Journal of the Ramanujan Mathematical Society

Open Access Open Access  Restricted Access Subscription Access
Open Access Open Access Open Access  Restricted Access Restricted Access Subscription Access

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


Affiliations
1 Department of Applied Mathematics, Indian Institute of Technology (ISM), Dhanbad 826 004, Jharkhand, India
     

   Subscribe/Renew Journal


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.
User
Subscription Login to verify subscription
Notifications
Font Size

  • Rupam Barman and Gautam Kalita, Certain values of Gaussian hypergeometric series and a family of algebraic curves, International Journal of Number Theory, 8(04) (2012) 945–961.

  • Rupam Barman and Gautam Kalita, Hypergeometric functions and a family of algebraic curves, The Ramanujan Journal, 2(28) (2012) 175–185.

  • Rupam Barman and Gautam Kalita, Elliptic curves and special values of Gaussian hypergeometric series, Journal of Number Theory, 133(9) (2013) 3099–3111.

  • Rupam Barman and Gautam Kalita, On the polynomial xd + ax + b over Fq and Gaussian hypergeometric series, International Journal of Number Theory, 09(07) (2013) 1753–1763.

  • Rupam Barman and Gautam Kalita, Hyperelliptic curves over Fq and Gaussian hypergeometric series, Journal of the Ramanujan Mathematical Society, 30(3) (2015) 331–348.

  • Jenny G. Fuselier, Hypergeometric functions over finite fields over Fp and relations to elliptic curves and modular forms, PhD thesis, Texas A&M University (2007).

  • John Greene, Hypergeometric functions over finite fields, Transactions of the American Mathematical Society, 301(1) (1987) 77–101.

  • Kenneth Ireland and Michael Rosen, A classical introduction to modern number theory, volume 84 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2 edition (1990).

  • Gautam Kalita, Values of Gaussian hypergeometric series and their connections to algebraic curves, International Journal of Number Theory, 14(01) (2018) 1–18.

  • Masao Koike, Orthogonal matrices obtained from hypergeometric series over finite fields and elliptic curves over finite fields, Hiroshima Mathematical Journal, 25(1) (1995) 43–52.

  • Serge Lang, Cyclotomic fields I and II, volume 121. Springer Science and Business Media (2012).

  • Catherine Lennon, Gaussian hypergeometric evaluations of traces of Frobenius for elliptic curves, Proceedings of the American Mathematical Society, 139(6) (2011) 1931–1938.

  • Rudolf Lidl and Harald Niederreiter, Finite fields, volume 20. Cambridge university press (1997).

  • Ken Ono, Values of Gaussian hypergeometric series, Transactions of the American Mathematical Society, 350(3) (1998) 1205–1223.


Abstract Views: 224

PDF Views: 0




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

Abstract Views: 224  |  PDF Views: 0

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