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Kumar, Ram
- Hypergeometric Functions and Algebraic Curves ye = xd + ax + b
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1 Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad 826 004, Jharkhand, IN
1 Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad 826 004, Jharkhand, IN
Source
Journal of the Ramanujan Mathematical Society, Vol 34, No 3 (2019), Pagination: 325-342Abstract
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 Ee,d : ye = xd + ax + b over a finite field Fq in terms of d Fd-1 Gaussian hypergeometric series with multiplicative characters of orders d and e(d − 1), and in terms of d-1Fd-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 Fq in terms of above hypergeometric series. As applications, we obtain some transformations and special values of 2F1 hypergeometric series.References
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