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arXiv · 1712.03004

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

Michael H. Mertens·Martin Raum
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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.

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Michael H. Mertens, Martin Raum. 2017-12-08. Modular forms of virtually real-arithmetic type I -- Mixed mock modular forms yield vector-valued modular forms. https://doi.org/10.4310/mrl.2021.v28.n2.a7

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