On a supercongruence conjecture of Rodriguez-Villegas

TL;DR

This paper proves a significant supercongruence conjecture linking hypergeometric series and modular form coefficients, advancing understanding of arithmetic properties of Calabi-Yau manifolds.

Contribution

It provides a proof for a key supercongruence conjecture connecting hypergeometric series with modular forms, a problem previously unresolved.

Findings

Confirmed a supercongruence conjecture for a specific hypergeometric series

Established a link between hypergeometric series values and modular form coefficients

Enhanced understanding of arithmetic properties of Calabi-Yau manifolds

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 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.