Visible Elements of $H^1(K,A)$

In [Maz99], Mazur introduced the following definition. Let $A$ be an abelian variety over an arbitrary field $K$.

Definition 1.1   Let $\iota: A\hookrightarrow J$ be an embedding of $A$ into an abelian variety $J$ over $K$. Then the visible subgroup of $H^1(K,A)$ with respect to the embedding $\iota$ is

$$\Vis_J(H^1(K,A)) = \Ker(H^1(K,A)\rightarrow {}H^1(K,J)).$$

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 [LT58]). To a cohomology class $c\in H^1(K,A)$, there is a corresponding variety $X$ over $K$ and a map $A\times X \rightarrow X$ 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 $\iota: A\rightarrow J$ is an embedding and $c\in \Vis_J(H^1(K,A))$. We have an exact sequence of abelian varieties $0\rightarrow A\rightarrow J\rightarrow C\rightarrow 0$, where $C=J/A$. A piece of the associated long exact sequence of Galois cohomology is

$$0 \rightarrow A(K) \rightarrow J(K)\rightarrow C(K) \rightarrow H^1(K,A) \rightarrow H^1(K,J) \rightarrow \cdots,$$

so there is an exact sequence

$$0 \rightarrow J(K)/A(K) \rightarrow C(K) \rightarrow \Vis_J(H^1(K,A)) \rightarrow 0.$$ (1.1)

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 $C(K)$.