  Let A be an abelian variety defined over a number field k and let F be a
finite Galois extension of k. Let p be a prime number. Then under certain
not-too-stringent conditions on A and F we compute explicitly the algebraic
part of the p-component of the equivariant Tamagawa number of the pair (
h^1(A/F)(1), Z[Gal(F/k)] ). By comparing the result of this computation with
the theorem of Gross and Zagier we are able to give the first verification of
the p-component of the equivariant Tamagawa number conjecture for an abelian
variety in the technically most demanding case in which the relevant
Mordell-Weil group has strictly positive rank and the relevant field extension
is both non-abelian and of degree divisible by p. More generally, our approach
leads us to the formulation of certain precise families of conjectural p-adic
congruences between the values at s=1 of derivatives of the Hasse-Weil
L-functions associated to twists of A, normalised by a product of explicit
equivariant regulators and periods, and to explicit predictions concerning the
Galois structure of Tate-Shafarevich groups. In several interesting cases we
provide theoretical and numerical evidence in support of these more general
predictions.
