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Introduction to Shimura varieties with bad reduction of parahoric type. (English) Zbl 1148.11028
Arthur, James (ed.) et al., Harmonic analysis, the trace formula, and Shimura varieties. Proceedings of the Clay Mathematics Institute 2003 summer school, Toronto, Canada, June 2–27, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3844-X/pbk). Clay Mathematics Proceedings 4, 583-642 (2005).
This article provides a survey of several important concepts related to Shimura varieties with parahoric level structure at a prime $$p$$ by using the Rapoport-Zink local model as the main tool. In particular, it discusses local models attached to general linear groups and symplectic groups. Their relations with Shimura varieties with parahoric level structure are illustrated by describing two examples, namely, the simple or fake unitary Shimura varieties with parahoric level structure and the Siegel modular varieties with $$\Gamma_0 (p)$$-level structure. The article also includes some applications of local models to problems involving flatness, stratifications of special fibers, and the determination of the semisimple local zeta functions for simple Shimura varieties.
For the entire collection see [Zbl 1083.11002].

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