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Explicit computations of Serre’s obstruction for genus-3 curves and application to optimal curves

Part of: Arithmetic algebraic geometry Curves

Published online by Cambridge University Press:  01 May 2010

Christophe Ritzenthaler
Christophe Ritzenthaler*
Affiliation:
Institut de mathématiques de Luminy, UMR 6206, 163 Avenue de Luminy Case 90713288 Marseille, France (email: ritzenth@iml.univ-mrs.fr)

Abstract

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Let k be a field of characteristic other than 2. There can be an obstruction to a principally polarized abelian threefold (A,a) over k, which is a Jacobian over , being a Jacobian over k; this can be computed in terms of the rationality of the square root of the value of a certain Siegel modular form. We show how to do this explicitly for principally polarized abelian threefolds which are the third power of an elliptic curve with complex multiplication. We use our numerical results to prove or refute the existence of some optimal curves of genus 3.

MSC classification

Secondary: 11G20: Curves over finite and local fields 14H45: Special curves and curves of low genus 14H40: Jacobians, Prym varieties
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 13 , January 2010 , pp. 192 - 207
Copyright
Copyright © London Mathematical Society 2010

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