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

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(F / k) ]) (h{1}(A-{/F})(1),\mathbb{Z}[{\rm Ga}(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 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.