Multiplicities in Selmer groups and root numbers of Artin twists

Abstract

Let K/F be a finite Galois extension of number fields and let σ be an absolutely irreducible, self-dual, complex valued representation of Gal(K/F). Let p be an odd prime and consider two elliptic curves E1,E2 defined over Q with good, ordinary reduction at primes above p and equivalent mod-p Galois representations. In this article, we study the variation of the parity of the multiplicities of σ in the representation space associated to the p∞-Selmer groups of E1 and E2 over K. We also compare the root numbers for the twists of E1 and E2 over F by σ and show that the p-parity conjecture holds for the twist of E1/F by σ if and only if it holds for the twist of E2/F by σ. We also express Mazur-Rubin-Nekovář's arithmetic local constants in terms of certain local Iwasawa invariants.