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

TL;DR

This paper reinterprets the equivariant Tamagawa number conjecture for abelian varieties over number fields as explicit p-adic congruences involving derivatives of twisted Hasse-Weil L-functions, enabling numerical checks and refined conjectures.

Contribution

It provides a new explicit p-adic congruence framework for the eTNC, connecting derivatives of L-functions with regulators, and refines existing conjectures of Mazur and Tate.

Findings

Reinterpretation of eTNC as p-adic congruences

Explicit formulas involving derivatives of twisted L-functions

Predictions that refine Mazur and Tate conjectures

Abstract

Let A be an abelian variety over a number field k and F a finite cyclic extension of k of p-power degree for an odd prime p. Under certain technical hypotheses, we obtain a reinterpretation of the equivariant Tamagawa number conjecture (eTNC) for A, F/k and p as an explicit family of p-adic congru- ences involving values of derivatives of the Hasse-Weil L-functions of twists of A, normalised by completely explicit twisted regulators. This reinterpretation makes the eTNC amenable to numerical verification and furthermore leads to explicit predictions which refine well-known conjectures of Mazur and Tate.