%0 Journal Article
%T Supercongruence conjectures of Rodriguez-Villegas
%A Dermot McCarthy
%J Mathematics
%D 2009
%I arXiv
%X 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.
%U http://arxiv.org/abs/0907.5089v1