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We consider refined conjectures of Birch and Swinnerton-Dyer type for theHasse-Weil-Artin L-series of abelian varieties over general number fields. Weshall, in particular, formulate several new such conjectures and establishtheir precise relation to previous conjectures, including to the relevantspecial case of the equivariant Tamagawa number conjecture. We also derive awide range of concrete interpretations and explicit consequences of theseconjectures that, in general, involve a thoroughgoing mixture of difficultarchimedean considerations related to refinements of the conjecture of Deligneand Gross and delicate p-adic congruence relations that involve thebi-extension height pairing of Mazur and Tate and are related to key aspects ofnon-commutative Iwasawa theory. In important special cases we provide strongevidence, both theoretical and numerical, in support of the conjectures. Wealso point out an inconsistency in a conjecture of Bradshaw and Stein regardingZhang's Theorem on Heegner points and suggest a possible correction.

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