论文标题
关于签名$（1，n-1）$的分支pel统一的Rapoport-Zink的共同体
On the cohomology of the ramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$
论文作者
论文摘要
在本文中，我们通过在其特殊纤维上使用bruhat-tits分层来研究签名$（1，n-1）$的损坏的pel统一rapoport-zink的共同体。因此，我们应用了我们在上一篇论文中为未受到的案例开发的相同方法。更确切地说，我们首先研究给定的封闭式乳头尖层的共同体。它与普通的deligne-lusztig品种是同构的，它通常不光滑，并且与有限的符号相似性相关。我们确定了弗罗贝尼乌斯的权重以及大多数在同一个同种学中发生的单位表示。该计算涉及与经典Deligne-lusztig品种相关的光谱序列，这些序列是从较小的符号基团的Coxeter品种寄生诱导的。特别是，所有的一项代表仅促成了两个cuspidal系列。然后，我们介绍了封闭的Bruhat-tits地层的分析管，该管子为Rapoport-Zink空间的通用纤维提供了开放的覆盖。使用相关的čeCh光谱序列，我们证明，如果$ n $足够大，那么在超级级别的Rapoport-Zink空间的某些共同体组就无法接受。最终，当分裂情况下的$ n = 2 $时，当$ n = 3 $，当$ n = 4 $在非切块案例中时，就自动形式而言，我们对相关的Shimura品种的超单词shimura品种的共同基因座进行了完整描述。特别是，某些自动形式表示，取决于$ p $。因此，尽管封闭的bruhat-titts层面不平滑，但在受损情况下，我们在受损情况下的计算恢复了未受到的案例的所有主要特征。
In this paper, we study the cohomology of the ramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ by using the Bruhat-Tits stratification on its special fiber. As such, we apply the same method that we developped for the unramified case in two previous papers. More precisely, we first investigate the cohomology of a given closed Bruhat-Tits stratum. It is isomorphic to a generalized Deligne-Lusztig variety which is in general not smooth, and is associated to a finite group of symplectic similitudes. We determine the weights of the Frobenius and most of the unipotent representations occuring in its cohomology. This computation involves the spectral sequence associated to a stratification by classical Deligne-Lusztig varieties, which are parabolically induced from Coxeter varieties of smaller symplectic groups. In particular, all the unipotent representations contribute to only two cuspidal series. Then, we introduce the analytical tubes of the closed Bruhat-Tits strata, which give an open cover of the generic fiber of the Rapoport-Zink space. Using the associated Čech spectral sequence, we prove that certain cohomology groups of the Rapoport-Zink space at hyperspecial level fail to be admissible if $n$ is large enough. Eventually, when $n=2$ in the split case, when $n=3$ and when $n=4$ in the non-split case, we give a complete description of the cohomology of the supersingular locus of the associated Shimura variety at hyperspecial level, in terms of automorphic representations. In particular, certain automorphic representations occur with a multiplicity depending on $p$. Thus, our computations in the ramified case recover all the main features of the unramified case, despite new technical difficulties due to the closed Bruhat-Tits strata not being smooth.
