Title: Explicit Hyperelliptic Curves With Real Multiplication and Permutation Polynomials
Abstract: Abstract 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(2 π / 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 p th 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.