Seminar: \(p\)-adic Hodge Theory & Arithmetic Geometry (Spring 2023)

Topic: The Fargues-Fontaine Curve and Application To \(p\)-adic representations

Let \(C\) be an algebraically closed perfectoid field over \(\mathbb F_p\). The Fargues-Fontaine curve \(X_C\) is a complete algebraic curve whose closed points parametrize the untilts of \(C\). In this seminar, we will detail the construction of Fargues-Fontaine curve, outline some key features of it and its relation to the theory of perfectoid spaces. Finally, we will discuss the Geometrization of \(p\)-adic Galois representations. I will start from basic definitions and foundations on the topic, hence people new to \(p\)-adic Hodge theory are welcome to participate.


In the first half of the semester, I will spend a dedicated amount of time going through the basics of \(p\)-adic Hodge Theory: in particular, I will be going through finite flat group schemes, \(p\)-divisible groups, Hodge-Tate decompositions, Fontaine's formalism on period rings, de Rham representations, crystalline representations, the plan is to spend roughly 7 weeks on the basics.

In the second half of the semester, I will be spending more time discussing the Fargues-Fontaine curve: I will outline the construction of both schematic and adic Fargues-Fontaine curve, then talk about the geometric structure: I will plan to spend roughly 6 weeks on the topic.

Tentative Syllabus:

See here

Logistics (To be continuously updated thorughout the semester)


Week 0 (01/18)
Xiaorun Wu
Logistics & Introduction
We had a short session today to go through the general logistics, as well as updated our syllabus with some new topcis proposed. For a detailed memo, please see below. The actual seminar begins next week (see the description for next week).
Meeting Memo
Week 1 (01/25)
Xiaorun Wu
Overview of \(p\)-adic Hodge Theory
In our first meeting, I would be providing a brief overview of what would be covered in this semester: first, I would be talking about the arithmetic and the geometric perspective of \(p\)-adic Hodge Theory, and the interplay between the two via the representation theory.
notes here
Week 2 (02/01)
Xiaorun Wu
Overview of Fargues–Fontaine curve, finite flat group schemes
Picking up from last time, we will first elaborate on Grothendieck mysterious functor, and discuss how Fontaine's formalism serve as a general framework for connecting the study of geometry of a proper smooth variety over a \(p\)-adic field and the construction of a dictionary that relates certain \(p\)-adic representation to various semilinear algebraic objects. Then I would provide a first glimpse into Fargues-Fontaine curve, which serves as the fundamental curve of \(p\)-adic Hodge Theory. I would briefly outline its construction, highlight some key features, discuss its relation with the perfectoid spaces, and mention about the geometrization if \(p\)-adic Galois representations. From this, we will see why this object plays a pivotal role in modern \(p\)-adic Hodge Theory. Finally, I will provide a brief introduction to finite flat group schemes, which we will elaborate next week.
notes here(updated 02-02)
Week 3 (02/08)
Hung Chiang
Finite flat group schemes
This week, we will be talking about finite flat group schemes. We will first offer basic definitions of group scheme, affine group schemes, and also finite flat group schemes. After that, we will go thorugh the definition of Cartier duality, together with some examples. Then, we will talk about Connected-étale sequence, and Deligne's Theorem. If time allows, we will talk about Frobenius Morphism.
notes (special thanks to Hung for it)
notes II
Week 4 (02/15)
Xiaorun Wu
\(p\)-divisible groups
Picking up from last week, we will first be talking about Frobenius morphism. Then we will be talking about \(p\)-divisible groups. First, we will offer general definitions and discuss some basic propeties associated with what was being covered last week. Then, we will be talking about Serre-Tate equivalence for connected \(p\)-divisible groups. If time allows, we will discuss about Dieudonné-Manin classification.
Week 5 (02/22)
Xiaorun Wu
Hodge-Tate Decomposition (Part I)
This week, we will finish Dieudonné-Manin classification. Then we will talk about Hodge-Tate decomposition. We will talk about the completed algebraic closure of a \(p\)-adic field.
Week 6 (03/01)
Xiaorun Wu
Hodge-Tate Decomposition (Part II)
This week, we will catch up a little on the progress of what we left last time. We will finish up the Dieudonné-Manin classification. Then we will talk about Hodge-Tate decomposition.
Week 7 (03/08)
Xiaorun Wu
Hodge-Tate Decomposition (Part III)
This week, we will finish our discussion on Hodge-Tate Decomposition, and talk a little bit more about generic fibers of \(p\)-divisible groups. If time allows, we will get started on period rings and functors. In particular, we will talk a little bit about Fontaine's formalism on period rings.
Week 8 (03/22)
Hung Chiang
Fontaine's formalism on period rings (Continue'd)
This week, we will finish our discussion on generic fibers of \(p\)-divisible groups, which contains the second most important theorem of this section--we will also shed some light on a fundamental theorem which provides a classification of \(p\)-divisible groups over \(\mathcal O_K\) when \(K\) is unramified over \(\mathbb Q_p\). Next, we will give the basic definition of Fontain's formalism on period rings, and offer a few examples.
Week 9 (03/31)
Xiaorun Wu
Formal properties of admissible representations, introduction to de Rham Representations
This week, we will discuss some of the formal properties of admissible representations. Then we will briefly switch gear, and talk about perfectoid fields and tilting. If time allows, this would lead to a discussion of de Rham period ring \(B_{dR}\).
Week 10 (04/05)
Xiaorun Wu
Filtered vector spaces, properties and de Rham representataion, & introduction to crystalline representations
Continue on last week, we will first discuss on de Rham period ring, showing that \(B_{dR}^+\) is a discrete valuation ring and that \(B_{dR}\) is the fraction field of \(B_{dR}^+\). We will then talk about \(B_{dR}\)-admissible representations. If time permits, we will give a brief introduction to the crystalline period ring.
Week 11 (04/12)
Xiaorun Wu
Crystalline representations
We first finish off our discussion on de Rham representations. Then we will define and study the crystalline period ring and crystalline repesentations. Towards the end of the this week's talk, we will define and discuss results about Frobenius automorphism, and state the fundamental exact sequence of \(p\)-adic hodge theory.
Week 12 (04/19)
Xiaorun Wu
Crystalline representations (III) & Fargues-Fontaine curve
We will continue our discussion on crystalline representations, proving similar results for \(B_{cris}\) as we did for the case of \(B_{dR}\). Towards the end of the section, we will see an important example on the Tate curve \(E_p\) is an elliptic curve over \(K\). If time allows, we will briefly mention about the untils of a perfectoid field, which sets up the discussion of Fargues-Fontaine curve.
Week 13 (04/26)
Xiaorun Wu
Fargues-Fontaine curve-introduction
This week we will continue on discussion of Fargues-Fontaine curve. First we will talk about untilts of a perfectoid field, then we will briefly discuss about the schematic Fargues-Fontaine curve. If time allows, we will talk about the adic Fargues-Fontaine curve.
Week 14 (05/01)
Xiaorun Wu
Fargues-Fontaine curve
We conclude this semester of seminar by talking about the adic Fargues-Fontaine curve. We will make some outline on future steps to be discussed, or potential areas of research interest related to the subject.
(there would be no notes this week)