Supercongruences using modular forms

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 p² or even p³. 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.