Introduction

If a genus 0 curve $X$ over  $\mathbf{Q}$ has a point in every local field $\mathbf{Q}_p$ and in $\mathbf{R}$, then it has a global point over  $\mathbf{Q}$. 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 $\mathbf{R}$, but has no $\mathbf{Q}$-point. The Shafarevich-Tate group of an elliptic curve $E$, denoted ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(E)$, 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 ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$ 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  ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$.

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 $H^1(K,A)$ visible in $J$ (with respect to $\iota$) is $\ker(H^1(K,A)\rightarrow H^1(K,J))$. 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 ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$, 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  $\mathbf{Q}$ 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 1 contains the definition of visibility for cohomology classes and elements of Shafarevich-Tate groups. Then in Section 1.3, we use a restriction of scalars construction to prove that every cohomology class is visible in some abelian variety. Next, in Section 2, we investigate the visibility dimension of cohomology classes. Section 3 contains a theorem that can be used to construct visible elements of Shafarevich-Tate groups. The final section, Section 4, contains examples and applications of our visibility results in the context of modular abelian varieties.