Visible Subgroups of Shafarevich-Tate Groups

Let $K$ be a number field and $\iota : A_{/K} \hookrightarrow C_{/K}$ be an embedding of an abelian variety into another abelian variety over $K$ .

Definition 3.0.1   The visible subgroup of $\H ^1(K,A)$ relative to $\iota$ is

$${\mathrm{Vis}}_{C} \H ^1(K,A) = {\mathrm{Ker}}\left(\iota_* : \H ^1(K,A)\to \H ^1(K,C)\right).$$

The visible subgroup of ${\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(K, A)$ relative to the embedding $\iota$ is

$${\mathrm{Vis}}_{C} {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(K, A) = {\mbox{{\fontencoding{OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(K, A)\cap {\mathrm{Vis}}_C \H ^1(K,A)$$

$$= {\mathrm{Ker}}\left({\mbox{{\fontencoding{OT2}\fontfamily{wncyr... ...OT2}\fontfamily{wncyr}\fontseries{m}\fontshape{n}\selectfont Sh}}}(K, C)\right)$$

Let $Q$ be the abelian variety $C/\iota(A)$ , which is defined over $K$ . The long exact sequence of Galois cohomology corresponding to the short exact sequence $0\to{} A\to{}C\to{}Q \to{}0$ gives rise to the following exact sequence

$$0 \to A(K) \to C(K) \to Q(K) \to {\mathrm{Vis}}_{C} \H ^1(K,A)\to 0.$$

The last map being surjective means that the cohomology classes of ${\mathrm{Vis}}_C \H ^1(K,A)$ are images of $K$ -rational points on $Q$ , which explains the meaning of the word visible in the definition. The group ${\mathrm{Vis}}_C \H ^1(K,A)$ is finite since it is torsion and since the Mordell-Weil group $Q(K)$ is finitely generated.

Remark 3.0.2   If $A_{/K}$ is an abelian variety and $c \in \H ^1(K,A)$ is any cohomology class, there exists an abelian variety $C_{/K}$ and an embedding $\iota : A \hookrightarrow C$ defined over $K$ , such that $c\in{\mathrm{Vis}}_C \H ^1(K,A)$ , i.e., $c$ is visible in $C$ (see [AS02, Prop. 1.3]). The $C$ of [AS02, Prop. 1.3] is the restriction of scalars of $A_L = A \times_K L$ down to $K$ , where $L$ is any finite extension of $K$ such that $c$ has trivial image in $\H ^1(L,A)$ .