On certain unitary group Shimura varieties
On certain unitary group Shimura varieties
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关于某些单一群志村品种
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发表时间：
2004
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通讯作者：
E. Mantovan
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作者：
E. Mantovan
- 本文研究了一类（PEL）型Shimura簇在素数p处的局部几何.我们开始研究在p处具有良好约化的Shimura簇的特殊纤维的牛顿多边形分层。每一层可以用相应的RapopartZink空间的约化纤维与某些光滑簇（我们称之为Igusa簇）的乘积以及依赖于该层的某个p-adic群Tα对它们的作用来描述。(The在这种情况下，Igusa变种的定义是基于Zink对Barsotti-Tate群的斜率过滤和Oort叶理的概念的结果。特别地，我们证明了利用Igusa簇和Rapoport-Zink空间的紧支集的étale上同调以及Tα的群同调来计算Newton多边形层的紧支集的étale上同调是可能的.进一步地，我们能够将Zagliki的上述构造局部地推广到特征零，并得到了在p处好约化和坏约化两种情况下Shimura簇的étale上同调的一个类似的描述。作为这种分析的结果，我们得到了Shimura簇的l-adic上同调的一个描述，在l-adic上同调的Igusa品种和Rapoport-Zink空间的紧支撑。2000年数学学科分类。- 11G18 14G35
— In this paper, we study the local geometry at a prime p of a certain class of (PEL) type Shimura varieties. We begin by studing the Newton polygon stratification of the special fiber of a Shimura variety with good reduction at p. Each stratum can be described in terms of the products of the reduced fiber of the corresponding RapoportZink space with some smooth varieties (we call the Igusa varieties), and of the action on them of a certain p-adic group Tα, which depends on the stratum. (The definition of the Igusa varieties in this context is based upon a result of Zink on the slope filtration of a Barsotti-Tate group and on the notion of Oort’s foliation.) In particular, we show that it is possible to compute the étale cohomology with compact supports of the Newton polygon strata, in terms of the étale cohomology with compact supports of the Igusa varieties and the Rapoport-Zink spaces, and of the group homology of Tα. Further more, we are able to extend Zariski locally the above constructions to characteristic zero and obtain an analoguous description for the étale cohomology of the Shimura varieties in both the cases of good and bad reduction at p. As a result of this analysis, we obtain a description of the l-adic cohomology of the Shimura varieties, in terms of the l-adic cohomology with compact supports of the Igusa varieties and of the Rapoport-Zink spaces. 2000 Mathematics Subject Classification. — 11G18,14G35.