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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Modèles entiers des courbes hyperelliptiques
sur un corps de valuation discrète


Author: Qing Liu
Journal: Trans. Amer. Math. Soc. 348 (1996), 4577-4610
MSC (1991): Primary 11G20, 14H25; Secondary 14G20
MathSciNet review: 1363944
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Abstract: Let $C$ be a hyperelliptic curve of genus $g\ge 1$ over a discrete valuation field $K$. In this article we study the models of $C$ over the ring of integers $ \mathcal {O}_{K}$ of $K$. To each Weierstrass model (that is a projective model arising from a hyperelliptic equation of $C$ 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 $C$ over $ \mathcal {O}_{K}$ dominates every minimal Weierstrass model. Some classical facts concerning Weierstrass models over $ \mathcal {O}_{K}$ of elliptic curves are generalized to hyperelliptic curves, and some others are proved in this new setting.


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Additional Information

Qing Liu
Affiliation: CNRS, Laboratoire de Mathématiques Pures, Université Bordeaux I, 351, Cours de la Libération, 33405 Talence Cedex, France
Email: liu@math.u-bordeaux.fr

DOI: http://dx.doi.org/10.1090/S0002-9947-96-01684-4
PII: S 0002-9947(96)01684-4
Keywords: Courbe hyperelliptique, modèle de Weierstrass, discriminant
Received by editor(s): August 22, 1995
Article copyright: © Copyright 1996 American Mathematical Society