Document Type
Article
Publication Date
12-24-2021
Abstract
We establish the supercongruences for the fourteen rigid hypergeometric Calabi–Yau threefolds over 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 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.
Publication Source (Journal or Book title)
Advances in Mathematics
Recommended Citation
Long, L., Tu, F., Yui, N., & Zudilin, W. (2021). Supercongruences for rigid hypergeometric Calabi–Yau threefolds. Advances in Mathematics, 393 https://doi.org/10.1016/j.aim.2021.108058