×

A supercongruence conjecture of Rodriguez-Villegas for a certain truncated hypergeometric function. (English) Zbl 1074.11045

Summary: F. Rodriguez-Villegas [Hypergeometric families of Calabi-Yau manifolds. Fields Inst. Commun. 38, 223–231 (2003; Zbl 1062.11038)] studied hypergeometric families of Calabi-Yau manifolds and found (numerically) many possible supercongruences. For example, he conjectured for every odd prime \(p\) that
\[ \sum_{n=0}^{p-1}\binom{2n}{n}^2 16^{-n}\equiv \left(\frac{-4}{p}\right)\pmod {p^2}. \]
Here, the author uses the theory of Gaussian hypergeometric series, the properties of the \(p\)-adic \(\Gamma\)-function, and a strange combinatorial identity to prove this conjecture.
In the paper reviewed above [Trans. Am. Math. Soc. 355, No. 3, 987–1007 (2003; Zbl 1074.11044)], the author has proved the other three congruences.

MSC:

11A07 Congruences; primitive roots; residue systems
11G20 Curves over finite and local fields
11T24 Other character sums and Gauss sums
11L10 Jacobsthal and Brewer sums; other complete character sums
33C20 Generalized hypergeometric series, \({}_pF_q\)
33E50 Special functions in characteristic \(p\) (gamma functions, etc.)
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ahlgren, S., Gaussian hypergeometric series and combinatorial congruences, (), 1-12 · Zbl 1037.33016
[2] Ahlgren, S.; Ono, K., A Gaussian hypergeometric series evaluation and apéry number congruences, J. reine ange. math., 518, 187-212, (2000) · Zbl 0940.33002
[3] Beukers, F., Another congruence for the apéry numbers, J. number theory, 25, 201-210, (1987) · Zbl 0614.10011
[4] Greene, J., Hypergeometric functions over finite fields, Trans. amer. math. soc., 301, 77-101, (1987) · Zbl 0629.12017
[5] Gross, B.; Koblitz, N., Gauss sums and the p-adic γ-function, Ann. math., 109, 569-581, (1979) · Zbl 0406.12010
[6] Ishikawa, T., On beurkers’ congruence, Kobe J. math., 6, 49-52, (1989)
[7] Kaar, M., Summation in finite terms, J. assoc. comput. Mach., 28, 2, 305-350, (1981)
[8] Schneider, C., An implementation of Karr’s summation algorithm in Mathematica, Sém. lothar. combin., S43b, 1-10, (1999) · Zbl 0941.68162
[9] F. Rodriguez-Villegas, Hypergeometric families of Calabi-Yau manifolds, preprint. · Zbl 1062.11038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.