Local root numbers of elliptic curves over dyadic fields

TL;DR

This paper investigates the local root numbers of elliptic curves over dyadic fields with additive, potentially good reduction, by analyzing the Galois extension generated by three-torsion points and providing a formula for the local root number.

Contribution

It introduces a new formula to compute the local root number of elliptic curves over dyadic fields considering their three-torsion Galois extensions.

Findings

Derived a formula for local root numbers over dyadic fields

Analyzed Galois extensions generated by three-torsion points

Enhanced understanding of elliptic curve behavior in dyadic fields

Abstract

We consider an elliptic curve over a dyadic field with additive, potentially good reduction. We study the finite Galois extension of the dyadic field generated by the three-torsion points of the elliptic curve. As an application, we give a formula to calculate the local root number of the elliptic curve over the dyadic field.