Congruences for \({}_3F_2\) hypergeometric functions over finite fields (Q1856410)

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.