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 | References | Similar Articles | Additional Information

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.

**1.**Oskar Bolza,*On binary sextics with linear transformations into themselves*, Amer. J. Math.**10**(1887), no. 1, 47–70. MR**1505464**, 10.2307/2369402**2.**J. W. S. Cassels and E. V. Flynn,*Prolegomena to a middlebrow arithmetic of curves of genus 2*, London Mathematical Society Lecture Note Series, vol. 230, Cambridge University Press, Cambridge, 1996. MR**1406090****3.**G. Cardona, J. González, J. C. Lario, and A. Rio,*On curves of genus 2 with Jacobian of 𝐺𝐿₂-type*, Manuscripta Math.**98**(1999), no. 1, 37–54. MR**1669607**, 10.1007/s002290050123**4.**Jun-ichi Igusa,*Arithmetic variety of moduli for genus two*, Ann. of Math. (2)**72**(1960), 612–649. MR**0114819****5.**Jean-François Mestre,*Construction de courbes de genre 2 à partir de leurs modules*, Effective methods in algebraic geometry (Castiglioncello, 1990) Progr. Math., vol. 94, Birkhäuser Boston, Boston, MA, 1991, pp. 313–334 (French). MR**1106431****6.**Bjorn Poonen,*Computational aspects of curves of genus at least 2*, Algorithmic number theory (Talence, 1996) Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 283–306. MR**1446520**, 10.1007/3-540-61581-4_63**7.**Jean-Pierre Serre,*Galois cohomology*, Springer-Verlag, Berlin, 1997. Translated from the French by Patrick Ion and revised by the author. MR**1466966**

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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:
http://dx.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