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Journal of the Ramanujan Mathematical Society

Volume 34, Issue 3, September 2019  pp. 325–342.

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

Authors:  Kewat, Pramod Kumar and Kumar, Ram
Author institution:Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad 826 004, Jharkhand, India

Summary:  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}:ye=xd+ax+b over a finite field Fq in terms of dFd-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 above hypergeometric series. As~applications, we obtain some transformations and special values of 2F{1} hypergeometric series.

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