The theory of vector-modular forms for the modular group

TL;DR

This work develops a comprehensive theory of vector-valued modular forms for the modular group by recasting vvmf as sections of flat vector bundles and linking them to genus-0 Fuchsian differential equations. It proves existence for arbitrary admissible data, establishes a Mittag-Leffler framework for principal parts, and reformulates the problem via Birkhoff–Grothendieck/Fuchsian ODEs, showing holomorphic vvmf form free modules over the ring of modular forms. Serre duality yields a sharp dimension formula, and the q-expansion coefficients live in controlled number fields, enabling explicit arithmetic control. The theory is shown to be effective in practice, with complete descriptions in low dimensions, a tightness property for $d<6$, and constructive methods for building higher-rank vvmf through tensor products and weight-shifts, broadening applications to RCFT and noncongruence settings.

Abstract

We explain the basic ideas, describe with proofs the main results, and demonstrate the effectiveness, of an evolving theory of vector-valued modular forms (vvmf). To keep the exposition concrete, we restrict here to the special case of the modular group. Among other things,we construct vvmf for arbitrary multipliers, solve the Mittag-Leffler problem here, establish Serre duality and find a dimension formula for holomorphic vvmf, all in far greater generality than has been done elsewhere. More important, the new ideas involved are sufficiently simple and robust that this entire theory extends directly to any genus-0 Fuchsian group.