Modèles entiers des courbes hyperelliptiques sur un corps de valuation discrète

Abstract

Let C C be a hyperelliptic curve of genus g ≥ 1 g\ge 1 over a discrete valuation field K K . In this article we study the models of C C over the ring of integers O K \mathcal {O}_{K} of K K . To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of C C with integral coefficients), one can associate a (valuation of) discriminant. Then we give a criterion for a Weierstrass model to have minimal discriminant. We show also that in the most cases, the minimal regular model of C C over O K \mathcal {O}_{K} dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over O K \mathcal {O}_{K} of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.