# 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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