Hypergeometric functions and algebraic curves ye = xd + ax plus 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 equivalent to 1(mod ed(d - 1)), we calculate the number of points on an algebraic curve E-e,E-d : y(e) = x(d) + ax + b over a finite field F-q in terms of F-d(d-1) Gaussian hypergeometric series with multiplicative characters of orders d and e(d - 1), and in terms of F-d-1(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,E-d over a finite field F-q in terms of above hypergeometric series. As applications, we obtain some transformations and special values of F-2(1) hypergeometric series.