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

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.