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

Abstract

We study the character sums \[ϕ_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}ϕ\left(x(x^{m}+a)(x^{n}+b)\right),\textrm{ and, } ψ_{(m,n)}(a,b)=\sum_{x\in\mathbb{F}_q}ϕ\left((x^{m}+a)(x^{n}+b)\right)\] where $\varphi$ 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.