Mathematical Research Letters

Volume 28 (2021)

Number 2

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

Pages: 511 – 561



Michael H. Mertens (Department of Mathematical Sciences, University of Liverpool, United Kingdom)

Martin Raum (Chalmers tekniska högskola och Göteborgs Universitet, Institutionen för Matematiska vetenskaper, Göteborg, Sweden)


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.

The research of the first-named author is supported by the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 - AQSER.

The second-named author was partially supported by Vetenskapsrådet Grant 2015-04139.

Received 25 February 2019

Accepted 31 July 2019

Published 13 May 2021