Abstract:
The Fourier coefficients of modular forms often count interesting arithmetic and combinatorial objects. For example, the inverted Dedekind eta function counts integer partitions. Such coefficients also tend to feature noteworthy divisibility properties and p-adic behavior. These properties can vary enormously in their difficulty: some are quite easy to prove, while others remain standing conjectures. A recent development is the discovery that the divisibility properties for the coefficients of one modular form can often manifest in the coefficients of another quite different form. This provides us with a useful tool for proving conjectured congruence families. We will show a few examples of this phenomenon, which we call "congruence multiplicity."