Hypergeometric functions and algebraic curves $y^e=x^d+ax+b$

Abstract

Let $q$ be a prime power and ${\mathbb{F}}_q$ be a finite field with $q$ elements. Let $e$ and $d$ be positive integers. In this paper, for $d\ge 2$ and $q\equiv 1(\mathrm{mod}ed(d-1))$, we calculate the number of points on an algebraic curve $E_{e,d}:y^e=x^d+ax+b$ over a finite field ${\mathbb{F}}_q$ in terms of ${}_d{F}_{d-1}$ Gaussian hypergeometric series with multiplicative characters of orders $d$ and $e(d-1)$, and in terms of ${}_{d-1}{F}_{d-2}$ Gaussian hypergeometric series with multiplicative characters of orders $ed(d-1)$ and $e(d-1)$. This helps us to express the trace of Frobenius endomorphism of an algebraic curve $E_{e,d}$ over a finite field ${\mathbb{F}}_q$ in terms of the above hypergeometric series. As applications, we obtain some transformations and special values of ${}_2{F}_1$ Gaussian hypergeometric series.