On Quadratic Twists of Hyperelliptic Curves - Mathematics > Number Theory

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Abstract: Let C be a hyperelliptic curve of good reduction defined over a discretevaluation field K with algebraically closed residue field k. Assume moreoverthat char k e 2. Given d \in K^*\K^*2, we introduce an explicit descriptionof the minimal regular model of the quadratic twist of C by d. As anapplication, we show that if C-Q is a nonsingular hyperelliptic curve given byy^2 = fx with f an irreducible polynomial, there exists a positive densityfamily of prime quadratic twists of C which are not everywhere locally soluble.