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

TL;DR

This paper proposes a refined version of the Birch and Swinnerton-Dyer conjecture tailored for abelian varieties over global function fields, linking L-series, Galois structures, and Selmer complexes.

Contribution

It introduces a new refined conjecture that extends the BSD conjecture to positive characteristic, incorporating congruences and Galois restrictions, with supporting evidence and proofs.

Findings

Provides a full proof of the refined conjecture in certain cases, assuming Tate-Shafarevich group finiteness.

Establishes a precise analogue of the equivariant Tamagawa number conjecture for function fields.

Offers strong evidence supporting the validity of the refined conjecture.

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.