Journal of the Ramanujan Mathematical Society

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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, India
     

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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.
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  • Hypergeometric Functions and Algebraic Curves ye = xd + ax + b

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Authors

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

Abstract


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