Congruences for critical values of higher derivatives of twisted Hasse-Weil L-functions, III

TL;DR

This paper interprets the $p$-component of the equivariant Tamagawa number conjecture for abelian varieties over number fields using integral congruences related to the Mazur-Tate height pairing, and provides numerical verifications.

Contribution

It offers a new interpretation of the conjecture in terms of integral congruences and presents the first numerical verifications in complex Galois module cases.

Findings

Established a link between the conjecture and integral congruences involving the height pairing.

Performed numerical computations confirming the conjecture in non-projective Galois module scenarios.

First verifications of the conjecture beyond projective Galois module cases.

Abstract

Let $A$ be an abelian variety defined over a number field $k$, let $p$ be an odd prime number and let $F/k$ be a cyclic extension of $p$-power degree. Under not-too-stringent hypotheses we give an interpretation of the $p$-component of the relevant case of the equivariant Tamagawa number conjecture in terms of integral congruence relations involving the evaluation on appropriate points of $A$ of the ${\rm Gal}(F/k)$-valued height pairing of Mazur and Tate. We then discuss the numerical computation of this pairing, and in particular obtain the first numerical verifications of this conjecture in situations in which the $p$-completion of the Mordell-Weil group of $A$ over $F$ is not a projective Galois module.