Incongruences for modular forms and applications to partition functions

TL;DR

This paper extends methods to analyze congruences in the coefficients of modular forms, applying these to partition functions and mock theta functions, revealing new arithmetic properties and congruence patterns.

Contribution

It generalizes the $q$-expansion approach to a broader class of modular forms and related functions, providing new results on their arithmetic congruences.

Findings

Extended congruence results for modular form coefficients

Analogous congruence results for Frobenius partitions

New insights into mock theta function coefficients

Abstract

The study of arithmetic properties of coefficients of modular forms $f(\tau) = \sum a(n)q^n$ has a rich history, including deep results regarding congruences in arithmetic progressions. Recently, work of C.-S. Radu, S. Ahlgren, B. Kim, N. Andersen, and S. L\"{o}brich have employed the $q$-expansion principle of P. Deligne and M. Rapoport in order to determine more about where these congruences can occur. Here, we extend the method to give additional results for a large class of modular forms. We also give analogous results for generalized Frobenius partitions and the two mock theta functions $f(q)$ and $\omega(q).$