Multiplicities in Selmer groups and root numbers for Artin twists

TL;DR

This paper investigates the parity of Selmer group multiplicities and root numbers for elliptic curves twisted by Artin representations, establishing equivalences and expressing local invariants in terms of Iwasawa theory.

Contribution

It introduces new relations between Selmer group multiplicities, root numbers, and local Iwasawa invariants for elliptic curves twisted by Artin representations.

Findings

Parity of Selmer multiplicities varies with Galois representations.

Root numbers for twisted elliptic curves are compared and related.

The p-parity conjecture equivalence is established for different elliptic curves.

Abstract

Let $K/F$ be a finite Galois extension of number fields and $\sigma$ be an absolutely irreducible, self-dual representation of $\mathrm{Gal}(K/F)$. Let $p$ be an odd prime and consider two elliptic curves $E_1, E_2$ 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 $\sigma$ in the representation space associated to the $p^\infty$-Selmer group of $E_i$ over $K$. We also compare the root numbers for the twist of $E_i/F$ by $\sigma$ and show that the $p$-parity conjecture holds for the twist of $E_1/F$ by $\sigma$ if and only if it holds for the twist of $E_2/F$ by $\sigma$. We also express Mazur-Rubin-Nekov\'a\v{r}'s arithmetic local constants in terms of certain local Iwasawa invariants.