Abelian varieties for function fields of curves over local fields

Abstract

Let $K$ be the function field of a smooth projective curve $X$ over a $p$-adic field or over $\mathbb{C}((t))$. We define Tate-Shafarevich groups of a commutative group scheme via cohomology classes locally trivial at each completion of $K$ coming from a closed point of $X$. We prove arithmetic duality theorems for Tate-Shafarevich groups of abelian varieties over $K$.