Unifying relaxed notions of modular forms

This is joint work with Michael Mertens

Elliptic modular forms are functions on the complex upper half plane that are invariant under a certain action of the special linear group with integer entries. During the past decade it has been à la mode to study relaxed notions of modularity. Relevant keywords in this context are mock modular forms and higher order modular forms. In this talk, we suggest a change of perspective on such generalizations. Most of the novel variants of modular forms (with one prominent exception) can be viewed as components of vector-valued modular forms of "virtually real- arithmetic type". On the one hand, we can integrate into our work results by Kuga and Shimura that hitherto seemed almost forgotten. On the other hand, we can subsume iterated Eichler-Shimura integrals, and thus provide spectral deformations by Poincaré series. We also obtain Petersson inner products for mixed mock modular forms.