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On a supercongruence conjecture of Rodriguez-Villegas. (English) Zbl 1354.11030

Summary: In examining the relationship between the number of points over \( \mathbb{F}_p\) on certain Calabi-Yau manifolds and hypergeometric series which correspond to a particular period of the manifold, F. Rodriguez-Villegas [in: Calabi-Yau varieties and mirror symmetry. Proceedings of the workshop on arithmetic, geometry and physics around Calabi-Yau varieties and mirror symmetry, Toronto, Canada, 2001. Providence, RI: American Mathematical Society (AMS). 223–231 (2003; Zbl 1062.11038)] identified numerically 22 possible supercongruences. We prove one of the outstanding supercongruence conjectures between a special value of a truncated generalized hypergeometric series and the \( p\)-th Fourier coefficient of a modular form.

MSC:

11F33 Congruences for modular and \(p\)-adic modular forms
33C20 Generalized hypergeometric series, \({}_pF_q\)
11T24 Other character sums and Gauss sums

Citations:

Zbl 1062.11038
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References:

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