Modular forms of virtually real-arithmetic type I: Mixed mock modular forms yield vector-valued modular forms
Journal article, 2021

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 SL (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.


Michael H. Mertens

University of Liverpool

Martin Raum

Chalmers, Mathematical Sciences, Algebra and geometry

Mathematical Research Letters

1073-2780 (ISSN)

Vol. 28 2 511-561

Subject Categories

Algebra and Logic

Discrete Mathematics

Mathematical Analysis



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