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Let A be an abelian variety defined over a number field k and let F be afinite Galois extension of k. Let p be a prime number. Then under certainnot-too-stringent conditions on A and F we compute explicitly the algebraicpart 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 withthe theorem of Gross and Zagier we are able to give the first verification ofthe p-component of the equivariant Tamagawa number conjecture for an abelianvariety in the technically most demanding case in which the relevantMordell-Weil group has strictly positive rank and the relevant field extensionis both non-abelian and of degree divisible by p. More generally, our approachleads us to the formulation of certain precise families of conjectural p-adiccongruences between the values at s=1 of derivatives of the Hasse-WeilL-functions associated to twists of A, normalised by a product of explicitequivariant regulators and periods, and to explicit predictions concerning theGalois structure of Tate-Shafarevich groups. In several interesting cases weprovide theoretical and numerical evidence in support of these more generalpredictions.

On Mordell-Weil groups and congruences between derivatives of twisted Hasse-Weil L-functions