Geodesic
Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials
Canadian journal of mathematics, Tome 43 (1991) no. 5, pp. 1055-1064

Voir la notice de l'article provenant de la source Cambridge University Press

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(e 2π 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 F l → F l for all prime numbers in certain congruence classes. 
Tautz, Walter; Top, Jaap; Verberkmoes, Alain. Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials. Canadian journal of mathematics, Tome 43 (1991) no. 5, pp. 1055-1064. doi: 10.4153/CJM-1991-061-x
@article{10_4153_CJM_1991_061_x,
     author = {Tautz, Walter and Top, Jaap and Verberkmoes, Alain},
     title = {Explicit {Hyperelliptic} {Curves} {With} {Real} {Multiplication} and {Permutation} {Polynomials}},
     journal = {Canadian journal of mathematics},
     pages = {1055--1064},
     year = {1991},
     volume = {43},
     number = {5},
     doi = {10.4153/CJM-1991-061-x},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-061-x/}
}
TY  - JOUR
AU  - Tautz, Walter
AU  - Top, Jaap
AU  - Verberkmoes, Alain
TI  - Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials
JO  - Canadian journal of mathematics
PY  - 1991
SP  - 1055
EP  - 1064
VL  - 43
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-061-x/
DO  - 10.4153/CJM-1991-061-x
ID  - 10_4153_CJM_1991_061_x
ER  - 
%0 Journal Article
%A Tautz, Walter
%A Top, Jaap
%A Verberkmoes, Alain
%T Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials
%J Canadian journal of mathematics
%D 1991
%P 1055-1064
%V 43
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-061-x/
%R 10.4153/CJM-1991-061-x
%F 10_4153_CJM_1991_061_x

[1] 1. van Geemen, B. and Werner, J., Nodal Quintics inP4, Math. Inst. R.U. Utrecht, 1989, preprint. Google Scholar

[2] 2. de Jong, J. and Noot, R., Jacobians with complex multiplication, Math. Inst. R.U. Utrecht, 1989, preprint. Google Scholar

[3] 3. Mestre, J.F., Courbes hyperelliptiques à multiplications réelles, C.R. Acad. Sci. Paris, 307 Série 1 (1988), 721–724. Google Scholar

[4] 4. Shimura, G. and Taniyama, Y., Complex multiplication abelian varieties and its applications to number theory, Publ. Math. Soc. Japan 6 (1961). Google Scholar

Cité par Sources :