Supercongruences between truncated \(_{2}F_{1}\) hypergeometric functions and their Gaussian analogs. (English) Zbl 1074.11044

Mortenson, Eric

Trans. Am. Math. Soc. 355, No. 3, 987-1007 (2003).

F. Rodriguez-Villegas [Hypergeometric families of Calabi-Yau manifolds, Yui, Noriko (ed.) et al., Calabi-Yau varieties and mirror symmetry. Providence, RI: American Mathematical Society (AMS). Fields Inst. Commun. 38, 223–231 (2003; Zbl 1062.11038)] conjectured a number of supercongruences for hypergeometric Calabi-Yau manifolds of dimension \(d\leq 3\). For manifolds of dimension \(d=1\), he observed four potential supercongruences. Later that author proved one of the four. Motivated by Rodriguez-Villegas’s work, in the paper under review the present author proves a general result on supercongruences between values of truncated \(_{2}F_{1}\) hypergeometric functions and Gaussian hypergeometric functions. As a corollary to that result, he proves the three remaining supercongruences.

Reviewer: Olaf Ninnemann (Berlin)

MSC:

11L10	Jacobsthal and Brewer sums; other complete character sums
11G20	Curves over finite and local fields
11A07	Congruences; primitive roots; residue systems
33C05	Classical hypergeometric functions, \({}_2F_1\)

Keywords:

supercongruences

Citations:

Zbl 1062.11038

References:

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