Abstract
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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brandt R. and Stichtenoth H. (1986). Die Automorphismengruppen hyperelliptischer Kurven. Manuscr. Math. 55(1): 83–92
Bujalance E., Cirre F. J., Gamboa J. M., Gromadzki G.: Symmetry types of hyperelliptic Riemann surfaces. Mm. Soc. Math. Fr. No. 86 (2001)
Clebsch A. (1872). Theorie der Binären Algebraischen Formen. Verlag von B.G. Teubner, Leipzig
Gutierrez J. (1991). A polynomial decomposition algorithm over factorial domains. Comptes Rendues Mathematiques, de Ac. de Sciences 13: 81–86
Gutierrez J., Rubio R. and Sevilla D. (2002). On multivariate rational function decomposition. J. Symb. Comput. 33(5): 546–562
Gutierrez J. and Shaska T. (2005). Hyperelliptic curves with extra involutions. LMS JCM. 8: 102–115
Gutierrez J., Sevilla D. and Shaska T. (2005). Hyperelliptic curves of genus 3 and their automorphisms. Lect. Notes Comput. 13: 109–123
Klein F. (1956). Lectures on the icosahedron and the solution of equations of the fifth degree. Dover, New York
Shaska T. (2004). Some special families of hyperelliptic curves. J. Algebra Appl. 3(1): 75–89
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)
Völklein, H.: Groups as Galois groups. An introduction. Cambridge Studies in Advanced Mathematics, 53. Cambridge University Press (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Sevilla, D., Shaska, T. Hyperelliptic curves with reduced automorphism group A 5 . AAECC 18, 3–20 (2007). https://doi.org/10.1007/s00200-006-0030-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00200-006-0030-9