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.


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.)
Full Text: DOI


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