Hypergeometric Supercongruences

Abstract

We discuss two related principles for hypergeometric supercongrences, one related to accelerated convergence and the other to the vanishing of Hodge numbers.

Special Hypergeometric Motives and Their L-Functions: Asai Recognition

“…The congruence (2.8) is in line with a general prediction of Roberts-Rodriguez-Villegas [36], though stated there for z = ±1 only.…”

On the Atkin and Swinnerton-Dyer type congruences for some truncated hypergeometric ${}_1F_0$ series

“…Li and Long [5] gave a nice survey on the Atkin and Swinnerton-Dyer type congruences. The reader may consult [8], [9], [11] for recent work. In particular, recently Sun [13] proposed many conjectured Atkin and Swinnerton-Dyer type congruences.…”

Supercongruences Occurred to Rigid Hypergeometric Type Calabi–Yau Threefolds

“…The more recent work [47] of Roberts, Rodriguez-Villegas and Watkins conjectures refined predictions related to (1); it also indicates a heuristic underlying the supercongruences that explains the exponent 3 of the modulus by the Hodge filtration of the corresponding hypergeometric motives. One part of the story is Dwork's unit-root theory in [19], in which the truncated hypergeometric series…”