\section{The Visibility Dimension}\label{sec:visdim}
Let~$A$ be an abelian variety over a field~$K$
and fix $c\in H^1(K,A)$.
\begin{definition}
The {\em visibility dimension} of~$c$ is the
minimum of the dimensions of the abelian varieties~$J$
such that~$c$ is visible in~$J$.
\end{definition}
In Section~\ref{sec:simple_bound} we prove an elementary lemma which,
when combined with the proof of Proposition~\ref{prop:allvisible},
gives an upper bound on the visibility dimension of~$c$ in terms of
the order of~$c$ and the dimension of~$A$. Then, in
Section~\ref{sec:visdim1}, we consider the visibility dimension in the
case when~$A=E$ is an elliptic curve. After summarizing the results
of Mazur and Klenke on the visibility dimension, we apply a theorem of
Cassels to deduce that the visibility dimension of $c\in \Sha(E)$ is
at most the order of~$c$.
\subsection{A Simple Bound}\label{sec:simple_bound}
The following elementary lemma, which the second author learned from
Hendrik Lenstra, will be used to give a bound on the visibility
dimension in terms of the order of~$c$ and the dimension of~$A$.
\begin{lemma}\label{lem:splitbound}
Let~$G$ be a group, ~$M$ be a finite (discrete) $G$-module,
and $c \in H^1(G,M)$. Then there is a subgroup $H$ of
$G$ such that $\res_H(c)=0$ and $\#(G/H) \leq \#M$.
\end{lemma}
\begin{proof}
Let $f:G \ra M$ be a cocycle corresponding to~$c$, so $f(\tau\sigma) =
f(\tau) + \tau f(\sigma)$ for all $\tau, \sigma\in G$. Let $H =
\ker(f) = \{\sigma \in G : f(\sigma) = 0\}$. The map $\tau H \mapsto
f(\tau)$ is a well-defined injection from the coset space $G/H$
to~$M$.
\end{proof}
The following is a general bound on the visibility dimension.
\begin{proposition}\label{prop:visdim}
The visibility dimension of any~$c\in H^1(K,A)$
is at most $d\cdot{}n^{2d}$
where~$n$ is the order of~$c$ and~$d$ is the dimension of~$A$.
\end{proposition}
\begin{proof}
The map $H^1(K,A[n])\ra H^1(K,A)[n]$ is surjective
and $A[n]$ has order $n^{2d}$,
so
Lemma~\ref{lem:splitbound} implies that there is an extension~$L$
of~$K$ of degree at most $n^{2d}$ such that $\res_L(c)=0$.
The proof of Proposition~\ref{prop:allvisible} implies
that~$c$ is visible in an abelian variety of dimension
$[L:K]\cdot \dim A\leq d n^{2d}$.
\end{proof}
\subsection{The Visibility Dimension for Elliptic Curves}
\label{sec:visdim1}
We now consider the case when $A=E$ is
an elliptic curve over a number field~$K$.
Mazur proved in \cite{mazur:visthree} that every nonzero $c\in
\Sha(E)[3]$ has visibility dimension~$2$ (note that
Proposition~\ref{prop:visdim} only implies that the visibility
dimension is $\leq 3$). Mazur's result is particularly nice because it
shows that~$c$ is visible in an abelian variety that is isogenous to
the product of two elliptic curves. Using similar techniques,
T.~Klenke proved in \cite{klenke:phd} that every nonzero $c\in
H^1(K,E)[2]$ has visibility dimension~$2$ (note that
Proposition~\ref{prop:visdim} only implies that the visibility
dimension of any $c\in H^1(K,E)[2]$ is $\leq 4$).
It is unknown whether the visibility dimension of every nonzero element of
$H^1(K,E)[3]$ is~$2$, and it is not known whether elements of
$\Sha(E)[5]$ must have visibility dimension~$2$.
When~$c$ lies in $\Sha(E)$ we use a classical result of Cassels to
strengthen the conclusion of Proposition~\ref{prop:visdim}.
\begin{proposition}\label{prop:visdimsha}
Let~$E$ be an elliptic curve over a number field~$K$
and let $c\in \Sha(E)$.
Then the visibility dimension of~$c$
is at most the order of~$c$.
\end{proposition}
\begin{proof}
Let $n$ be the order of~$c$.
In view of the restriction of scalars construction in the proof of
Proposition~\ref{prop:allvisible}, it suffices to show that there is
an extension~$L$ of~$K$ of degree~$n$ such that $\res_L(c)=0$.
Without the hypothesis that~$c$ lies in $\Sha(E)$, such an
extension~$L$ might not exist, as Cassels observed in
\cite{cassels:arithmeticV}. However, in that
same paper, Cassels proved that such
an~$L$ exists when $c\in \Sha(E)$
(see also \cite{coneil} for another proof).% (see the last remark in \S2).
\comment{
Let~$X$ be a genus one curve in the torsor class
corresponding to~$c$. The long exact sequence associated to
$$0\ra \Pic^0(X_{\Kbar}) \ra \Pic(X_{\Kbar})
\xrightarrow{ \text{deg} } \Z\ra 0$$
begins
$$0\ra H^0(K,\Pic^0(X_{\Kbar})) \ra H^0(K,\Pic(X_{\Kbar}))
\xrightarrow{ \text{deg} } \Z\xrightarrow{ \delta } H^1(K,E) \ra
\cdots,$$
and $\delta(1)=c$ has order~$n$.
Letting $\Princ(X_{\Kbar})$ denote the principal divisors on $X_{\Kbar}$,
we have an exact sequence
$$0\ra \Princ(X_{\Kbar}) \ra \Div(X_{\Kbar}) \ra \Pic(X_{\Kbar})\ra 0,$$
from which we obtain the exact sequence
$$\Div(X) \ra H^0(K,\Pic(X_{\Kbar})) \ra H^1(K,\Princ(X_{\Kbar})).$$
Since $\Princ(X_{\Kbar})=K(\overline{X})^*/\Kbar^*$,
Hilbert's theorem 90 produces
an injection
$$H^1(K,\Princ(X_{\Kbar}))\hookrightarrow H^2(K,\Kbar^*)=\Br(K),$$
so
$\coker(\Div(X)\ra H^0(K,\Pic(X_{\Kbar})))$ is
isomorphic to the image of $H^0(K,\Pic(X_{\Kbar}))$ in $\Br(K)$.
Because~$X$ has a point everywhere locally, this image is locally
zero; hence, by the local-to-global principle for the Brauer
group, this image is globally zero.
In other words, every $K$-rational divisor class on~$X$
contains a $K$-rational divisor.
We now show that there is a point on~$X$ defined over an extension of degree
at most~$n$. Since $n\in \ker(\delta)$, there exists $D\in \Div(X)$ which maps
to $n \in \Z$ under the degree map.
By the Riemann-Roch theorem, there is an effective divisor linearly
equivalent to~$D$. Since this divisor is effective and of degree~$n$,
each point in the support of~$D$ is defined over an extension~$L$ of~$K$
of degree at most~$n$ (alternatively, the residue field of each
scheme-theoretic
point is of degree at most~$n$). Thus the index of~$c$ is at most~$n$
(recall that $X(L)\neq \emptyset$ if and only if $\res_L(c)=0$).
This completes the proof because the order of~$c$, which is~$n$,
divides the index of~$c$, which is at most~$n$.
}
\end{proof}
\begin{remark}
In contrast to the case of dimension~$1$, it seems to be an open
problem to determine whether or not elements of $\Sha(A)[n]$ split
over an extension of degree~$n$.
\end{remark}
\section{Construction of Visible Elements}
\label{sec:construction}
The goal of this section is to state and prove
the main result of this paper, which we use to
construct visible elements of Shafarevich-Tate groups and
sometimes give a nontrivial lower bound for the order of the
Shafarevich-Tate group of an abelian variety, thus
providing new evidence for the conjecture of Birch and
Swinnerton-Dyer (see Section~\ref{sec:exvissha389E} and
\cite{agashe-stein:shacomp}). The Tamagawa numbers $c_{A,v}$ and $c_{B,v}$
will be defined in Section~\ref{sec:tamagawa} below.
\begin{theorem}\label{thm:shaexists}
Let~$A$ and~$B$ be abelian subvarieties of an abelian
variety~$J$ over a number field~$K$ such that $A\intersect B$ is finite.
Let~$N$ be an integer divisible by the residue characteristics
of primes of bad reduction for~$B$.
Suppose~$n$ is an integer such that for each prime $p\mid n$,
we have $e_p2$ because $1\leq e_p