Fredrik Strömberg

A reduction algorithm for Hilbert modular groups

Given a group acting on a topological space it often useful to have a “nice” set of representatives, a so-called fundamental domain, for this action. In practice it is also useful to not only know that such a domain exists, but also to know exactly how to reduce a given point to its representative.
For the modular group, PSL(2,Z), a number of fundamental domains and associated reduction algorithms have been known for a long time and are relatively simple to describe.
In the case of the Hilbert modular group PSL(2,OK), where OK is the ring of integers of a totally real number field, the fundamental domain is harder to describe geometrically but an algorithmic description has been known in principle since works of Blumenthal, Maass and others. Until recently, however, no explicit (finite-time) reduction algorithm has been known in the case of class number greater than one.
The aim of this talk is to present some of the motivations and the recent development and implementation of a new reduction algorithm for Hilbert modular groups, valid for any class number and degree.