  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.
