On Quadratic Twists of Hyperelliptic Curves

TL;DR

This paper provides an explicit description of minimal regular models for quadratic twists of hyperelliptic curves over local fields and demonstrates the existence of many twists that are not locally soluble over Q.

Contribution

It introduces a new explicit method to describe minimal regular models of quadratic twists of hyperelliptic curves over local fields.

Findings

Explicit description of minimal regular models for quadratic twists.

Existence of a positive density family of quadratic twists not locally soluble.

Application to hyperelliptic curves over Q with irreducible defining polynomials.

Abstract

Let C be a hyperelliptic curve of good reduction defined over a discrete valuation field K with algebraically closed residue field k. Assume moreover that char k \ne 2. Given d \in K^*\K^*2, we introduce an explicit description of the minimal regular model of the quadratic twist of C by d. As an application, we show that if C/Q is a nonsingular hyperelliptic curve given by y^2 = f(x) with f an irreducible polynomial, there exists a positive density family of prime quadratic twists of C which are not everywhere locally soluble.