Modular symbols and Hecke operators

TL;DR

This paper surveys methods for computing Hecke operators on the cohomology of arithmetic groups, extending classical modular symbol algorithms and highlighting recent unpublished results.

Contribution

It generalizes classical modular symbol techniques to broader contexts and compiles recent advances and unpublished work in the field.

Findings

Generalization of modular symbol algorithms

Effective techniques for Hecke operator computation

Integration of recent unpublished results

Abstract

We survey techniques to compute the action of the Hecke operators on the cohomology of arithmetic groups. These techniques can be seen as generalizations in different directions of the classical modular symbol algorithm, due to Manin and Ash-Rudolph. Most of the work is contained in papers of the author and the author with Mark McConnell. Some results are unpublished work of Mark McConnell and Robert MacPherson.