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Steinberg homology, modular forms, and real quadratic fields. (English) Zbl 07334491
Summary: We compare the homology of a congruence subgroup \(\Gamma\) of \(\mathrm{GL}_2(\mathbb{Z})\) with coefficients in the Steinberg modules over \(\mathbb{Q}\) 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 \(H_0(\Gamma,\mathrm{St}(\mathbb{Q}^2; R))\) generated by classes \(z_\beta\) indexed by \(\beta\in E\smallsetminus\mathbb{Q}\). We investigate this image. When \(R=\mathbb{C},H_0(\Gamma,\mathrm{St}(\mathbb{Q}^2;\mathbb{C}))\) 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 \(\beta^\prime\). Assuming GRH we prove that the image of \(\psi_{{\Gamma},E}\) equals the entire cuspidal part.
When \(R=\mathbb{Z}\), we have an integral version of the situation. We define the cuspidal part of the Steinberg homology, \(H_0^{\mathrm{cusp}}(\Gamma,\mathrm{St}(\mathbb{Q}^2;\mathbb{Z}))\). Assuming GRH we prove that for any congruence subgroup, \(\psi_{{\Gamma},E}\) always has finite index in \(H_0^{\mathrm{cusp}}(\Gamma,\mathrm{St}(\mathbb{Q}^2;\mathbb{Z}))\), and if \(\Gamma=\Gamma_1(N)^\pm\) or \(\Gamma_1(N)\), then the image is all of \(H_0^{\mathrm{cusp}}(\Gamma,\mathrm{St}(\mathbb{Q}^2;\mathbb{Z}))\). If \(\Gamma= \Gamma_0 (N)^\pm\) or \(\Gamma_0(N)\), we prove (still assuming GRH) an upper bound for the size of \(H_0^{\mathrm{cusp}}(\Gamma,\mathrm{St}(\mathbb{Q}^2; \mathbb{Z}))/\mathrm{Im}(\psi_{{\Gamma},E})\). 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 \(H_0^{cusp}(\Gamma,\mathrm{St}(\mathbb{Q}^2;\mathbb{Z}))\) for these groups, for general \(N\).
20J06 Cohomology of groups
11F67 Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols
11F75 Cohomology of arithmetic groups
SageMath; ecdata; Magma; PFPK
Full Text: DOI
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