Leaves in parahoric reductions of Shimura varieties

• Applicant: Professor Dr. Torsten Wedhorn
• Subject Area: Mathematics
• Term: from 2016 to 2019
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 317095975

One of the central programs in Arithmetic Geometry is the web of conjectures and results predicted by the Langlands philosophy. In the last decade at almost every ICM a fields medal was awarded for work in this area (2002 Laurent Lafforgue, 2010 Ngo Bao Chau, 2014 Manjul Bhargava). The long-standing expectations that reductions of Shimura varieties are an essential tool in approaching the problems within the Langlands program has been impressively confirmed at the latest when Harris and Taylor proved the local Langlands conjecture at the end of the 1990's. Since then reductions of Shimura varieties and variants thereof have been a central tool for several substantial progresses (such as the work of Fargues, Boyer, Shin, or Scholze). In recent years there has also been a rapid development of tools in the field. Examples are the construction of reductions of very general classes of Shimura varities (e.g., by Vasiu, Kisin, and Pappas), the study of several important stratifications of general classes of reductions (e.g., by Haines, Görtz, Kottwitz, Rapoport, Viehmann, Wedhorn, Wortmann, Hamacher), or the construction of automorphic invariants (e.g., by Nicole, Goldring, Boxer, Koskivirta, Wedhorn and for perfectoizations of Shimura varieties by Caraiani and Scholze).One of the finest invariants is the isomorphism class of the $p$-divisible group together with its additional structure attached to a point in the reduction of the Shimura variety. The subsets where this invariant is constant are called leaves. In special cases much is known about these subsets (e.g. for moduli spaces of polarized abelian varieties by the work of Oort) but in full generality the structure of the leaves is still an open problem in particular in bad reduction. In this project the leaves will be studied in the very general case of parahoric reductions of Shimura varieties of abelian type.

• DFG Programme: Research Grants