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

Abstract

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.