*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/9780691202082.003.0012
- Subject:
- Mathematics, Geometry / Topology

This chapter analyzes shtukas with one leg over a geometric point in detail, and discusses the relation to (integral) p-adic Hodge theory. It focuses on the connection between shtukas with one leg ...
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This chapter analyzes shtukas with one leg over a geometric point in detail, and discusses the relation to (integral) p-adic Hodge theory. It focuses on the connection between shtukas with one leg and p-divisible groups, and recovers a result of Fargues which states that p-divisible groups are equivalent to one-legged shtukas of a certain kind. In fact this is a special case of a much more general connection between shtukas with one leg and proper smooth (formal) schemes. Throughout, the goal is to fix an algebraically closed nonarchimedean field. The chapter then provides an overview of shtukas with one leg and p-divisible groups.Less

This chapter analyzes shtukas with one leg over a geometric point in detail, and discusses the relation to (integral) *p*-adic Hodge theory. It focuses on the connection between shtukas with one leg and *p*-divisible groups, and recovers a result of Fargues which states that *p*-divisible groups are equivalent to one-legged shtukas of a certain kind. In fact this is a special case of a much more general connection between shtukas with one leg and proper smooth (formal) schemes. Throughout, the goal is to fix an algebraically closed nonarchimedean field. The chapter then provides an overview of shtukas with one leg and *p*-divisible groups.

*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 **Q**p, 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:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691202082.003.0017
- Subject:
- Mathematics, Geometry / Topology

This chapter describes the v-topology. It develops a powerful technique for proving results about diamonds. There is a topology even finer than the pro-étale topology, the v-topology, which is ...
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This chapter describes the v-topology. It develops a powerful technique for proving results about diamonds. There is a topology even finer than the pro-étale topology, the v-topology, which is reminiscent of the fpqc topology on schemes but which is more “topological” in nature. The class of v-covers is extremely general, which will reduce many proofs to very simple base cases. The chapter provides a sample application of this philosophy by establishing a general classification of p-divisible groups over integral perfectoid rings in terms of Breuil-Kisin-Fargues modules. Another use of the v-topology is to prove that certain pro-étale sheaves on Perf are diamonds without finding an explicit pro-étale cover.Less

This chapter describes the v-topology. It develops a powerful technique for proving results about diamonds. There is a topology even finer than the pro-étale topology, the v-topology, which is reminiscent of the fpqc topology on schemes but which is more “topological” in nature. The class of v-covers is extremely general, which will reduce many proofs to very simple base cases. The chapter provides a sample application of this philosophy by establishing a general classification of *p*-divisible groups over integral perfectoid rings in terms of Breuil-Kisin-Fargues modules. Another use of the v-topology is to prove that certain pro-étale sheaves on Perf are diamonds without finding an explicit pro-étale cover.

*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.