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

Abstract

Let q be a prime power and Fq be a finite field with q elements. Let e and d be positive integers. In this paper, for d ≥ 2 and q ≡ 1(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 Fq 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 Fq in terms of the above hypergeometric series. As applications, we obtain some transformations and special values of 2 F 1 Gaussian hypergeometric series.