Cohomology of Abelian Varieties Over Function Fields

Abstract

Let k be a field of algebraic functions in one variable over an algebraically closed constant field k., and let A be an abelian variety defined over k. We consider the first cohomology group H1(k, A), i.e., the group of isomorphism classes of principal homogeneous spaces for A over k. For each place p of k, A is also defined over the completion ka of k at 4, and there is a canonical map of H1(k, A) into H1(k., A), and hence a canonical map