Hyperelliptic curves with reduced automorphism group A 5

  • David Sevilla
  • Tanush Shaska


We study genus g hyperelliptic curves with reduced automorphism group A 5 and give equations y 2 = f(x) for such curves in both cases where f(x) is a decomposable polynomial in x 2 or x 5. For any fixed genus the locus of such curves is a rational variety. We show that for every point in this locus the field of moduli is a field of definition. Moreover, there exists a rational model y 2 = F(x) or y 2 = x F(x) of the curve over its field of moduli where F(x) can be chosen to be decomposable in x 2 or x 5. While similar equations have been given in (Bujalance et al. in Mm. Soc. Math. Fr. No. 86, 2001) over \({\mathbb R}\) , this is the first time that these equations are given over the field of moduli of the curve.


Modulus Space Automorphism Group Branch Point Isomorphism Class Binary Form 
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  1. 1.
    Brandt R. and Stichtenoth H. (1986). Die Automorphismengruppen hyperelliptischer Kurven. Manuscr. Math. 55(1): 83–92 CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Bujalance E., Cirre F. J., Gamboa J. M., Gromadzki G.: Symmetry types of hyperelliptic Riemann surfaces. Mm. Soc. Math. Fr. No. 86 (2001)Google Scholar
  3. 3.
    Clebsch A. (1872). Theorie der Binären Algebraischen Formen. Verlag von B.G. Teubner, Leipzig zbMATHGoogle Scholar
  4. 4.
    Gutierrez J. (1991). A polynomial decomposition algorithm over factorial domains. Comptes Rendues Mathematiques, de Ac. de Sciences 13: 81–86 MathSciNetzbMATHGoogle Scholar
  5. 5.
    Gutierrez J., Rubio R. and Sevilla D. (2002). On multivariate rational function decomposition. J. Symb. Comput. 33(5): 546–562 CrossRefMathSciNetGoogle Scholar
  6. 6.
    Gutierrez J. and Shaska T. (2005). Hyperelliptic curves with extra involutions. LMS JCM. 8: 102–115 MathSciNetzbMATHGoogle Scholar
  7. 7.
    Gutierrez J., Sevilla D. and Shaska T. (2005). Hyperelliptic curves of genus 3 and their automorphisms. Lect. Notes Comput. 13: 109–123 MathSciNetCrossRefGoogle Scholar
  8. 8.
    Klein F. (1956). Lectures on the icosahedron and the solution of equations of the fifth degree. Dover, New York zbMATHGoogle Scholar
  9. 9.
    Shaska T. (2004). Some special families of hyperelliptic curves. J. Algebra Appl. 3(1): 75–89 CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Shaska, T.: Computational aspects of hyperelliptic curves, Computer mathematics. In: Proceedings of the sixth Asian symposium (ASCM 2003), Beijing, China, April 17–19, 2003. World Scientific. River Edge Lect. Notes Ser. Comput. 10, 248–257 (2003)Google Scholar
  11. 11.
    Völklein, H.: Groups as Galois groups. An introduction. Cambridge Studies in Advanced Mathematics, 53. Cambridge University Press (1996)Google Scholar

Copyright information

© Springer-Verlag 2006

Authors and Affiliations

  1. 1.Department of Computer Science and Software EngineeringConcordia UniversityMontrealCanada
  2. 2.Department of Mathematics and StatisticsOakland UniversityRochesterUSA

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