Gisbert Wüstholz and Clemens Fuchs (eds)
- Published in print:
- 2019
- Published Online:
- May 2020
- ISBN:
- 9780691193779
- eISBN:
- 9780691197548
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691193779.003.0002
- Subject:
- Mathematics, Geometry / Topology
This chapter discusses Peter Scholze's minicourse on local Shimura varieties. The goal of these lectures is to describe a program to construct local Langlands correspondence. The construction is ...
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This chapter discusses Peter Scholze's minicourse on local Shimura varieties. The goal of these lectures is to describe a program to construct local Langlands correspondence. The construction is based on cohomology of so-called local Shimura varieties and generalizations thereof. It was predicted by Robert Kottwitz that for each local Shimura datum, there exists a so-called local Shimura variety, which is a pro-object in the category of rigid analytic spaces. Thus, local Shimura varieties are determined by a purely group-theoretic datum without any underlying deformation problem. This is now an unpublished theorem, by the work of Fargues, Kedlaya–Liu, and Caraiani–Scholze. The chapter then explains the approach to local Langlands correspondence via cohomology of Lubin–Tate spaces as well as Rapoport–Zink spaces. It also introduces a formal deformation problem and describes properties of the corresponding universal deformation formal scheme.Less
This chapter discusses Peter Scholze's minicourse on local Shimura varieties. The goal of these lectures is to describe a program to construct local Langlands correspondence. The construction is based on cohomology of so-called local Shimura varieties and generalizations thereof. It was predicted by Robert Kottwitz that for each local Shimura datum, there exists a so-called local Shimura variety, which is a pro-object in the category of rigid analytic spaces. Thus, local Shimura varieties are determined by a purely group-theoretic datum without any underlying deformation problem. This is now an unpublished theorem, by the work of Fargues, Kedlaya–Liu, and Caraiani–Scholze. The chapter then explains the approach to local Langlands correspondence via cohomology of Lubin–Tate spaces as well as Rapoport–Zink spaces. It also introduces a formal deformation problem and describes properties of the corresponding universal deformation formal scheme.
Peter Scholze and Jared Weinstein
- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- chapter
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0002
- Subject:
- Mathematics, Geometry / Topology
This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic ...
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This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic varieties. The goal is to construct a category of adic spaces which contains both formal schemes and rigid-analytic spaces as full subcategories. Just as formal schemes are built out of affine formal schemes associated to adic rings, and rigid-analytic spaces are built out of affinoid spaces associated to affinoid algebras, adic spaces are built out of affinoid adic spaces, which are associated to pairs of topological rings. The affinoid adic space associated to such a pair is the adic spectrum. The chapter then looks at Huber rings and defines the set of continuous valuations on a Huber ring, which constitute the points of an adic space.Less
This chapter reviews the theory of adic spaces as developed by Huber. There are two familiar categories of geometric objects which arise in nonarchimedean geometry: formal schemes and rigid-analytic varieties. The goal is to construct a category of adic spaces which contains both formal schemes and rigid-analytic spaces as full subcategories. Just as formal schemes are built out of affine formal schemes associated to adic rings, and rigid-analytic spaces are built out of affinoid spaces associated to affinoid algebras, adic spaces are built out of affinoid adic spaces, which are associated to pairs of topological rings. The affinoid adic space associated to such a pair is the adic spectrum. The chapter then looks at Huber rings and defines the set of continuous valuations on a Huber ring, which constitute the points of an adic space.
