Let $d\geq2$ be an integer. Denote by $E_d$ and $E'_{ d}$ the hyperelliptic curves over $\mathbb{ F}_q$ given by $$E_d: y^2=x^d+ax+b~~~ \text{ and} ~~~E'_d: y^2=x^d+ax^{ d-1}+b,$$ respectively. We explicitly find the number of $\mathbb{ F}_q$-points on $E_d$ and $E'_d$ in terms of special values of ${ _{ d}}F_{ d-1}$ and ${ _{ d-1}}F_{ d-2}$ 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 \cite[p. 204]{ ono2} on special values of ${ _{ n+1}}F_n$ Gaussian hypergeometric series for $n > 2$. We also show that the results of Lennon \cite{ lennon1} and the authors \cite{ BK3} on trace of Frobenius of elliptic curves follow from the main results.
Categories: math.NT