%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % visibility_of_sha.tex % % % % Visibility of Shafarevich-Tate groups of Abelian Varieties % % % % AUTHORS: William A. Stein and Amod Agashe. % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\documentclass[jnth, draft]{apjrnl} \documentclass[jnth]{apjrnl} \usepackage{amsmath} \usepackage{amssymb} \newcommand{\done}{{\bf [[DONE]]}} \newcommand{\notdone}{{\bf [[NOT DONE!!!]]}} \newcommand{\sM}{\boldsymbol{\mathcal{M}}} \newcommand{\sS}{\boldsymbol{\mathcal{S}}} \newcommand{\sB}{\boldsymbol{\mathcal{B}}} \newcommand{\Adual}{A^{\vee}} \newcommand{\Bdual}{B^{\vee}} \newcommand{\defn}[1]{{\em #1}} \newcommand{\an}{{\rm an}} \newcommand{\e}{\mathbf{e}} \DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts \newcommand{\textcyr}[1]{% {\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}% \selectfont #1}} \newcommand{\Sha}{{\mbox{\textcyr{Sh}}}} \newcommand{\Shaan}{\Sha_{\mbox{\tiny \rm an}}} \newcommand{\Gam}{\Gamma} \newcommand{\cA}{\mathcal{A}} \newcommand{\cAdual}{\mathcal{A}^{\vee}} \newcommand{\cJ}{\mathcal{J}} \newcommand{\h}{\mathfrak{h}} \newcommand{\U}{\mathcal{U}} \newcommand{\Kbar}{\overline{K}} \newcommand{\q}{\mathbf{q}} \newcommand{\union}{\cup} \newcommand{\intersect}{\cap} \newcommand{\cross}{\times} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcommand{\alp}{\alpha} \newcommand{\tensor}{\otimes} \newcommand{\vphi}{\varphi} \newcommand{\comment}[1]{} \newcommand{\Q}{\mathbf{Q}} \newcommand{\R}{\mathbf{R}} \newcommand{\D}{{\mathbb D}} \newcommand{\cC}{\mathcal{C}} \newcommand{\K}{{\mathbb K}} \newcommand{\C}{\mathbf{C}} \newcommand{\Qbar}{\overline{\Q}} \newcommand{\T}{\mathbf{T}} \newcommand{\Z}{\mathbf{Z}} \newcommand{\F}{\mathbf{F}} \newcommand{\Fbar}{\overline{\F}} \newcommand{\Fp}{\F_p} \newcommand{\Ad}{Ad} \newcommand{\A}{\mathcal{A}} \newcommand{\isom}{\cong} \newcommand{\ncisom}{\approx} \newcommand{\cB}{\mathcal{B}} \newcommand{\p}{\mathfrak{p}} \newcommand{\m}{\mathfrak{m}} \newcommand{\magma}{{\sc Magma}} \renewcommand{\Re}{\mbox{\rm Re}} \renewcommand{\L}{\mathcal{L}} \renewcommand{\l}{\ell} \renewcommand{\t}{\tau} \renewcommand{\P}{\mathbf{P}} \renewcommand{\O}{\mathcal{O}} \renewcommand{\a}{\mathfrak{a}} \DeclareMathOperator{\Ann}{Ann} \DeclareMathOperator{\ch}{char} \DeclareMathOperator{\coker}{coker} \DeclareMathOperator{\Br}{Br} \DeclareMathOperator{\Gal}{Gal} \DeclareMathOperator{\Pic}{Pic} \DeclareMathOperator{\Vis}{Vis} \DeclareMathOperator{\Res}{Res} \DeclareMathOperator{\id}{id} \DeclareMathOperator{\ind}{ind} \DeclareMathOperator{\new}{new} \DeclareMathOperator{\op}{op} \DeclareMathOperator{\tor}{tor} \DeclareMathOperator{\SL}{SL} \DeclareMathOperator{\Spec}{Spec} \DeclareMathOperator{\Div}{Div} \DeclareMathOperator{\res}{res} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\GL}{GL} \DeclareMathOperator{\ur}{ur} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator{\Hom}{Hom} %%%% Theoremstyles \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{claim}[theorem]{Claim} \newtheorem{lemma}[theorem]{Lemma} \newtheorem{conjecture}[theorem]{Conjecture} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{algorithm}[theorem]{Algorithm} \newtheorem{question}[theorem]{Question} \theoremstyle{remark} \newtheorem{goal}[theorem]{Goal} \newtheorem{remark}[theorem]{Remark} \newtheorem{remarks}[theorem]{Remarks} \newtheorem{example}[theorem]{Example} \newtheorem{exercise}[theorem]{Exercise} \numberwithin{equation}{section} \usepackage[all]{xy} \DeclareMathOperator{\Princ}{Princ} \font\cyrbig=wncyr10 scaled \magstep 1 \newcommand{\BigSha}{\mbox{\cyrbig X}} \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