Steinberg homology, modular forms, and real quadratic fields

TL;DR

This paper explores the relationship between homology of congruence subgroups, modular forms, and real quadratic fields, revealing deep connections and conjecturing unconditional results based on extensive computational evidence.

Contribution

It establishes new links between Steinberg homology, modular forms, and quadratic fields, including results conditional on GRH and extensive computational analysis.

Findings

The image of the connecting homomorphism equals the entire cuspidal part under GRH.

Finite index of the image in integral Steinberg homology for certain groups.

Computational evidence suggests the image is not always the entire homology group.

Abstract

We compare the homology of a congruence subgroup Gamma of GL_2(Z) with coefficients in the Steinberg modules over Q and over E, where E is a real quadratic field. If R is any commutative base ring, the last connecting homomorphism psi_{Gamma,E} in the long exact sequence of homology stemming from this comparison has image in H_0(Gamma, St(Q^2;R)) generated by classes z_\beta indexed by beta in E \ Q. We investigate this image. When R=C, H_0(Gamma, St(Q^2;C)) is isomorphic to a space of classical modular forms of weight 2, and the image lies inside the cuspidal part. In this case, z_beta is closely related to periods of modular forms over the geodesic in the upper half plane from beta to its conjugate beta'. Assuming GRH we prove that the image of $\psi_{\Gamma,E}$ equals the entire cuspidal part. When R=Z, we have an integral version of the situation. We define the cuspidal part of the Steinberg homology, H_0^cusp(Gamma, St(Q^2;Z)). Assuming GRH we prove that for any congruence subgroup, psi_{Gamma,E} always has finite index in H_0^cusp(Gamma, St(Q^2;Z)), and if Gamma=Gamma_1(N)^pm or \Gamma_1(N), then the image is all of H_0^cusp(Gamma, St(Q^2;Z)). If Gamma=Gamma_0(N)^pm or Gamma_0(N), we prove (still assuming GRH) an upper bound for the size of H_0^cusp(Gamma, St(Q^2;Z))/image(psi_{Gamma,E}). We conjecture that the results in this paragraph are true unconditionally. We also report on extensive computations of the image of psi_{Gamma,E} that we made for Gamma=Gamma_0(N)^pm and Gamma=Gamma_0(N). Based on these computations, we believe that the image of psi_{Gamma,E} is not all of H_0^cusp(Gamma, St(Q^2;Z)) for these groups, for general N.