arXiv 2003.046112020-03-10math.NTmath.KT

Eisenstein级数与$SL_n/\mathbb{Q}$的算术子群的上同调

Eisenstein series and the top degree cohomology of arithmetic subgroups of $SL_n/\mathbb{Q}$

Joachim Schwermer

备注：: 27 pages, no figures

摘要：

特殊线性$\mathbb{Q}$群$\mathsf{G} = SL_n$的无挠算术子群$Γ$的上同调$H^*(Γ, E) $可以用$Γ$的自守谱来解释。在这个框架内，上同调分解为尖点上同调和爱森斯坦上同调。后一个空间根据$\mathsf{G}$的伴随真抛物$\mathbb{Q}$ -子群的类$\{\mathsf{P}\}$分解。每个被加数$H^*_{\mathrm{\{P\}}}(Γ, E)$都是由附着在$\{\mathsf{P}\}$中元素的Levi分量上的尖点自守形式上的爱森斯坦级数（或其残基）建立的。上同调$H^*(Γ, E) $在上同调维数$\mathrm{cd}(Γ) = \frac{n(n-1)}{2}$给出的次数以上为零。我们关心的是在这个最高度上的上同调的内部结构。一方面，我们显式地描述了关联类$\{\mathsf{P}\}$，其中相应的被加数$H^{\mathrm{cd}(Γ)}_{\mathrm{\{\mathsf{P}\}}}(Γ, E)$为零。另一方面，在其余的情况下，我们的关联类，构建各种家庭的非零Eisenstein上同调类的跨度$H^{\mathrm{cd}(Γ)}_{\mathrm{\{\mathsf{Q}\}}}(Γ, \mathbb{C})$。最后，在主同余子群$Γ(q)$，$q = p^ν > 5$，$p\geq 3$是素数的情况下，我们给出了这些空间的大小的下界，如果甚至没有一个精确的公式，它的尺寸为某些关联类$\{\mathsf{Q}\}$。

英文摘要：

The cohomology $H^*(Γ, E) $ of a torsion-free arithmetic subgroup $Γ$ of the special linear $\mathbb{Q}$-group $\mathsf{G} = SL_n$ may be interpreted in terms of the automorphic spectrum of $Γ$. Within this framework, there is a decomposition of the cohomology into the cuspidal cohomology and the Eisenstein cohomology. The latter space is decomposed according to the classes $\{\mathsf{P}\}$ of associate proper parabolic $\mathbb{Q}$-subgroups of $\mathsf{G}$. Each summand $H^*_{\mathrm{\{P\}}}(Γ, E)$ is built up by Eisenstein series (or residues of such) attached to cuspidal automorphic forms on the Levi components of elements in $\{\mathsf{P}\}$. The cohomology $H^*(Γ, E) $ vanishes above the degree given by the cohomological dimension $\mathrm{cd}(Γ) = \frac{n(n-1)}{2}$. We are concerned with the internal structure of the cohomology in this top degree. On the one hand, we explicitly describe the associate classes $\{\mathsf{P}\}$ for which the corresponding summand $H^{\mathrm{cd}(Γ)}_{\mathrm{\{\mathsf{P}\}}}(Γ, E)$ vanishes. On the other hand, in the remaining cases of associate classes we construct various families of non-vanishing Eisenstein cohomology classes which span $H^{\mathrm{cd}(Γ)}_{\mathrm{\{\mathsf{Q}\}}}(Γ, \mathbb{C})$. Finally, in the case of a principal congruence subgroup $Γ(q)$, $q = p^ν > 5$, $p\geq 3$ a prime, we give lower bounds for the size of these spaces if not even a precise formula for its dimension for certain associate classes $\{\mathsf{Q}\}$.

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