For non-negative integers \(l_{1}, l_{2},\ldots , l_{n}\), we define character sums \(\varphi _{(l_{1}, l_{2},\ldots , l_{n})}\) and \(\psi _{(l_{1}, l_{2},\ldots , l_{n})}\) over a finite field which are generalizations of Jacobsthal and modified Jacobsthal sums, respectively. We express these character sums in terms of Greene’s finite field hypergeometric series. We then express the number of points on the hyperelliptic curves \(y^2=(x^m+a)(x^m+b)(x^m+c)\) and \(y^2=x(x^m+a)(x^m+b)(x^m+c)\) over a finite field in terms of the character sums \(\varphi _{(l_{1}, l_{2}, l_{3})}\) and \(\psi _{(l_{1}, l_{2}, l_{3})}\), and finally obtain expressions in terms of the finite field hypergeometric series.

对于非负整数\（l_ {1}，l_ {2}，\ ldots，l_ {n} \），我们定义字符和\（\ varphi _ {（l_ {1}，l_ {2}，\ ldots ，l_ {n}）} \）和\（\ psi _ {（l_ {1}，l_ {2}，\ ldots，l_ {n}）} \））在有限字段上，它们是Jacobsthal和修改的Jacobsthal的概括总和。我们用格林的有限域超几何级数来表达这些特征和。然后我们表示超椭圆曲线\（y ^ 2 =（x ^ m + a）（x ^ m + b）（x ^ m + c）\）和\（y ^ 2 = x（x ^ m + a）（x ^ m + b）（x ^ m + c）\）在字符和\（\ varphi _ {（l_ {1}，l_ {2}，l_ { 3}）} \）和\（\ psi _ {（l_ {1}，l_ {2}，l_ {3}）} \\），最后根据有限域超几何级数获得表达式。