算术群上同调的计算，第1部分
Computations of Cohomology of Arithmetic Groups, Part 1
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中文总结 AI 辅助
本文提出一种更简单、快速且结构化的方法，计算算术群$GL_4({\mathbb Z})$等在不同最高权表示系数下的边界与Eisenstein上同调，并提出了类似Serre对偶的猜想，经30个表示族验证。
中文摘要 AI 辅助
本文的关键部分是对$GL_4({\mathbb Z})$在任意最高权表示系数下的边界上同调和Eisenstein上同调的计算。我们开发的方法使我们能够以另一种方式计算$SL_3({\mathbb Z})$和$GL_3({\mathbb Z})$在任意最高权表示系数下的上同调。与\cite{BHHM}相比，这种方法更简单、更快且更具结构性。我们陈述了$GL_m({\mathbb Z})$的边界上同调的一种对偶性，类似于Serre对偶，其中对偶层是行列式表示的幂。我们将此对偶性细化到边界上同调谱序列$E_\infty^{p,q}$层面上的对偶，并将其作为一个猜想陈述。然而，所有的计算，共30个不同的表示族，都满足此猜想。我们计算了$GL_4({\mathbb Z})$在对称幂及其与行列式表示扭曲的系数下的Eisenstein上同调。对于其他几种表示，我们基于几个猜想计算了Eisenstein上同调。基于这些猜想，人们可以在大多数情况下计算Eisenstein上同调。这些内容将包含在论文的下一版本中。
英文摘要
The key part of the current paper is the computation of boundary and Eisenstein cohomology of $GL_4({\mathbb Z})$ with coefficient in any highest weight representations. The method we develop let us compute in an alternative way the cohomology of $SL_3({\mathbb Z})$ and of $GL_3({\mathbb Z})$ with coefficients in any highest weight representation. This is done in a simpler, faster and in a more structured way compared to \cite{BHHM}. We state a duality for the boundary cohomology of $GL_m({\mathbb Z})$ of the type of Serre's duality, where the dualizing sheaf is a power of the determinant representation. We refine this duality to a duality on the level of the spectral sequence for the boundary cohomology $E_\infty^{p,q}$. We state it as a conjecture. However, all the computations, 30 different families of representations, satisfy this conjecture. We compute the Eisenstein cohomology of $GL_4({\mathbb Z})$ with coefficients in the symmetric powers and their twist by the determinant representation. For several other representations, we compute the Eisenstein cohomology, based a few conjectures. Based on those conjectured, one can compute the Eisenstein cohomology in most of the cases. They will be included in the next version of the paper.
