Regulator constants and the parity conjecture

TL;DR

This paper proves the p-parity conjecture for certain twists of elliptic curves and abelian varieties, establishing a link between group representations, regulator constants, and local invariants.

Contribution

It introduces new methods connecting permutation representations, regulator constants, and local invariants to prove the p-parity conjecture for a broad class of twists.

Findings

Proves the p-parity conjecture for twists of elliptic curves under specific conditions.

Establishes relations between permutation representations and regulator constants.

Connects local root numbers with Tamagawa numbers through these relations.

Abstract

The p-parity conjecture for twists of elliptic curves relates multiplicities of Artin representations in p-infinity Selmer groups to root numbers. In this paper we prove this conjecture for a class of such twists. For example, if E/Q is semistable at 2 and 3, K/Q is abelian and K^\infty is its maximal pro-p extension, then the p-parity conjecture holds for twists of E by all orthogonal Artin representations of Gal(K^\infty/Q). We also give analogous results when K/Q is non-abelian, the base field is not Q and E is replaced by an abelian variety. The heart of the paper is a study of relations between permutation representations of finite groups, their "regulator constants", and compatibility between local root numbers and local Tamagawa numbers of abelian varieties in such relations.