Modular forms, hypergeometric functions and congruences

TL;DR

This paper explores the connection between modular forms and hypergeometric functions to establish elementary congruences for specific binomial coefficient sums, advancing understanding of their arithmetic properties.

Contribution

It introduces new elementary congruences for binomial coefficient sums using the theory of modular forms and hypergeometric functions, extending previous work on their arithmetic properties.

Findings

Established elementary congruences for binomial coefficient sums

Analyzed Fourier coefficients of modular forms for congruence properties

Connected hypergeometric functions with modular form theory

Abstract

Using the theory of Stienstra and Beukers, we prove various elementary congruences for the numbers \sum \binom{2i_1}{i_1}^2\binom{2i_2}{i_2}^2...\binom{2i_k}{i_k}^2, where k,n \in N, and the summation is over the integers i_1, i_2, ...i_k >= 0 such that i_1+i_2+...+i_k=n. To obtain that, we study the arithmetic properties of Fourier coefficients of certain (weakly holomorphic) modular forms.