Generalizations of Jacobsthal sums and hypergeometric series over finite fields

For non-negative integers $l_{1}, l_{2},\ldots, l_{n}$, we define character sums $φ_{(l_{1}, l_{2},\ldots, l_{n})}$ and $ψ_{(l_{1}, l_{2},\ldots, l_{n})}$ 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 $φ_{(l_{1}, l_{2}, l_{3})}$ and $ψ_{(l_{1}, l_{2}, l_{3})}$, and finally obtain expressions in terms of the finite field hypergeometric series.

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