Visible Elements of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$

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 Shafarevich-Tate group of $A$ is

$${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontsh... ...ectfont Sh}}}(A) := \Ker\left(H^1(K,A) \rightarrow \prod_{v} H^1(K_v,A)\right),$$

where the product is over all places of $K$.

Definition 1.2   If $\iota: A\hookrightarrow J$ is an embedding, then the visible subgroup of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(A)$ with respect to $\iota$ is

$$\Vis_J({\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}... ...ncoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(J)).$$