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 This is now an unpublished theorem, by the work of Fargues, Kedlaya–Liu, and Caraiani–Scholze.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 This is now an unpublished theorem, by the work of Fargues, Kedlaya–Liu, and Caraiani–Scholze.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/9780691202082.003.0025
- Subject:
- Mathematics, Geometry / Topology
This chapter explains an application of the theory developed in these lectures towards the problem of understanding integral models of local Shimura varieties. As a specific example, it resolves ...
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This chapter explains an application of the theory developed in these lectures towards the problem of understanding integral models of local Shimura varieties. As a specific example, it resolves conjectures of Kudla-Rapoport-Zink and Rapoport-Zink, that two Rapoport-Zink spaces associated with very different PEL data are isomorphic. The basic reason is that the corresponding group-theoretic data are related by an exceptional isomorphism of groups, so such results follow once one has a group-theoretic characterization of Rapoport-Zink spaces. The interest in these conjectures comes from the observation of Kudla-Rapoport-Zink that one can obtain a moduli-theoretic proof of Čerednik's p-adic uniformization for Shimura curves using these exceptional isomorphisms. The chapter defines integral models of local Shimura varieties as v-sheaves.Less
This chapter explains an application of the theory developed in these lectures towards the problem of understanding integral models of local Shimura varieties. As a specific example, it resolves conjectures of Kudla-Rapoport-Zink and Rapoport-Zink, that two Rapoport-Zink spaces associated with very different PEL data are isomorphic. The basic reason is that the corresponding group-theoretic data are related by an exceptional isomorphism of groups, so such results follow once one has a group-theoretic characterization of Rapoport-Zink spaces. The interest in these conjectures comes from the observation of Kudla-Rapoport-Zink that one can obtain a moduli-theoretic proof of Čerednik's p-adic uniformization for Shimura curves using these exceptional isomorphisms. The chapter defines integral models of local Shimura varieties as v-sheaves.
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/9780691202082.003.0024
- Subject:
- Mathematics, Geometry / Topology
This chapter specializes the theory back to the case of local Shimura varieties, and explains the relation with Rapoport-Zink spaces. It begins with a local Shimura datum. A local Shimura datum is a ...
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This chapter specializes the theory back to the case of local Shimura varieties, and explains the relation with Rapoport-Zink spaces. It begins with a local Shimura datum. A local Shimura datum is a triple (G, b, µ) consisting of a reductive group G over Qp, a conjugacy class µ of minuscule cocharacters. Rapoport-Zink spaces are moduli of deformations of a fixed p-divisible group. After reviewing these, the chapter shows that the diamond associated with the generic fiber of a Rapoport-Zink space is isomorphic to a moduli space of shtukas of the form with µ minuscule. It then extends the results to general EL and PEL data.Less
This chapter specializes the theory back to the case of local Shimura varieties, and explains the relation with Rapoport-Zink spaces. It begins with a local Shimura datum. A local Shimura datum is a triple (G, b, µ) consisting of a reductive group G over Qp, a conjugacy class µ of minuscule cocharacters. Rapoport-Zink spaces are moduli of deformations of a fixed p-divisible group. After reviewing these, the chapter shows that the diamond associated with the generic fiber of a Rapoport-Zink space is isomorphic to a moduli space of shtukas of the form with µ minuscule. It then extends the results to general EL and PEL data.
Peter Scholze and Jared Weinstein
- Published in print:
- 2020
- Published Online:
- January 2021
- ISBN:
- 9780691202082
- eISBN:
- 9780691202150
- Item type:
- book
- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.001.0001
- Subject:
- Mathematics, Geometry / Topology
This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory ...
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This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, the author introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. This book shows that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. The book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained.Less
This book presents an important breakthrough in arithmetic geometry. In 2014, this book's author delivered a series of lectures at the University of California, Berkeley, on new ideas in the theory of p-adic geometry. Building on his discovery of perfectoid spaces, the author introduced the concept of “diamonds,” which are to perfectoid spaces what algebraic spaces are to schemes. The introduction of diamonds, along with the development of a mixed-characteristic shtuka, set the stage for a critical advance in the discipline. This book shows that the moduli space of mixed-characteristic shtukas is a diamond, raising the possibility of using the cohomology of such spaces to attack the Langlands conjectures for a reductive group over a p-adic field. The book follows the informal style of the original Berkeley lectures, with one chapter per lecture. It explores p-adic and perfectoid spaces before laying out the newer theory of shtukas and their moduli spaces. Points of contact with other threads of the subject, including p-divisible groups, p-adic Hodge theory, and Rapoport-Zink spaces, are thoroughly explained.
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/9780691202082.003.0001
- Subject:
- Mathematics, Geometry / Topology
This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case ...
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This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.Less
This introductory chapter provides an overview of Drinfeld's work on the global Langlands correspondence over function fields. Whereas the global Langlands correspondence is largely open in the case of number fields K, it is a theorem for function fields, due to Drinfeld and L. Lafforgue. The key innovation in this case is Drinfeld's notion of an X-shtuka (or simply shtuka). The Langlands correspondence for X is obtained by studying moduli spaces of shtukas. A large part of this course is about the definition of perfectoid spaces and diamonds. There is an important special case where the moduli spaces of shtukas are classical rigid-analytic spaces. This is the case of local Shimura varieties. Some examples of these are the Rapoport-Zink spaces.
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/9780691202082.003.0018
- Subject:
- Mathematics, Geometry / Topology
This chapter explores v-sheaves associated with perfect and formal schemes. The more general formalism of v-sheaves makes it possible to consider not only analytic adic spaces as diamonds, but also ...
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This chapter explores v-sheaves associated with perfect and formal schemes. The more general formalism of v-sheaves makes it possible to consider not only analytic adic spaces as diamonds, but also certain non-analytic objects as v-sheaves. The chapter first analyzes the behavior on topological spaces. Let X be any pre-adic space over Zp. This is not a diamond, but the chapter shows that it is a v-sheaf. It assesses some properties of this construction. The chapter then looks at applications to local models and integral models of Rapoport-Zink spaces. By passage to the maximal unramified extension and Galois descent, one can assume that k is algebraically closed.Less
This chapter explores v-sheaves associated with perfect and formal schemes. The more general formalism of v-sheaves makes it possible to consider not only analytic adic spaces as diamonds, but also certain non-analytic objects as v-sheaves. The chapter first analyzes the behavior on topological spaces. Let X be any pre-adic space over Zp. This is not a diamond, but the chapter shows that it is a v-sheaf. It assesses some properties of this construction. The chapter then looks at applications to local models and integral models of Rapoport-Zink spaces. By passage to the maximal unramified extension and Galois descent, one can assume that k is algebraically closed.
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/9780691202082.003.0021
- Subject:
- Mathematics, Geometry / Topology
This chapter reviews affine flag varieties. It generalizes some of the previous results to the case where G over Zp is a parahoric group scheme. In fact, slightly more generally, it allows the case ...
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This chapter reviews affine flag varieties. It generalizes some of the previous results to the case where G over Zp is a parahoric group scheme. In fact, slightly more generally, it allows the case that the special fiber is not connected, with connected component of the identity G? being a parahoric group scheme. This case comes up naturally in the classical definition of Rapoport-Zink spaces. The chapter first discusses the Witt vector affine flag variety over Fp. This is an increasing union of perfections of quasiprojective varieties along closed immersions. In the case that G° is parahoric, one gets ind-properness.Less
This chapter reviews affine flag varieties. It generalizes some of the previous results to the case where G over Zp is a parahoric group scheme. In fact, slightly more generally, it allows the case that the special fiber is not connected, with connected component of the identity G? being a parahoric group scheme. This case comes up naturally in the classical definition of Rapoport-Zink spaces. The chapter first discusses the Witt vector affine flag variety over Fp. This is an increasing union of perfections of quasiprojective varieties along closed immersions. In the case that G° is parahoric, one gets ind-properness.