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Hyperelliptic Curves with Extra Involutions

Published online by Cambridge University Press:  01 February 2010

J. Gutierrez  and
T. Shaska
J. Gutierrez
Affiliation:
Department of Mathematics, Universidad de Cantabria, E-39071, Santander, Spain, jaime@matesco.unican.es
T. Shaska
Affiliation:
Department of Mathematics, 300 Brink Hall, University of Idaho, USA, tshaska@uidaho.edu

Abstract

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The purpose of this paper is to study hyperelliptic curves with extra involutions. The locus Lg of such genus-g hyperelliptic curves is a g-dimensional subvariety of the moduli space of hyperelliptic curves Hg. The authors present a birational parameterization of Lg via dihedral invariants, and show how these invariants can be used to determine the field of moduli of points p ∈ Lg. They conjecture that for p ∈ Hg with |Aut(p)| > 2, the field of moduli is a field of definition, and they prove this conjecture for any point p ∈ Lg such that the Klein 4-group is embedded in the reduced automorphism group of p. Further, for g = 3, they show that for every moduli point p ∈ H3 such that |Aut(p)| > 4, the field of moduli is a field of definition. A rational model of the curve over its field of moduli is provided.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 8 , 2005 , pp. 102 - 115
Copyright
Copyright © London Mathematical Society 2005

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