Hypergeometric functions and algebraic curves ye = xd + ax + b

Abstract

Let q be a prime power and IF q be a finite field with q elements. Let e and d be positive integers. In this paper, for d 2: 2 and q = l(mod ed(d - 1)), we calculate the number of points on an algebraic curve Ee,d : ye = xd + ax + b over a finite field IFq in terms of dFd-1 Gaussian hypergeometric series with multiplicative characters of orders d and e(d - 1 ), and in terms of d-1 Fd-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 Ee,d over a finite field IF q in terms of above hypergeometric series. As applications, we obtain some transformations and special values of 2F1 hypergeometric series.