Supercongruences for truncated hypergeometric series and p-adic gamma function

Abstract

We prove three more general supercongruences between truncated hypergeometric series and p-adic gamma function from which some known supercongruences follow. A supercongruence conjectured by Rodriguez--Villegas and proved by E. Mortenson using the theory of finite field hypergeometric series follows from one of our more general supercongruences. We also prove a supercongruence for 7F6 truncated hypergeometric series which is similar to a supercongruence proved by L. Long and R. Ramakrishna.

Some Numeric Hypergeometric Supercongruences

“…For most of the listed modular forms in this paper, they will be identified by their labels in L-Functions and Modular Forms Database (LMFDB) [24]. For instance, f 8.6.a.a denotes a level-8 weight 6 Hecke eigenform, η(z) 8 η(4z) 4 + 8η(4z) 12 in terms of the Dedekind eta function, its label at LMFDB is 8.6.a.a.…”

“…The proof of Theorem 2 uses the finite hypergeometric functions (see Definition 3 below) originated in Katz's work [21] and modified by McCarthy [33] and a p-adic perturbation method, which was used in [28] and described explicitly in [29] by the author and Ramakrishna. The method was also used in other papers, we list a few here: [43] by Swisher, [27] by Liu, [20] by He,[4] by Barman and Saikia,and [34] by Mao and Pan. Essentially one regroups the the corresponding character sum into a major term, that is F (α, β; 1) p−1 − a p (f α ), and graded error terms.…”