Congruences for \({}_3F_2\) hypergeometric functions over finite fields
type: Item
instance of: scholarly article
title: Congruences for \({}_3F_2\) hypergeometric functions over finite fields
zbMATH Open document ID: 1026.11091
author: Ken Ono
author: David Penniston
published in: Illinois Journal of Mathematics
publication date: 2003-12-12T00:00:00Z
review text: This paper is concerned with congruences for the values of certain Gaussian hypergeometric series \({}_3F_2(\lambda)_p\) over finite fields \(\mathbb F_p\) of characteristic \(p\). Some recent works [see, e.g., \textit{S. D. Ahlgren} and \textit{K. Ono}, J. Reine Angew. Math. 518, 187-212 (2000; Zbl 0940.33002); \textit{S. D. Ahlgren}, in: Symbolic computation, number theory, special functions, physics and combinatorics (Gainesville, FL, 1999), Kluwer Acad. Publ., Dordrecht, 1-12 (2001) (for the entire collection, see Zbl 0980.00026); \textit{E. Mortenson}, Trans. Am. Math. Soc. 355, 987-1007 (2003; Zbl 1074.11044); \textit{M. Koike}, Hiroshima Math. J. 22, 461-467 (1992; Zbl 0784.11057)] have shown that special values of these functions are congruent to Apéry-type numbers modulo \(p\). In this paper the authors derive a family of congruences of a different sort; the new congruences relate values of these functions to a simple polynomial in the characteristic. Many families of these congruences are given. As just one example of this phenomenon, if \(p=7\) or \(p\geq 13\) then we have \[ {}_3F_2\fracwithdelims(){2673}{2048}_p\equiv\fracwithdelims(){-2}p (1+p^{2}+p^{3})\pmod {20}. \] The proof depends on an analysis of the torsion structure of the elliptic curves to which the Gaussian hypergeometric series are related. In particular, the modulus of each congruence is closely related to the order of the torsion subgroup of the relevant elliptic curve.
reviewed by: Scott Ahlgren
Mathematics Subject Classification ID: 11T24
Mathematics Subject Classification ID: 33E50
zbMATH DE Number: 1865525
zbMATH Keywords: Gaussian hypergeometric series
zbMATH Keywords: elliptic curves
zbMATH Keywords: congruences
zbMATH Keywords: finite fields
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1865525