论文标题
Steinberg同源性，模块化形式和实际二次领域
Steinberg homology, modular forms, and real quadratic fields
论文作者
论文摘要
我们将GL_2（Z）的一致性亚组伽玛与Q和E上的Steinberg模块中的系数进行比较，而E是一个真正的二次二次字段。如果r是任何可交换的基基环，则在此比较中以h_0（gamma，st（q^2; r））的图像在e \ q中由z_βIndex产生的h_0（gamma，st（q^2; r））中的最后一个同源的同源序列中的最后一个连接同形的同构{gamma {gamma，e}。 当r = c，h_0（gamma，st（q^2; c））是同构与经典模块化形式的重量2的空间，图像位于尖端部分。在这种情况下，z_beta与从β到其共轭β的上半平面上的模块化形式的时期密切相关。假设GRH我们证明$ψ_{γ，e} $的图像等于整个Cuspidal部分。 当r = z时，我们有一个不可或缺的情况。我们定义了Steinberg同源性的Cuspidal部分，H_0^cusp（gamma，st（q^2; z））。假设GRH我们证明，对于任何一致性子组，psi_ {gamma，e}始终具有H_0^cusp（gamma，st（q^2; z））中的有限索引，并且如果gamma = gamma = gamma_1（n）^pm orγ_1（n），则图像是H_0^Cusp（qusp（qusp（qum）），则为k^2;如果gamma = gamma_0（n）^pm或gamma_0（n），我们证明（仍然假设GRH）是H_0^cusp（gamma，st（q^2; z））/image（psi_ {psi_ {gamma，e}）的上限。我们猜想本段中的结果是无条件的。 我们还报告了我们为gamma = gamma_0（n）^pm和gamma = gamma_0（n）制作的psi_ {gamma，e}图像的大量计算。基于这些计算，我们认为对于一般N。
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_βindexed 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 $ψ_{Γ,E}$ 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 Γ_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.
