On Some Hypergeometric Supercongruence Conjectures of Long

TL;DR

This paper extends p-adic techniques to prove additional hypergeometric supercongruences conjectured by Long, advancing understanding of their connection to modular forms and Galois representations.

Contribution

It generalizes existing methods to prove six new supercongruences and proposes a conjectural framework linking these to modularity and Galois representations.

Findings

Proved six new supercongruences conjectured by Long.

Extended Dwork's p-adic techniques to broader cases.

Conjectured a generalization linking supercongruences to modularity.

Abstract

In 2003, Rodriguez Villegas conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin. Using p-adic techniques of Dwork, they reduce the original supercongruences to related congruences which involve only the hypergeometric series. We generalize their techniques to consider six further supercongruences recently conjectured by Long. In particular we prove an analogous version of Long, Tu, Yui, and Zudilin's reduced congruences for each of these six cases. We also conjecture a generalization of Dwork's work which has been observed computationally and which would, together with a proof of modularity for Galois representations associated to our hypergeometric data, yield a full proof of Long's conjectures.