Certain values of Gaussian hypergeometric series and a family of algebraic curves

TL;DR

This paper explores the connection between the number of points on certain algebraic curves over finite fields and Gaussian hypergeometric series, providing new evaluations and extending previous results.

Contribution

It establishes a relation between algebraic curve point counts and hypergeometric series, offers an alternative proof of McCarthy's result, and evaluates specific hypergeometric series values.

Findings

Relation between point counts and hypergeometric series

New evaluations of Gaussian hypergeometric series

Extension of Ono's hypergeometric value result

Abstract

Let $\lambda \in \mathbb{Q}\setminus \{0, -1\}$ and $l \geq 2$. Denote by $C_{l,\lambda}$ the nonsingular projective algebraic curve over $\mathbb{Q}$ with affine equation given by $$y^l=(x-1)(x^2+\lambda).$$ In this paper we give a relation between the number of points on $C_{l, \lambda}$ over a finite field and Gaussian hypergeometric series. We also give an alternate proof of a result of McCarthy (2010). We find some special values of ${_{3}}F_2$ and ${_{2}}F_1$ Gaussian hypergeometric series. Finally we evaluate the value of ${_{3}}F_2(4)$ which extends a result of Ono (1998).