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

We study the character sums $${\varphi }_{(m,n)}(a,b)=\sum _{x\in {\mathbb{F}}_q}\varphi (x(x^m+a)(x^n+b)),\text{ and, }{\psi }_{(m,n)}(a,b)=\sum _{x\in {\mathbb{F}}_q}\varphi ((x^m+a)(x^n+b))$$ 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.