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

TL;DR

This paper computes the number of points on certain hyperelliptic curves over finite fields using Gaussian hypergeometric series, solving a problem posed by Ken Ono and connecting to previous work on elliptic curves.

Contribution

It explicitly relates point counts on hyperelliptic curves to special values of Gaussian hypergeometric series, extending known results to higher genus curves.

Findings

Explicit formulas for point counts on hyperelliptic curves in terms of hypergeometric series.

Connection of results to previous work on elliptic curve Frobenius traces.

Solution to a problem posed by Ken Ono regarding 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.