Rational Points over Finite Fields on a Family of Higher Genus Curves and Hypergeometric Functions

Abstract

In this paper we investigate the relation between the number of rational points over a finite field $\mathbb{F}_{p^n}$ on a family of higher genus curves and their periods in terms of hypergeometric functions. For the case $y^\ell = x(x-1)(x-\lambda)$ we find a closed form in terms of hypergeometric functions associated with the periods of the curve. For the general situation $y^\ell = x^{a_1}(x-1)^{a_2}(x-\lambda)^{a_3}$ we show that the number of rational points is a linear combination of hypergeometric series, and we provide an algorithm to determine the coefficients involved.