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

TY - JOUR

T1 - On a refinement of the Birch and Swinnerton-Dyer Conjecture in positive characteristic

AU - Burns, David

AU - Kakde, Mahesh

AU - Kim, Wansu

PY - 2025

Y1 - 2025

N2 - We formulate a refined version of the Birch and Swinnerton-Dyer conjecture 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.

AB - We formulate a refined version of the Birch and Swinnerton-Dyer conjecture 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.

M3 - Article

SN - 1937-0652

JO - Algebra and Number Theory

JF - Algebra and Number Theory

ER -