Supercongruence conjectures of Rodriguez-Villegas

Details Supercongruence conjectures of Rodriguez-Villegas Supercongruence conjectures of Rodriguez-Villegas

2009-07-29

arxiv.org/abs/0907.5089v1

Mathematics

Number Theory

63 pages

Scientific paper

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, Rodriguez-Villegas identified 22 possible supercongruences. We provide a framework of congruences covering all 22 cases. Using this framework we prove one of the outstanding supercongruence conjectures between a special value of a truncated ordinary hypergeometric series and the $p$-th Fourier coefficient of a modular form. In the course of this work we also establish two new binomial coefficient-harmonic sum identities.

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