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

TL;DR

This paper computes the number of points on certain algebraic curves over finite fields using Gaussian hypergeometric series, linking algebraic geometry with special functions and deriving new transformations and values.

Contribution

It introduces explicit formulas for point counts on curves $y^e=x^d+ax+b$ over finite fields using hypergeometric series, expanding the connection between algebraic curves and special functions.

Findings

Derived formulas for point counts in terms of hypergeometric series

Expressed Frobenius trace in terms of hypergeometric series

Obtained new transformations and special values of Gaussian hypergeometric series

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.