Local data of elliptic curves under quadratic twist

TL;DR

This paper studies how the local data, including Tamagawa numbers, of elliptic curves over local fields change under quadratic twists, introducing strongly-minimal models to facilitate explicit calculations.

Contribution

It introduces the concept of strongly-minimal models for elliptic curves over local fields and provides explicit criteria to determine local data of quadratic twists.

Findings

Explicit conditions for local data of quadratic twists.

Determination of minimal discriminant valuation and conductor exponent.

Analysis focused on residue characteristic 2, especially over bQ_2.

Abstract

Let $K$ be the field of fractions of a complete discrete valuation ring with a perfect residue field. In this article, we investigate how the Tamagawa number of $E/K$ changes under quadratic twist. To accomplish this, we introduce the notion of a strongly-minimal model for an elliptic curve $E/K$, which is a minimal Weierstrass model satisfying certain conditions that lead one to easily infer the local data of $E/K$. Our main results provide explicit conditions on the Weierstrass coefficients of a strongly-minimal model of $E/K$ to determine the local data of a quadratic twist $E^{d}/K$. We note that when the residue field has characteristic $2$, we only consider the special case $K=\mathbb{Q}_{2}$. In this setting, we also determine the minimal discriminant valuation and conductor exponent of $E$ and $E^d$ from further conditions on the coefficients of a strongly-minimal model for $E$.