Vector-valued modular forms with an unnatural boundary

TL;DR

This paper characterizes a special class of vector-valued modular forms that are logarithmic, holomorphic, and can be extended beyond the upper half-plane, revealing new insights into their analytic continuation properties.

Contribution

It provides a complete characterization of vector-valued modular forms with an unnatural boundary, expanding understanding of their analytic continuation beyond the upper half-plane.

Findings

Identifies conditions for analytic continuation of these modular forms.

Classifies all such forms with an unnatural boundary.

Enhances understanding of the boundary behavior of vector-valued modular forms.

Abstract

We characterize all logarithmic, holomorphic vector-valued modular forms which can be analytically continued to a region strictly larger than the upper half-plane.