Hyperelliptic Curves Over and 120125Q and Gaussian Hypergeometric Series

Abstract

Let da��2 be an integer. Denote by Ed and E'd the hyper-elliptic curves over 𝔽q given by Ed- y2 = xd + ax + b and E'd- y2 = xd + axda��1 + b, respectively. We explicitly find the number of 𝔽q-points on Ed and E'd in terms of special values of d Fd-1 and d-1Fd-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 n+1Fn 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.