Modular forms, hypergeometric functions and congruences

Abstract

Let

$$A_k(n)=\sum_{\substack{i_1,i_2,\ldots, i_k \ge0\\ i_1+i_2+\cdots+ i_k=n}}\binom{2i_1}{i_1}^2 \binom{2i_2}{i_2}^2\cdots \binom{2i_k}{i_k}^2, \quad \textrm{for } k,n\in\mathbb{N}. $$

Using the theory of Stienstra and Beukers (Math. Ann., 271:269–304, 1985), we prove that the numbers A ₃(n) and A ₃(n−1)−16A ₃(n−2) satisfy three term congruence relations similar to those satisfied by Apery numbers. Moreover, for k≥3 and p prime, we prove divisibility by p of some simple linear combinations of the numbers A _k (n), for $n\in \mathbb{N}$.

To obtain this, we study the arithmetic properties of the Fourier coefficients of certain holomorphic and weakly holomorphic modular forms.