Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-24T00:11:51.050Z Has data issue: false hasContentIssue false

Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials

Published online by Cambridge University Press:  20 November 2018

Walter Tautz
Affiliation:
Department of Mathematics and Statistics, Queen's University, Kingston, Ontario, K7L 3N6
Jaap Top
Affiliation:
Erasmus Univ. Rotterdam, Vakgroep Wiskunde, Postbus 1738, 3000 DR Rotterdam, The Netherlands
Alain Verberkmoes
Affiliation:
Mathematisch Instituut, Rijksuniversiteit Utrecht, P.O. Box 80.010, 3508 TA Utrecht, The Netherlands
Rights & Permissions [Opens in a new window]

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 aim of this paper is to present a very explicit construction of one parameter families of hyperelliptic curves C of genus (p−1 )/ 2, for any odd prime number p, with the property that the endomorphism algebra of the jacobian of C contains the real subfield Q(2 cos(2π/p)) of the cyclotomic field Q(ei/p).

Two proofs of the fact that the constructed curves have this property will be given. One is by providing a double cover with the pth roots of unity in its automorphism group. The other is by explicitly writing down equations of a correspondence in C x C which defines multiplication by 2cos(2π/ p) on the jacobian of C. As a byproduct we obtain polynomials which define bijective maps FF for all prime numbers in certain congruence classes.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 1991

References

1. van Geemen, B. and Werner, J., Nodal Quintics inP4, Math. Inst. R.U. Utrecht, 1989, preprint.Google Scholar
2. de Jong, J. and Noot, R., Jacobians with complex multiplication, Math. Inst. R.U. Utrecht, 1989, preprint.Google Scholar
3. Mestre, J.F., Courbes hyperelliptiques à multiplications réelles, C.R. Acad. Sci. Paris, 307 Série 1 (1988), 721724.Google Scholar
4. Shimura, G. and Taniyama, Y., Complex multiplication abelian varieties and its applications to number theory, Publ. Math. Soc. Japan 6 (1961).Google Scholar