On refined conjectures of Birch and Swinnerton-Dyer type for Hasse-Weil-Artin L-series

TL;DR

This paper formulates new refined conjectures related to the Birch and Swinnerton-Dyer conjecture for Hasse-Weil-Artin L-series of abelian varieties over number fields, linking them to existing conjectures and providing evidence.

Contribution

It introduces several new conjectures, establishes their relations to known conjectures, and offers theoretical and numerical evidence supporting these conjectures.

Findings

Formulation of new refined BSD-type conjectures

Connections established with the equivariant Tamagawa number conjecture

Evidence provided in special cases supporting the conjectures

Abstract

We consider refined conjectures of Birch and Swinnerton-Dyer type for the Hasse-Weil-Artin L-series of abelian varieties over general number fields. We shall, in particular, formulate several new such conjectures and establish their precise relation to previous conjectures, including to the relevant special case of the equivariant Tamagawa number conjecture. We also derive a wide range of concrete interpretations and explicit consequences of these conjectures that, in general, involve a thoroughgoing mixture of difficult archimedean considerations related to refinements of the conjecture of Deligne and Gross and delicate p-adic congruence relations that involve the bi-extension height pairing of Mazur and Tate and are related to key aspects of non-commutative Iwasawa theory. In important special cases we provide strong evidence, both theoretical and numerical, in support of the conjectures. We also point out an inconsistency in a conjecture of Bradshaw and Stein regarding Zhang's Theorem on Heegner points and suggest a possible correction.