Root numbers and parity of local Iwasawa invariants

TL;DR

This paper investigates the relationship between root numbers and the parity of local Iwasawa invariants for elliptic curves over number fields, focusing on their $p$-Selmer ranks and Galois representations.

Contribution

It establishes a comparison framework for $p$-Selmer ranks and root numbers of elliptic curves with equivalent mod $p$ Galois representations over number fields.

Findings

Comparison of $p$-Selmer ranks for elliptic curves with similar Galois representations

Analysis of global and local root number relations

Insights into parity of local Iwasawa invariants

Abstract

Given two elliptic curves $E_1$ and $E_2$ defined over the field of rational numbers, $\mathbb{Q}$, with good reduction at an odd prime $p$ and equivalent mod $p$ Galois representation, we compare the $p$-Selmer rank, global and local root numbers of $E_1$ and $E_2$ over number fields.