Supercongruences and complex multiplication

Abstract

We study congruences involving truncated hypergeometric series of the formwhere p is a prime and m, s, r are positive integers. These truncated hypergeometric series are related to the arithmetic of a family of algebraic varieties and exhibit Atkin and Swinnerton-Dyer type congruences. In particular, when r = 3, they are related to K3 surfaces. For special values of λ, with s = 1 and r = 3, our congruences are stronger than what can be predicted by the theory of formal groups because of the presence of elliptic curves with complex multiplications. They generalize a conjecture made by Rodriguez-Villegas for the λ = 1 case and confirm some other supercongruence conjectures at special values of λ.

q-Analogues of two “divergent” Ramanujan-type supercongruences

“…In many cases the method to prove an ordinary congruence cannot be generalized directly to prove the corresponding q-congruence. For instance, no one knows how to extend the p-adic analysis in [13] to the q-case. On the other hand, sometimes studying q-congruences will enable us to find new basic hypergeometric series identities [9,11].…”

Supercongruences for truncated hypergeometric series and p-adic gamma function

“…Recently, L. Long and R. Ramakrishna [18] prove several supercongruences using a technique which relies on the relations between the classical and p-adic gamma functions. They also prove a conjecture of J. Kibelbek [14] and a strengthened version of a conjecture of van Hamme. For instance, they prove the following supercongruence modulo p 6 [18, theorem 2] which is stronger than a prediction of van Hamme in [13].…”

A modular supercongruence for _6F_5: An Apéry-like story

“…In addition, (1.1) is one of van Hamme's original 13 Ramanujan-type supercongruences (see [12, (M.2)]). For further details on this and related topics we refer to [9,13,21,29].…”