On Quadratic Twists of Hyperelliptic Curves

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.