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

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.