Vector-valued Modular Forms, Computational Considerations

Licentiate thesis, 2020

In the following thesis we give a thorough self-contained introduction to vector-valued modular forms with an eye to representation theoretic aspects. We also examine the mathematical details of an implementation that we provide for an algorithm that computes bases of certain spaces of vector-valued modular forms in terms of a theorem due to Raum and Xià.

Our argument consists of the following key steps. Firstly, we compute Fourier series for vector-valued Eisenstein series, including the case of quasi-modular Eisenstein series of weight $2$ (chapter 3.1-3.2). Secondly, we connect the spaces that occur in work of Raum and Xià in a precise and computable way to the spaces of vector-valued modular forms under consideration. This is done by means of tracing out all relevant mappings in a large commutative diagram which also showcases the effective complexity of our approach.

We provide an implementation ModularForms.jl in the programming language Julia that incorporates the algorithms described in this thesis to determine spaces of vector-valued modular forms. This package includes related auxiliary functionality, thus extending the Hecke.jl/Nemo.jl ecosystem, which we built our work upon.