Character Sums, Gaussian Hypergeometric Series, and a Family of Hyperelliptic Curves

TL;DR

This paper expresses certain character sums related to hyperelliptic curves in terms of Gaussian hypergeometric series over finite fields, enabling explicit computation of rational points on these curves and their Jacobians.

Contribution

It provides new formulas linking character sums to hypergeometric series, facilitating point counting on hyperelliptic curves over finite fields.

Findings

Explicit formulas for character sums in terms of hypergeometric series

Determination of the number of rational points on hyperelliptic curves

Insights into the structure of Jacobian varieties of these curves

Abstract

We study the character sums \[\phi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left(x(x^{m}+a)(x^{n}+b)\right),\textrm{ and, } \psi_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}\phi\left((x^{m}+a)(x^{n}+b)\right)\] where $\phi$ is the quadratic character defined over $\mathbb{F}_q$. These sums are expressed in terms of Gaussian hypergeometric series over $\mathbb{F}_q$. Then we use these expressions to exhibit the number of $\mathbb{F}_q$-rational points on families of hyperelliptic curves and their Jacobian varieties.