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

Abstract

AbstractLetAbe an abelian variety defined over a number fieldkand letFbe a finite Galois extension ofk. Letpbe a prime number. Then under certain not-too-stringent conditions onAandFwe compute explicitly the algebraic part of thep-component of the equivariant Tamagawa number of the pair(h^{1}(A_{/F})(1),\mathbb{Z}[{\rm 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 thep-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 byp. More generally, our approach leads us to the formulation of certain precise families of conjecturalp-adic congruences between the values ats=1of derivatives of the Hasse–WeilL-functions associated to twists ofA, 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.