论文标题
Eisenstein系列和$ sl_n/\ mathbb {q} $的算术亚组的顶级学位共同体
Eisenstein series and the top degree cohomology of arithmetic subgroups of $SL_n/\mathbb{Q}$
论文作者
论文摘要
特殊线性$ \ mathbb {q} $ - 组$ \ mathsf {g} = sl_n $的$ h^*（γ，e）的$可以用自动型$ $γ$来解释。在此框架内，共同体分解为Cuspidal共同体学和Eisenstein的共同体学。后一个空间是根据类$ \ {\ Mathsf {p} \} $分解的，$ \ Mathsf {g} $每个汇总$ h^*_ {\ mathrm {\ {p \}}}}}（γ，e）$由Eisenstein系列（或此类残留物）建立在cuspidal自动形式上的cuspidal自动形式，该形式是$ \ \ \ {\ nathsf {p p} $的元素组成部分。 $ h^*（γ，e）$消失于同胞维度$ \ mathrm {cd}（γ）= \ frac {n（n-1）} {2} $的程度。我们关注此最高程度上的同种学的内部结构。一方面，我们明确地描述了$ \ {\ Mathsf {p} \} $的关联类，相应的summand $ h^{\ mathrm {cd}（γ）} _ {\ mathrm {\ mathrm {\ mathrm {\ mathsf {\ mathsf {\ mathsf {p} {p} {p} \ \}}}}}另一方面，在关联类的剩余案例中，我们构建了跨越$ h^{\ mathrm {cd}（γ）} _ {\ mathrm {\ mathrm {\ mathrm {\ mathsf {\ mathsf {q}}}}} {c <mathbb的各种非消失的Eisenstein共同体学类别的各种家族。最后，在主要的一致性子组$γ（q）$，$ q = p = p^ν> 5 $，$ p \ geq 3 $ a prime中，我们为这些空间的大小提供了较低的界限，即使不是某些相关类$ \ \ \ \ {\ nathsf {q}} \ \} $的精确公式。
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 $\{\mathsf{Q}\}$.
