Speaker
Martin Raum
Abstract
There are two established exact methods to compute Fourier expansions of modular forms: Via modular symbols and via trace formulas. Due to their tight connection to cohomology of groups and modular varieties, they inherit some of the common limitations. In particular, general cusp expansions and vector-valued notions of modular forms are impossible to access through them. They would be needed to compute, for instance, non-quadratic twists of elliptic curves or to assemble Siegel and Hermitian modular forms. This opens a niche for one further type of algorithm, which covers these aspects.
We discuss how the Rankin–Selberg convolution provides a whole class of algorithms which deliver Fourier expansions of modular forms including all their cusp expansions. An implementation of the case of elliptic modular forms reveals a shift away from usual performance issues around linear algebra to more group and representation theoretic concerns.