Steinberg homology, modular forms, and real quadratic fields

Abstract

We compare the homology of a congruence subgroup Γ of GL2(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 ψΓ,E in the long exact sequence of homology stemming from this comparison has image in H0(Γ,St(Q2;R)) generated by classes zβ indexed by β∈E∖Q. We investigate this image.When R=C, H0(Γ,St(Q2;C)) is isomorphic to a space of classical modular forms of weight 2, and the image lies inside the cuspidal part. In this case, zβ is closely related to periods of modular forms over the geodesic in the upper half plane from β to its conjugate β′. Assuming GRH we prove that the image of ψΓ,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, H0cusp(Γ,St(Q2;Z)). Assuming GRH we prove that for any congruence subgroup, ψΓ,E always has finite index in H0cusp(Γ,St(Q2;Z)), and if Γ=Γ1(N)± or Γ1(N), then the image is all of H0cusp(Γ,St(Q2;Z)). If Γ=Γ0(N)± or Γ0(N), we prove (still assuming GRH) an upper bound for the size of H0cusp(Γ,St(Q2;Z))/Im(ψΓ,E). We conjecture that the results in this paragraph are true unconditionally.We also report on extensive computations of the image of ψΓ,E that we made for Γ=Γ0(N)± and Γ=Γ0(N). Based on these computations, we believe that the image of ψΓ,E is not all of H0cusp(Γ,St(Q2;Z)) for these groups, for general N.