Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-14T05:43:29.405Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

References

1.Baily, W., ‘On the automorphism group of a generic curve of genus > 2’, J. Math. Kyoto Univ. 1 (1961–1962) 101–108; correction, 325.Google Scholar
2.Baily, W. L., ‘On the theory of theta functions, the moduli of abelian varieties and the moduli of curves’, Ann. of Math. 75 (1967) 342–381.CrossRefGoogle Scholar
3.Brandt, R. and Stichtenoth, H., ‘Die Automorphismengruppen hyperelliptischer Kurven’, Manuscripta Math 55 (1986) 83–92.CrossRefGoogle Scholar
4.Bujulance, E., Gamboa, J. and Gromadzki, G., ‘The full automorphism groups of hyperelliptic Riemann surfaces’, Manuscripta Math. 79 (1993) 267–282.CrossRefGoogle Scholar
5.Cardona, G. and Quer, J., ‘Field of moduli and field of definition for curves of genus 2’, Computational aspects of algebraic curves, Lecture Notes Ser. Comput. (World Sci. Publishing, to appear).Google Scholar
6.Débes, P. and Douai, J.-C., ‘Algebraic covers: Field of moduli versus field of definition’, Ann. Sci. Ecóle Norm. Sup. (4) 30 (1997) 303–338.Google Scholar
7.Débes, P. and Douai, J.-C., ‘Local-global principle for algebraic covers’, Israel. Math. 103 (1998) 237–257.CrossRefGoogle Scholar
8.Deligne, P. and Mumford, D., ‘The irreducibility of the space of curves of given genus’, Publ. Math. Hautes Études Sci. 36 (1969) 75–109.Google Scholar
9.Earle, C. J., ‘On the moduli of closed Riemann surfaces with symmetries’, Advances in the theory of Riemann surfaces, Ann. of Math. Stud. 66 (ed. Ahlfors, L. V., et al. , Princeton, 1971) 119–130.Google Scholar
10.Gathen, J. Von Zur, ‘Functional decomposition of polynomials: the tame case’, J. Symbolic Comput. 9 (1990) 281–299.CrossRefGoogle Scholar
11.Gutierrez, J., ‘A polynomial decomposition algorithm over factorial domains’, C. R. Acad. Sci. Paris 13 (1991) 81–86.Google Scholar
12.Gutierrez, J., Sevilla, D. and Shaska, T., ‘Hyperelliptic curves of genus 3 with prescribed automorphism group’, Computational aspects of algebraic curves, Lecture Notes Ser. Comput. (World Sci. Publishing, to appear).Google Scholar
13.Igusa, J., ‘Arithmetic variety of moduli for genus 2’, Ann. of Math. (2) 72 (1960) 612–649.CrossRefGoogle Scholar
14.Krishnamorthy, V., Shaska, T. and Völklein, H., ‘Invariants of binary forms’, Dev. Math. 12 (2005) 101–122.Google Scholar
15.Magaard, K., Shaska, T., Shpectorov, S. and Völklein, H., ‘The locus of curvesith prescribed automorphism groups’, Communications in arithmetic fundamental groups (Kyoto, 1999/2001), Sūrikaisekikenkyūsho Kōkyūroku 1267 (Kyoto Univ., Kyoto, 2002) 112–141.Google Scholar
16.Mestre, J.-F., ‘Construction de courbes de genre 2 á partir de leurs modules’, Effective methods in algebraic geometry, Progr. Math. 94 (ed. Mora, T. and Traverso, C., Basel, Birkhaüser, 1991)313–334.CrossRefGoogle Scholar
17.Shaska, T., ‘Genus 2 curves with (3,3)-split Jacobian and large automorphism group’, Algorithmic number theory (Sydney, 2002), Lecture Notes in Comput. Sci. 2369 (ed. Fieker, C., et al. , Springer, Berlin, 2002) 205–218.CrossRefGoogle Scholar
18.Shaska, T., ‘Computational aspects of hyperelliptic curves’, Computer mathematics, Proc. 6th Asian symposium (ASCM 2003), Beijing, China, April 17–19, 2003, Lect. Notes Ser. Comput. 10 (ed. Li, Z, et al. , World Scientific, River Edge, NJ, 2003) 248–257.CrossRefGoogle Scholar
19.Shaska, T., ‘Determining the automorphism group of hyperelliptic curves’, Proc. 2003 Intl Symposium on Symbolic and Algebraic Computation (ACM Press, 2003) 248–254.Google Scholar
20.Shaska, T., ‘Some special families of hyperelliptic curves’, J. Algebra Appl. 3 (2004) 75–89.CrossRefGoogle Scholar
21.Shaska, T. and Völklein, H., ‘Elliptic subfields and automorphisms of genus 2 function fields’, Algebra, arithmetic and geometry with applications (West Lafayette, IN, 2000) (Springer, Berlin, 2004) 687–707.CrossRefGoogle Scholar
22.Shimura, G., ‘On the field of rationality for an abelian variety’, Nagoya Math. J. 45 (1972)167–178.CrossRefGoogle Scholar
23.Weil, A., ‘The field of definition of a variety’, Amer. J. Math. 78 (1956) 509–524.CrossRefGoogle Scholar
24.Wolfart, J., ‘The "obvious" part of Belyi's theorem and Riemann surfaces with many automorphisms’, Geometric Galois action, London Math. Soc. Lecture Note Ser. (Cambridge Univ. Press, 1997).Google Scholar