On the Notion of Visibility of Torsors

Abstract

Abstract. Let J be an abelian variety and A be an abelian subvariety of J, both defined over Q. Let x be an element of H 1 (Q, A). Then there are at least two definitions of x being visible in J: one asks that the torsor corresponding to x be isomorphic over Q to a subvariety of J, and the other asks that x be in the kernel of the natural map H 1 (Q, A) → H 1 (Q, J). In this article, we clarify the relation between the two definitions.