Incongruences for modular forms and applications to partition functions

Abstract

The study of arithmetic properties of coefficients of modular forms f(τ)=∑a(n)qn 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. Löbrich have employed the q-expansion theory of P. Deligne and M. Rapoport in order to determine more about where these congruences can occur. Here, we apply the method to a large class of modular forms, and in particular to several noteworthy examples, including generalized Frobenius partitions and the two mock theta functions f(q) and ω(q).