Hypergeometric functions and algebraic curves $y^e=x^d+ax+b$ (1604.07613v5)

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 $d\geq2$ and $q\equiv1(\mathrm{mod}~ed(d-1))$, 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 $_dF{d-1}$ 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 $_2F{1}$ Gaussian hypergeometric series.