# Supercongruences for rigid hypergeometric Calabi--Yau threefolds

## TL;DR

This paper proves supercongruences for fourteen rigid hypergeometric Calabi--Yau threefolds over the rationals, confirming conjectures by Rodriguez-Villegas using $p$-adic and motive-based methods.

## Contribution

It establishes the supercongruences for these Calabi--Yau threefolds through novel applications of $p$-adic theory and hypergeometric motives, confirming longstanding conjectures.

## Key findings

- Supercongruences verified for 14 Calabi--Yau threefolds
- Methods include $p$-adic unit roots and hypergeometric motives
- Results confirm Rodriguez-Villegas' conjectures

## Abstract

We establish the supercongruences for the fourteen rigid hypergeometric Calabi--Yau threefolds over $\mathbb Q$ conjectured by Rodriguez-Villegas in 2003. Our first method is based on Dwork's theory of $p$-adic unit roots and it allows us to establish the supercongruences between the truncated hypergeometric series and the corresponding unit roots for ordinary primes. The other method makes use of the theory of hypergeometric motives, in particular, adapts the techniques from the recent work of Beukers, Cohen and Mellit on finite hypergeometric sums over $\mathbb Q$. Essential ingredients in executing the both approaches are the modularity of the underlying Calabi--Yau threefolds and a $p$-adic perturbation method applied to hypergeometric functions.