Computations directly on the cuspidal cohomology of congruence subgroups of $\mathrm{SL}(3, \mathbb{Z})$

TL;DR

This paper develops a new method to compute Hecke operator actions directly on the cuspidal cohomology of congruence subgroups of SL(3, Z), providing new data on nonzero cuspidal classes at various levels.

Contribution

It introduces a novel computational approach for directly analyzing cuspidal cohomology, extending previous methods and enabling new data collection.

Findings

Identified seven new levels with nonzero cuspidal classes.

Calculated local factors for five levels.

Extended understanding of the structure of cuspidal cohomology in SL(3, Z).

Abstract

Ash, Grayson, and Green [J. Number Theory 19 (1984), pp. 412-436] compute the action of Hecke operators on a certain subspace of the cohomology of low-level congruence subgroups of $\mathsf{SL}(3, \mathbb{Z})$. This subspace contains the cuspidal cohomology, which is of primary interest. We extend their work, introducing a method that allows for computing the action of Hecke operators directly on the cuspidal cohomology. Using this method, we obtain data for prime level less than 3500, finding seven additional levels at which nonzero cuspidal classes appear and calculating local factors for five of these levels.