# School - Perfectoids

## Schedule :

### Monday

09h00-10h15 : O. Brinon, Perfectoid rings 1

10h30-10h45 : Coffee break

10h45-12h00 : A. Vezzani, Adic Spaces 1

12:30 : Lunch

14h00-15h15 : O. Brinon, Perfectoid rings 2

15:30-15h45 : Coffee break

15h45-17h00 : A. Vezzani, Adic Spaces 2

### Tuesday

09h00-10h15 : A. Vezzani, Adic Spaces 3

10:30-10h45 : Coffee break

10h45-12h00 : O. Brinon, Perfectoid rings 3

12:30 : Lunch

14h00-15h15 : A. Vezzani, Adic Spaces 4

15:30-15h45 : Coffee break

15h45-17h00 : O. Brinon, Perfectoid rings 4

### Wednesday

09h00-10h15 : M. Morrow, Perfectoid spaces 1

10:30-10h45 : Coffee break

10h45-12h00 : M. Morrow, Perfectoid spaces 2

12h30-18h00 : Visit of Saint-Malo

### Thursday

9h00-10h15 : M. Morrow, Perfectoid spaces 3

10:30-10h45 : Coffee break

10h45-12h00 : M. Morrow, Perfectoid spaces 4

12:30 : Lunch

14h00-15h00 : J. Anschütz, Prismatic Dieudonné theory I

15:15-15h45 : Coffee break

15h30-16h30 : A.-C. Le Bras, Prismatic Dieudonné theory II

16h45-17h45 : J. Ludwig, Perfectoid Shimura varieties and applications

*20h00-22h:00 : Conference Dinner*

### Friday

9h30-10h30 : K. Cesnavicius, Purity for flat cohomology

10:30-11:00 : Coffee break

11h00-12h00: A. Caraiani, Vanishing theorems for Shimura varieties at unipotent level

12:30 : Lunch

## Abstracts :

Johannes ANSCHÜTZ

Title: Prismatic Dieudonné theory I

Abstract: This is the first of two talks on prismatic Dieudonné theory.
After recalling crystalline Dieudonné theory, we will give a short
presentation of the theory of prisms and prismatic cohomology as
developed by Bhatt/Scholze. In the second talk (given by Arthur-César Le
Bras) this formalism will be used to develop prismatic Dieudonné theory.

Olivier BRINON

Mini-course: Perfectoid rings

Abstract: As an introduction to Morrow's lectures, I will define perfectoid Tate rings, give their basic properties and explain the tilting correspondance. I will also state the almost purity theorem, and explain the case of perfectoid fields.

Ana CARAIANI

Title: Vanishing theorems for Shimura varieties at unipotent level

Abstract: I will discuss a vanishing theorem for the compactly supported cohomology of a Shimura variety of Hodge type at unipotent level, assuming that the corresponding group splits over Q_p. This is based on joint work with Daniel Gulotta and Christian Johansson.

Kęstutis ČESNAVIČIUS

Title: Purity for flat cohomology

Abstract: The absolute cohomological purity for étale cohomology of Gabber--Thomasson implies that an étale cohomology class on a regular scheme extends uniquely over a closed subscheme of large codimension. I will discuss the corresponding phenomenon for flat cohomology. The talk is based on joint work with Peter Scholze.

Arthur-César LE BRAS

Title: Prismatic Dieudonné theory II

Abstract: This is the second of two talks on prismatic Dieudonné theory.
We will use the theory of prisms and prismatic cohomology as presented
in the first talk to develop prismatic Dieudonné theory over
quasi-syntomic rings. Under some mild assumption, we obtain
classification results for p-divisible groups over such rings. In the
end we will explain how to recover known results of Breuil, Kisin and
Lau for rings of integers in p-adic fields.

Judith LUDVIG

Title: Perfectoid Shimura varieties and applications

Abstract: Consider a tower of Shimura varieties, where the level at a fixed prime p grows. Scholze has shown that in many cases the limit carries the structure of a perfectoid space. I will explain this result focussing mostly on the modular curve case. We will then discuss an application to the mod p local Langlands correspondence of GL(n) over a p-adic field based on joint work with Christian Johansson.

Matthew MORROW

Mini-course: Adic Spaces

Abstract: Continuing Brinon’s talks, I will present a proof of the tilting correspondence and almost purity theorem for perfectoid Tate algebras. This will require first using the theory of adic spaces from Vezzani’s talks to introduce and study perfectoid spaces. Then we will give an introduction to the Fargues—Fontaine curve.

Alberto VEZZANI

Mini-course: Adic Spaces

Abstract: We introduce the basics of the theory of Adic Spaces introduced by Huber, including definitions of affinoid fields and rings, criteria for sheafyness and properties of the analytic and étale topologies. We give explicit examples and connect it to the theory of formal schemes, Tate's rigid analytic varieties and Berkovich's spaces.