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Hypergeometric functions and a family of algebraic curves

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Abstract

Let λ∈ℚ∖{0,1} and l≥2, and denote by C l,λ the nonsingular projective algebraic curve over ℚ with affine equation given by

$$y^l=x(x-1)(x-\lambda).$$

In this paper, we define Ω(C l,λ ) analogous to the real periods of elliptic curves and find a relation with ordinary hypergeometric series. We also give a relation between the number of points on C l,λ over a finite field and Gaussian hypergeometric series. Finally, we give an alternate proof of a result of Rouse (Ramanujan J. 12(2):197–205, 2006).

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Authors and Affiliations

  1. Department of Mathematical Sciences, Tezpur University, 784028, Napaam, Sonitpur, Assam, India

    Rupam Barman & Gautam Kalita

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  1. Rupam Barman
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  2. Gautam Kalita
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Correspondence to Rupam Barman.

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Barman, R., Kalita, G. Hypergeometric functions and a family of algebraic curves. Ramanujan J 28, 175–185 (2012). https://doi.org/10.1007/s11139-011-9345-7

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  • DOI: https://doi.org/10.1007/s11139-011-9345-7

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