Generalizations of Jacobsthal sums and hypergeometric series over finite fields

TL;DR

This paper introduces generalized Jacobsthal sums over finite fields, expresses them via hypergeometric series, and applies these to count points on specific hyperelliptic curves, linking character sums to hypergeometric functions.

Contribution

It defines new generalized character sums over finite fields and connects them to hypergeometric series and point counts on hyperelliptic curves, extending previous work on Jacobsthal sums.

Findings

Expressed character sums in terms of Greene's hypergeometric series

Derived formulas for point counts on hyperelliptic curves using these sums

Established new links between character sums and hypergeometric functions

Abstract

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.