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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Curves of genus 2 with group of automorphisms isomorphic to $ D_8$ or $ D_{12}$


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 $ \mathcal M_2$. The locus of curves with group of automorphisms isomorphic to one of the dihedral groups $ D_8$ or $ D_{12}$ is a one-dimensional subvariety.

In this paper we classify these curves over an arbitrary perfect field $ k$ of characteristic $ \operatorname{char} k\neq2$ in the $ D_8$ case and $ \operatorname{char} k\neq2,3$ in the $ D_{12}$ case. We first parameterize the $ \overline k$-isomorphism classes of curves defined over $ k$ by the $ k$-rational points of a quasi-affine one-dimensional subvariety of $ \mathcal M_2$; then, for every curve $ C/k$ representing a point in that variety we compute all of its $ k$-twists, which is equivalent to the computation of the cohomology set $ H^1(G_k,\operatorname{Aut}(C))$.

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 $ \operatorname{GL}_2(\overline k)$. In particular, we give two generic hyperelliptic equations, depending on several parameters of $ k$, that by specialization produce all curves in every $ k$-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: http://dx.doi.org/10.1090/S0002-9947-07-04111-6
PII: S 0002-9947(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