We compare the homology of a congruence subgroup Γ 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 ${\psi }_{\mathrm{\Gamma },E}$ in the long exact sequence of homology stemming from this comparison has image in $H_0(\mathrm{\Gamma },\mathrm{St}({\mathbb{Q}}^2;R))$ generated by classes $z_{\beta }$ indexed by $\beta \in E\setminus \mathbb{Q}$. We investigate this image.

When $R=\mathbb{C}$, $H_0(\mathrm{\Gamma },\mathrm{St}({\mathbb{Q}}^2;\mathbb{C}))$ 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 β to its conjugate ${\beta }^{\prime }$. Assuming GRH we prove that the image of ${\psi }_{\mathrm{\Gamma },E}$ 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, $H_0^{\mathrm{cusp}}(\mathrm{\Gamma },\mathrm{St}({\mathbb{Q}}^2;\mathbb{Z}))$. Assuming GRH we prove that for any congruence subgroup, ${\psi }_{\mathrm{\Gamma },E}$ always has finite index in $H_0^{\mathrm{cusp}}(\mathrm{\Gamma },\mathrm{St}({\mathbb{Q}}^2;\mathbb{Z}))$, and if $\mathrm{\Gamma }={\mathrm{\Gamma }}_1{(N)}^{\pm }$ or ${\mathrm{\Gamma }}_1(N)$, then the image is all of $H_0^{\mathrm{cusp}}(\mathrm{\Gamma },\mathrm{St}({\mathbb{Q}}^2;\mathbb{Z}))$. If $\mathrm{\Gamma }={\mathrm{\Gamma }}_0{(N)}^{\pm }$ or ${\mathrm{\Gamma }}_0(N)$, we prove (still assuming GRH) an upper bound for the size of $H_0^{\mathrm{cusp}}(\mathrm{\Gamma },\mathrm{St}({\mathbb{Q}}^2;\mathbb{Z}))/\mathrm{Im}({\psi }_{\mathrm{\Gamma },E})$. We conjecture that the results in this paragraph are true unconditionally.

We also report on extensive computations of the image of ${\psi }_{\mathrm{\Gamma },E}$ that we made for $\mathrm{\Gamma }={\mathrm{\Gamma }}_0{(N)}^{\pm }$ and $\mathrm{\Gamma }={\mathrm{\Gamma }}_0(N)$. Based on these computations, we believe that the image of ${\psi }_{\mathrm{\Gamma },E}$ is not all of $H_0^{\mathrm{cusp}}(\mathrm{\Gamma },\mathrm{St}({\mathbb{Q}}^2;\mathbb{Z}))$ for these groups, for general N.

我们比较了一个全等子群Γ的同源性 ${\mathrm{GL}}_2（\mathbb{ž}）$ Steinberg模块中的系数超过 $\mathbb{问}$在E上，其中E是一个实数二次方。如果R是任何交换基环，则最后一个连接同态${\psi }_{\mathrm{\Gamma }，Ë}$ 从这个比较中得出的同源性的长而精确的序列中 $H_0（\mathrm{\Gamma }，\mathrm{圣}（{\mathbb{问}}^2;\mathrm{[R}））$ 由类生成 ${ž}_{\beta }$ 被索引 $\beta \in Ë\setminus \mathbb{问}$。我们调查这张图片。

什么时候 $\mathrm{[R}=\mathbb{C}$， $H_0（\mathrm{\Gamma }，\mathrm{圣}（{\mathbb{问}}^2;\mathbb{C}））$与权重2的经典模块化形式的空间同构，并且图像位于尖齿部分内。在这种情况下，${ž}_{\beta }$与上半平面中从β到共轭的测地线上的模块化形式的周期密切相关${\beta }^{\prime }$。假设GRH，我们证明${\psi }_{\mathrm{\Gamma }，Ë}$ 等于整个尖部。

什么时候 $\mathrm{[R}=\mathbb{ž}$，我们有此情况的完整版本。我们定义了斯坦伯格同源性的尖锐部分，$H_0^{\mathrm{尖顶}}（\mathrm{\Gamma }，\mathrm{圣}（{\mathbb{问}}^2;\mathbb{ž}））$。假设GRH，我们证明对于任何同余子组，${\psi }_{\mathrm{\Gamma }，Ë}$ 始终具有有限的索引 $H_0^{\mathrm{尖顶}}（\mathrm{\Gamma }，\mathrm{圣}（{\mathbb{问}}^2;\mathbb{ž}））$， 而如果 $\mathrm{\Gamma }={\mathrm{\Gamma }}_{\mathrm{1个}}{（ñ）}^{\pm }$ 要么 ${\mathrm{\Gamma }}_{\mathrm{1个}}（ñ）$，那么图像就是全部 $H_0^{\mathrm{尖顶}}（\mathrm{\Gamma }，\mathrm{圣}（{\mathbb{问}}^2;\mathbb{ž}））$。如果$\mathrm{\Gamma }={\mathrm{\Gamma }}_0{（ñ）}^{\pm }$ 要么 ${\mathrm{\Gamma }}_0（ñ）$，我们证明（仍然假设GRH）为的大小的上限 $H_0^{\mathrm{尖顶}}（\mathrm{\Gamma }，\mathrm{圣}（{\mathbb{问}}^2;\mathbb{ž}））/\mathrm{林}（{\psi }_{\mathrm{\Gamma }，Ë}）$。我们推测本段中的结果是无条件的。

我们还报告了对图像的大量计算 ${\psi }_{\mathrm{\Gamma }，Ë}$ 我们为 $\mathrm{\Gamma }={\mathrm{\Gamma }}_0{（ñ）}^{\pm }$ 和 $\mathrm{\Gamma }={\mathrm{\Gamma }}_0（ñ）$。基于这些计算，我们认为${\psi }_{\mathrm{\Gamma }，Ë}$ 不是全部 $H_0^{\mathrm{尖顶}}（\mathrm{\Gamma }，\mathrm{圣}（{\mathbb{问}}^2;\mathbb{ž}））$对于这些基团，一般Ñ。