Quotient Buildings

A MAGMA framework for computing quotients of Bruhat-Tits buildings over function fields modulo congruence subgroups and their Hecke operators.

This is the source code of algorithms developed in my Master's thesis Algorithmen zur Berechnung arithmetischer Quotienten von Bruhat-Tits-Gebäuden zu PGL_d+1(K_inf) (available in German only). Example output can be found in Chapter 5.

Usage

See the examples building.m and harmonic.m for basic usage of this framework. They contain complete programs to compute quotient buildings and Hecke operators respectively for the congruence subgroup Gamma_0(N). If you only want to run computations, adjust the parameters within these files and run magma building.m or magma harmonic.m respectively.

The scripts log2tex.py and log2txt.py convert logs generated by harmonic.m into the respective formats. They are expected to be run inside directory lib.

Custom congruence subgroups

It is also possible to use this framework to compute quotients modulo arbitrary congruence subgroups. Refer to the implementation of QuotientGamma0N and the definition of QuotientBuildingRec in lib/buildingQuotient.m for how to achieve this. You will need to adjust the creation of the QuotientBuildingRec structure as follows:

• Mandatory fields are representatives, q, d, N and subgroupCondition.
• If you also provide reductionFunc, the computation of the quotient building will be much faster. Hecke operatores are not implemented in the case where this function is missing.

Higher dimensions

For dimensions d greater than 2, the following functions need minor adjustments:

QuotientCusp, PermutationsAtVertex, HeckeLeftCosets.

Unit Tests

Run unit tests using magma unit_tests.m. Some of those will use optional dependencies if available, see folder optional_dependencies for detail.

Precomputed examples

The folder examples contains some precomputed tables of Hecke operators, in plain-text and LaTeX format. However, only the LaTeX files contain characteristic polynomials factorized into irreducibles.

Dependencies

• code was tested using Magma V2.21-4
• JavaView: http://javaview.de/ (for visualising 2-dimensional quotients)
• GraphViz: http://www.graphviz.org/ (for visualising 1-dimensional quotients)
• The script spring_layout.py needs NetworkX (https://networkx.github.io) installed. It is invoked by QuotientToJavaView in buildingQuotient.m if parameter Python is set to true.