Supercongruences Occurred to Rigid Hypergeometric Type Calabi–Yau Threefolds

Abstract

We establish the supercongruences for the fourteen rigid hypergeometric Calabi-Yau threefolds over 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 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.

A q-microscope for supercongruences

“…Proof. We substitute a = q n into (23). Then the left-hand side of (23) terminates at k = (n − 1)/2; therefore, it is exactly the left-hand side of (25), while the right-hand side of (23) becomes q −(n−1)…”

Hypergeometric Supercongruences

“…For example, the case α = (1/5, 2/5, 3/5, 4/5) was proved by McCarthy [6], the corresponding modular form having level 25 [9]. Just before submitting this note, Long, Tu, Yui, and Zudilin [4] announced two different proofs of (1) for all fourteen cases in CY 3.…”

A Hypergeometric Version of the Modularity of Rigid Calabi-Yau Manifolds

“…This counting naturally leads to representations of a(p) by means of finitefield hypergeometric functions -due to J. Greene[11], D. McCarthy[18] and, in a greater generality, F. Beukers, H. Cohen, A. Mellit[4] -the representations that are used in the proof of Observation 1 in the case r = t = 1 2 . All 14 cases in the observation, namely the modulo p 3 supercongruences, are now proved simultaneously and rigorously in the joint paper[17] with L. Long, F.-T. Tu and N. Yui.…”