Characteristic polynomials of the curve v2=up−au−b over finite fields of characteristic p

For the hyperelliptic curve v2=up−au−b over the field Fq with odd characteristic p, Duursma and Sakurai (2000) obtained its characteristic polynomial for q=p, a=1 and b∈Fq, and You et al. (2011) determined the characteristic polynomial for q=p2 and all a,b∈Fq. The current paper explicitly determines the characteristic polynomials for q=ps and all a,b∈Fq for an arbitrary integer s coprime to p. Jacobian group orders over Fqn, for any n⩾1, can be computed easily from these characteristic polynomials.