Hyperelliptic curves over ${\mathbb{F}}_q$ and Gaussian hypergeometric series

Abstract

Let $d\ge 2$ be an integer. Denote by $E_d$ and $E_d^{\prime }$ the hyperelliptic curves over ${\mathbb{F}}_q$ given by $$E_d:y^2=x^d+ax+b\text{and}{E}_d^{\prime }:y^2=x^d+a{x}^{d-1}+b,$$ respectively. We explicitly find the number of ${\mathbb{F}}_q$-points on $E_d$ and $E_d^{\prime }$ 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.