Characteristic polynomials of the curve v~2 = u~p -au-b over finite fields of characteristic p

摘要

For the hyperelliptic curve v~2 = u~p - au - b over the field F_q with odd characteristic p, Duursma and Sakurai (2000) obtained its characteristic polynomial for q = p, a = 1 and b e F,, and You et al. (2011) determined the characteristic polynomial for q = p2 and all a,b∈F_q. The current paper explicitly determines the characteristic polynomials for q = p~s and all a, b∈F_q for an arbitrary integer s coprime to p. Jacobian group orders over F_qn, for any n ≥ 1, can be computed easily from these characteristic polynomials.