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Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials

Published online by Cambridge University Press:  20 November 2018

Walter Tautz ,
Jaap Top  and
Alain Verberkmoes
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
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Abstract

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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(e2π i/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 Fℓ → Fℓ for all prime numbers in certain congruence classes.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 43 , Issue 5 , 01 October 1991 , pp. 1055 - 1064
Copyright
Copyright © Canadian Mathematical Society 1991

References

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