On the refined `Birch--Swinnerton-Dyer type' conjectures of Mazur and Tate

TL;DR

This paper proves significant parts of refined conjectures by Mazur and Tate related to the Birch and Swinnerton-Dyer conjecture, using the equivariant Tamagawa Number conjecture framework, and achieves results even more precise than originally predicted.

Contribution

It advances the understanding of refined BSD conjectures by proving substantial parts using the equivariant Tamagawa Number conjecture approach, with results surpassing previous predictions.

Findings

Proved key parts of Mazur and Tate's refined conjectures

Achieved results more precise than original predictions

Linked refined conjectures to the equivariant Tamagawa Number conjecture

Abstract

We prove a substantial part of conjectures of Mazur and Tate that refine the conjecture of Birch and Swinnerton-Dyer. Our approach, which also leads to some results even finer than the predictions of Mazur and Tate, is via the `rank-zero component' of the relevant case of the equivariant Tamagawa Number conjecture.