Modularity from $q$-series

TL;DR

The paper introduces a novel criterion to certify modularity for vectors of holomorphic $q$-series using only $q$-series algebra, first-order $q$-differential systems, and analytic continuation. By defining being 'good for a finite orbit datum' and proving an equivalence with vector-valued modularity, it recasts modularity as a gluing problem of local cuspidal data into a global system with monodromy. The Rogers–Ramanujan pair is shown to be modular at level 5 without modular input, via a finite orbit analysis and explicit monodromy; the method extends to Andrews–Gordon series, yielding explicit level-$2k{+}1$ modularity.

Abstract

In 1975, G. E. Andrews challenged the mathematics community to address L. Ehrenpreis' problem, which was to directly prove the modularity of the Rogers-Ramanujan $q$-series' 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.

Theorems & Definitions (21)

• Theorem 1: Rogers-Ramanujan vector-valued modularity
• Remark
• Definition : Good vector of $q$-series
• Remark
• Remark
• Remark
• Theorem 2: Modularity from $q$-series
• Remark
• Remark
• Theorem 3