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

TL;DR

This paper verifies the p-component of the equivariant Tamagawa number conjecture for abelian varieties with positive rank over non-abelian extensions and explores p-adic congruences between derivatives of twisted Hasse-Weil L-functions.

Contribution

It provides the first explicit verification of the conjecture in complex non-abelian cases and formulates new conjectural p-adic congruences related to derivatives of L-functions.

Findings

Verification of the p-component of the Tamagawa number conjecture for certain abelian varieties.

Formulation of conjectural p-adic congruences between derivatives of twisted L-functions.

Numerical evidence supporting the new conjectural relationships.

Abstract

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.