Hypergeometric Functions and Algebraic Curves ye = xd + ax + b

Let q be a prime power and F_q 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-1F_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 F_q in terms of above hypergeometric series. As applications, we obtain some transformations and special values of ₂F₁ hypergeometric series.