Supercongruences for rigid hypergeometric Calabi--Yau threefolds

Ling Long, Fang-Ting Tu, Noriko Yui, Wadim Zudilin

Published 2017-05-04Version 1

We establish the supercongruences for the fourteen rigid hypergeometric Calabi--Yau threefolds over $\mathbb Q$ conjectured by Rodriguez-Villegas in 2003. Two different approaches are implemented, and they both successfully apply to all the fourteen supercongruences. Our first method is based on Dwork's theory of $p$-adic unit roots, and it allows us to establish the supercongruences 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.