Hypergeometric functions and a family of algebraic curves
type: Item
instance of: scholarly article
title: Hypergeometric functions and a family of algebraic curves
zbMATH Open document ID: 1264.11054
author: Rupam Barman
author: Gautam Kalita
published in: The Ramanujan Journal
publication date: 2012-10-01T00:00:00Z
full work available at URL: https://arxiv.org/abs/1208.0492
review text: For \(\lambda\in\mathbb Q\backslash\{0,1\}\) and \(l\geq 2\), let \(C_{l,\lambda}\) be the nonsingular projective curve over \(\mathbb Q\) with affine equation \[ y^l=x(x-1)(x-\lambda). \] The paper in review defines a quantity \(\Omega(C_{l,\lambda})\) that generalizes the real periods of elliptic curves (the case for elliptic curves is \(l=2\)), and gives relations between \(\Omega(C_{l,\lambda})\), the number of points on \(C_{l,\lambda}\), and the Gaussian hypergeometric series. These relations generalize those for the elliptic curves in [\textit{K. Ono}, Trans. Am. Math. Soc. 350, No. 3, 1205--1223 (1998; Zbl 0910.11054)]. Finally, the paper gives an alternative proof of a result of \textit{J. Rouse} [Ramanujan J. 12, No. 2, 197--205 (2006; Zbl 1193.11057)].
Mathematics Subject Classification ID: 11G20
Mathematics Subject Classification ID: 33C20
zbMATH DE Number: 6088841
zbMATH Keywords: algebraic curves
zbMATH Keywords: hypergeometric series
reviewed by: Kit Ho Mak
MaRDI profile type: Publication
OpenAlex ID: W2030387385
arXiv ID: 1208.0492
cites work: Q4222737
cites work: Q4760447
cites work: Hypergeometric Functions Over Finite Fields
cites work: Hypergeometric functions over finite fields and representations of \(\mathrm{SL}(2,q)\)
cites work: A character sum evaluation and Gaussian hypergeometric series
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cites work: 3F2 HYPERGEOMETRIC SERIES AND PERIODS OF ELLIPTIC CURVES
cites work: Values of Gaussian hypergeometric series
cites work: Hypergeometric functions and elliptic curves
cites work: Q5508372
DOI: 10.1007/S11139-011-9345-7
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description: scientific article; zbMATH DE number 6088841