  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.
