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The product structure of Newton strata in the good reduction of Shimura varieties of Hodge type. (English) Zbl 1430.14058

A main tool to study the arithmetic of Shimura varieties is the Newton stratification. The almost product structure of Newton strata allows us to study their geometry and cohomology in terms of Igusa varieties and Rapoport-Zink spaces. This is well understood in the special case of Shimura varieties of PEL type. E. Mantovan [Duke Math. J. 129, No. 3, 573–610 (2005; Zbl 1112.11033)] gave the construction of the almost product structure of Newton strata for Shimura variety of PEL type with hyperspecial level structure at \(p\) and derived a formula for thecohomology of a local system over Shimura varieties and Rapoport-Zink spaces associated to a representation of the linear algebraic group attached to the Shimura variety. A. Caraiani and P. Scholze [Ann. Math. (2) 186, No. 3, 649–766 (2017; Zbl 1401.11108)] gave an infinite version of Mantovan’s almost product structure.
In the present paper, the author extends this structure to the larger class of Shimura varieties of Hodge type. He gives the almost product structure in the special fibre and Caraiani-Scholze type product structure for Shimura varieties of Hodge type.
Reviewer: Lei Yang (Beijing)

MSC:

14G35 Modular and Shimura varieties
11G18 Arithmetic aspects of modular and Shimura varieties
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