Supercongruences for rigid hypergeometric Calabi--Yau threefolds

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.

42 pages

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Cited by in corpus (10)

• A $q$-microscope for supercongruences
• Dwork-type supercongruences through a creative $q$-microscope
• $p$-adic analogues of hypergeometric identities
• A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds
• The Explicit Hypergeometric-Modularity Method I
• Computing Special $L$-Values of Certain Modular Forms with Complex Multiplication
• Ghosts and congruences for $p^s$-approximations of hypergeometric periods
• ($q$-)Supercongruences hit again
• Some Numeric Hypergeometric Supercongruences
• A Whipple ${}_7{F}_6$ formula revisited