Visible Elements of the Shafarevich-tate Group

We study a subgroup of the Shafarevich-Tate group of an abelian variety known as the visible subgroup. We explain the geometric intuition behind this subgroup, prove its finiteness and describe several techniques for exhibiting visible elements. Two important results are proved one what we call the visualization theorem, which asserts that every element of the Shafarevich-Tate group of an abelian variety becomes visible somewhere. Although the theorem is not original, the proof is original and is based on the explicit use of principal homogeneous spaces. The second result is the visibility theorem, stating that under certain conditions, one can inject a weak Mordell-Weil group into the Shafarevich-Tate group. Finally, we present two applications which provide evidence for the Birch and Swinnerton-Dyer conjecture one example, in which the visibility theorem applies directly, and one, where visibility occurs only after raising the level of the modular Jacobian.