A Weil-etale version of the Birch and Swinnerton-Dyer conjecture

We'll explain the Birch and Swinnerton-Dyer Conjecture for abelian varieties over global fields. If the field is of characteristic p, we give a reformulation in terms of Weil-etale cohomology of the Neron-model and show that it holds if the Tate-Shafarevich group is finite.