Supercongruences using modular forms

Abstract

Many generating series of combinatorially interesting numbers have the property that the sum of the terms of order $<p$ at some suitable point is congruent to a zero of a zeta-function modulo infinitely many primes $p$. Surprisingly, very often these congruences turn out to hold modulo $p^2$ or even $p^3$. 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.

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