Hyperelliptic curves over Fq and Gaussian hypergeometric series

Let d >= 2 be an integer. Denote by E-d and E'(d) the hyperelliptic curves over F-q given by E-d : y(2) = x(d) + ax + b and E'(d) : y(2) = x(d) + ax(d-1) + b, respectively. We explicitly find the number of F-q -points on E-d and E'(d) in terms of special values of F-d(d-1) and F-d-1(d-2) Gaussian hypergeometric series with characters of orders d - 1, d, 2(d - 1), 2d, and 2d(d - 1) as parameters. This gives a solution to a problem posed by Ken Ono [17, p. 204] on special values of F-n+ 1(n) Gaussian hypergeometric series for n > 2. We also show that the results of Lennon [14] and the authors [4] on trace of Frobenius of elliptic curves follow from the main results.