On a refinement of the Birch and Swinnerton-Dyer Conjecture in positive characteristic

Abstract

We formulate a refined version of the Birch and Swinnerton-Dyer conjecture for abelian varieties over global function fields. This refinement incorporates both families of congruences between the leading terms of Artin-Hasse-Weil $L$-series and also strong restrictions on the Galois structure of natural Selmer complexes and constitutes a precise analogue for abelian varieties over function fields of the equivariant Tamagawa number conjecture for abelian varieties over number fields. We then provide strong supporting evidence for this conjecture including giving a full proof, modulo only the assumed finiteness of Tate-Shafarevich groups, in an important class of examples.