Modularity from $q$-series

Abstract

In 1975, G. E. Andrews challenged the mathematics community to address L. Ehrenpreis&#39; problem, which was to directly prove the modularity of the Rogers-Ramanujan $q$-series&#39; summatory forms. This question is important because many different $q$-series appearing in combinatorics, representation theory, and physics often seem to be mysteriously modular, yet there is no general test to confirm this directly from the exotic $q$-series expressions. In this note, we answer the challenge. We use $q$-series algebra, first-order $q$-differential systems, and analytic continuation with monodromy to give a criterion that decides when such series are modular. Specifically, we establish a necessary and sufficient condition for a vector of holomorphic $q$-series on $|q|<1$ to form a vector-valued modular function without modular input, providing a clear path to modularity for strange $q$-series.