Variétés Abéliennes sur les corps de fonctions de courbes sur des corps locaux

Let K be the function field of a smooth projective curve X over a p-adic field or over 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.