Character sums, Gaussian hypergeometric series, and a family of hyperelliptic curves

We study the character sums phi((m,n))(a,b) = Sigma(x is an element of Fq) phi(x(x(m) + a)(x(n) + b)) and psi((m,n))(a, b) = Sigma(x is an element of Fq) phi(x(x(m) + a)(x(n) + b)), where phi is the quadratic character defined over (Fq). These sums are expressed in terms of Gaussian hypergeometric series over (Fq). Then we use these expressions to exhibit the number of (Fq)-rational points on families of hyperelliptic curves and their Jacobian varieties.