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Explicit Galois obstruction and descent for hyperelliptic curves with tamely cyclic reduced automorphism group

Authors: Reynald Lercier, Christophe Ritzenthaler and Jeroen Sijsling
Journal: Math. Comp. 85 (2016), 2011-2045
MSC (2010): Primary 14Q05, 13A50, 14H10, 14H25, 14H37
Published electronically: September 17, 2015
MathSciNet review: 3471117
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This paper is devoted to the study of the Galois descent obstruction for hyperelliptic curves of arbitrary genus whose reduced automorphism groups are cyclic of order coprime to the characteristic of their ground field. We give an explicit and effectively computable description of this obstruction. Along the way, we obtain an arithmetic criterion for the existence of a so-called hyperelliptic descent.

We define homogeneous dihedral invariants for general hyperelliptic curves, and show how the obstruction can be expressed in terms of these invariants. If this obstruction vanishes, then the homogeneous dihedral invariants can also be used to explicitly construct a model over the field of moduli of the curve; if not, then one still obtains a hyperelliptic model over a degree $2$ extension of the field of moduli.

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Additional Information

Reynald Lercier
Affiliation: DGA MI La Roche Marguerite, 35174 Bruz, France; IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France
MR Author ID: 602270
ORCID: 0000-0002-0531-8945

Christophe Ritzenthaler
Affiliation: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France
MR Author ID: 702917

Jeroen Sijsling
Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom
MR Author ID: 974789
ORCID: 0000-0002-0632-9910

Keywords: Hyperelliptic curve, Galois descent, field of definition, field of moduli, invariants, genus 3
Received by editor(s): February 20, 2013
Received by editor(s) in revised form: August 10, 2014, and January 6, 2015
Published electronically: September 17, 2015
Additional Notes: The authors acknowledge support by grant ANR-09-BLAN-0020-01, and by the research programme Investissements d’avenir (ANR-11-LABX-0020-01) of the Centre Henri Lebesgue. The third author was additionally supported by a Marie Curie Fellowship IEF-GA-2011-299887.
Article copyright: © Copyright 2015 American Mathematical Society