Congruences via modular forms

Abstract

Abstract. We prove two congruences for the coefficients of power series expansions in t of modular forms where t is a modular function. As a result, we settle two recent conjectures of Chan, Cooper and Sica. Additionally, we provide tables of congruences for numbers which appear in similar power series expansions and in the study of integral solutions of Apéry-like differential equations.

Modular forms, hypergeometric functions and congruences

“…Many authors have subsequently studied arithmetic properties of B(n) ′ s and discovered similar three term congruence relations for other Apery like numbers. For related work see [1,3,6,7,10,11].…”

Congruences Among Power Series Coefficients of Modular Forms

“…In our theorems, we have only examined modular forms on genus zero Γ 0 (N ). However, many examples exist for modular forms on subgroups of higher genus (see [6,11]). An infinite family of examples on a subgroup of higher genus would be interesting.…”

“…Our result was proved by Jarvis and Verrill in the case where the sum begins with n = 1 in (3.2). In [6] it is mentioned that Stienstra has indicated that the result follows in the case where n = 0 by formal group theory. We give a proof here for completeness.…”

Supercongruences for polynomial analogs of the Apéry numbers

“…(k + 1) 2 C(n; k + 1), (28) with C(n; k) as in (27). For the second congruence, we used the presence of the term [mn 1 ] q [mn 3 ] q , which is divisible by Φ m (q) 2 , together with…”

“…A major motivation for this paper is the observation of R. Osburn and B. Sahu [28] that all Apéry-like numbers appear to satisfy supercongruences. However, despite recent progress [29], it remains open to show that, for instance, the Almkvist-Zudilin numbers [4, sequence (4.12) (δ)], [11], [10]…”