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

We formulate a refined version of the Birch and Swinnerton-Dyer conjecture<br/>for abelian varieties over global function fields. This refinement incorporates both new families of algebraic relations between leading terms (at s “ 1) of Hasse-Weil-Artin L series and restrictions on the Galois structure of Selmer complexes, and constitutes a natural analogue for abelian varieties over function fields of the equivariant Tamagawa number conjecture for abelian varieties over number fields. We provide strong supporting evidence for the conjecture including giving a full proof, modulo only the assumed finiteness of Tate-Shafarevich groups, in an important class of examples.