Supercongruences using modular forms
Abstract
Many generating series of combinatorially interesting numbers have the property that the sum of the terms of order at some suitable point is congruent to a zero of a zeta-function modulo infinitely many primes . Surprisingly, very often these congruences turn out to hold modulo or even . We call such congruences supercongruences and in the past 15 years an abundance of them have been discovered. In this paper we show that a large proportion of them can be explained by the use of modular functions and forms.
Cite
@article{arxiv.2403.03301,
title = {Supercongruences using modular forms},
author = {Frits Beukers},
journal= {arXiv preprint arXiv:2403.03301},
year = {2025}
}
Comments
36 pages, this is an updated version. Several references have been added. We also improved the presentation significantly and extended our results to modular groups which are Atkin-Lehner extensions. We tried not to change the numbering of Theorems, lemmas, etc. The only changes are the numbers 1.24 and beyond in the introduction of the first version