Modular forms of virtually real-arithmetic type I -- Mixed mock modular forms yield vector-valued modular forms

TL;DR

This paper introduces virtually real-arithmetic types to unify various relaxed modularity notions, demonstrating their connection to vector-valued modular forms and establishing rationality of Fourier and Taylor coefficients.

Contribution

It defines a new class of representations called virtually real-arithmetic types, extending the framework of modular forms to include non-reducible cases.

Findings

Virtually real-arithmetic types are generally not completely reducible.

Fourier and Taylor coefficients of associated modular forms are rational.

Unified framework encompasses mock and higher order modular forms.

Abstract

The theory of elliptic modular forms has gained significant momentum from the discovery of relaxed yet well-behaved notions of modularity, such as mock modular forms, higher order modular forms, and iterated Eichler-Shimura integrals. Applications beyond number theory range from combinatorics, geometry, and representation theory to string theory and conformal field theory. We unify these relaxed notions in the framework of vector-valued modular forms by introducing a new class of $\mathrm{SL}_{2}(\mathbb{Z})$-representations: virtually real-arithmetic types. The key point of the paper is that virtually real-arithmetic types are in general not completely reducible. We obtain a rationality result for Fourier and Taylor coefficients of associated modular forms.