Supercongruence conjectures of Rodriguez-Villegas

TL;DR

This paper develops a comprehensive framework for supercongruences related to Calabi-Yau manifolds and hypergeometric series, proving a key conjecture and establishing new binomial coefficient identities.

Contribution

It introduces a unifying framework for 22 supercongruences and proves a significant conjecture linking hypergeometric series to modular form coefficients.

Findings

Proved a major supercongruence conjecture.

Established two new binomial coefficient-harmonic sum identities.

Provided a unified approach to all 22 supercongruences.

Abstract

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.