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

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- by Reynald Lercier, Christophe Ritzenthaler and Jeroen Sijsling PDF
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**85**(2016), 2011-2045 Request permission

## Abstract:

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
- Email: reynald.lercier@m4x.org
**Christophe Ritzenthaler**- Affiliation: IRMAR, Université de Rennes 1, Campus de Beaulieu, 35042 Rennes, France
- MR Author ID: 702917
- Email: ritzenth@iml.univ-mrs.fr
**Jeroen Sijsling**- Affiliation: Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom
- MR Author ID: 974789
- ORCID: 0000-0002-0632-9910
- Email: sijsling@gmail.com
- 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. - © Copyright 2015 American Mathematical Society
- Journal: Math. Comp.
**85**(2016), 2011-2045 - MSC (2010): Primary 14Q05, 13A50, 14H10, 14H25, 14H37
- DOI: https://doi.org/10.1090/mcom3032
- MathSciNet review: 3471117