Vector-valued Modular Forms, Computational Considerations
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.
Chalmers, Matematiska vetenskaper, Algebra och geometri
Chalmers tekniska högskola
Opponent: Peter Bruin, Universiteit Leiden