Cohomology of congruence subgroups of SL_3(Z), Steinberg modules, and real quadratic fields

TL;DR

This paper studies the homology of congruence subgroups of SL_3(Z) with coefficients in Steinberg modules over real quadratic fields, proposing a conjecture about the structure of certain cohomology subspaces and supporting it with computational evidence.

Contribution

It introduces a conjecture that a specific cohomology subspace is independent of the quadratic field and relates to cuspidal cohomology, supported by computational data.

Findings

Evidence from computer calculations supports the conjecture.

The subspace H(Gamma,E) appears to be independent of E.

Heuristic arguments suggest the conjecture's plausibility.

Abstract

We investigate the homology of a congruence subgroup Gamma of SL_3(Z) with coefficients in the Steinberg modules St(Q^3) and St(E^3), where E is a real quadratic field and the coefficients are Q. By Borel-Serre duality, H_0(Gamma, St(Q^3)) is isomorphic to H^3(Gamma,Q). Taking the image of the connecting homomorphism H_1(Gamma, St(E^3)/St(Q^3)) \to H_0(Gamma, St(Q^3)), followed by the Borel-Serre isomorphism, we obtain a naturally defined Hecke-stable subspace H(Gamma,E) of H^3(Gamma,Q). We conjecture that H(Gamma,E) is independent of E and consists of the cuspidal cohomology H_cusp^3(Gamma,Q) plus a certain subspace of H^3(Gamma, Q)$ that is isomorphic to the sum of the cuspidal cohomologies of the maximal faces of the Borel-Serre boundary. We report on computer calculations of H(Gamma,E) for various Gamma, E which provide evidence for the conjecture. We give a partial heuristic for the conjecture.