Curves of genus 2 with group of automorphisms isomorphic to or

Authors:
Gabriel Cardona and Jordi Quer

Journal:
Trans. Amer. Math. Soc. **359** (2007), 2831-2849

MSC (2000):
Primary 11G30, 14G27

Published electronically:
January 4, 2007

MathSciNet review:
2286059

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Abstract: The classification of curves of genus 2 over an algebraically closed field was studied by Clebsch and Bolza using invariants of binary sextic forms, and completed by Igusa with the computation of the corresponding three-dimensional moduli variety . The locus of curves with group of automorphisms isomorphic to one of the dihedral groups or is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field of characteristic in the case and in the case. We first parameterize the -isomorphism classes of curves defined over by the -rational points of a quasi-affine one-dimensional subvariety of ; then, for every curve representing a point in that variety we compute all of its -twists, which is equivalent to the computation of the cohomology set .

The classification is always performed by explicitly describing the objects involved: the curves are given by hyperelliptic models and their groups of automorphisms represented as subgroups of . In particular, we give two generic hyperelliptic equations, depending on several parameters of , that by specialization produce all curves in every -isomorphism class.

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Additional Information

**Gabriel Cardona**

Affiliation:
Departament Ciències Matemàtiques i Inf., Universitat de les Illes Balears, Ed. Anselm Turmeda, Campus UIB, Carretera Valldemossa, km. 7.5, E-07122 – Palma de Mallorca, Spain

Email:
gabriel.cardona@uib.es

**Jordi Quer**

Affiliation:
Departament Matemàtica Aplicada II, Universitat Politècnica de Catalunya, Ed. Omega, Campus Nord, Jordi Girona, 1-3, E-08034 – Barcelona, Spain

Email:
jordi.quer@upc.edu

DOI:
https://doi.org/10.1090/S0002-9947-07-04111-6

Keywords:
Curves of genus $2$,
twists of curves

Received by editor(s):
November 24, 2003

Received by editor(s) in revised form:
June 7, 2005

Published electronically:
January 4, 2007

Additional Notes:
The authors were supported by Grants BFM-2003-06768-C02-01 and SGR2005-00443

Article copyright:
© Copyright 2007
American Mathematical Society