Seminar: \(p\)adic Hodge Theory & Arithmetic Geometry (Spring 2023)
Topic: The FarguesFontaine Curve and Application To \(p\)adic representations
Let \(C\) be an algebraically closed perfectoid field over \(\mathbb F_p\). The FarguesFontaine 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 FarguesFontaine 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.
Plan:
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, HodgeTate 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 FarguesFontaine curve: I will outline the construction of both schematic and adic FarguesFontaine 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)

When: Wednesdays, 4:205:50 PM ET

Where: Room 622

Organizer: Xiaorun Wu

References:

Main References

Serin Hong, Course Notes on \(p\)adic Hodge Theory (MATH 679, University of Michigan, Spring 2020)

JeanMarc Fontaine and Yi Ouyang, Theory of \(p\)adic Hodge Theory

Oliver Brionon and Brian Conrad, CMI Summer School Notes on \(p\)adic Hodge Theory

\(p\)adic Hodge Theory

[Dem86] Michel Demazure  Lectures on pdivisible groups, Lecture Notes in Mathematics, vol. 302, SpringerVerlag, Berlin, 1986, Reprint of the 1972 original. MR 883960

[HT01] Michael Harris and Richard Taylor  The geometry and cohomology of some simple Shimura varieties, , Annals of Mathematics Studies, vol. 151, Princeton University Press, Princeton, NJ, 2001, With an appendix by Vladimir G. Berkovich. MR 1876802

[Pin04] Richard Pink  Finite Group Schemes, link

[Ber08] Laurent Berger  Construction of \((\phi, \gamma)\)modules: \(p\)adic representations and \(b\)pairs, Algebra & Number Theory 2 (2008), no. 1, 91–120

[Bha] Bhagav Bhatt  The HodgeTate decomposition via perfectoid spaces, link

FarguesFontaine Curve

[JL01] Jacob Lurie  Lectures on the FarguesFontine Curve, link

[Far16] Laurent Fargues  Geometrization of the local Langlands correspondence: an overview, link

[FF12] Laurent Fargues and JeanMarc Fontaine  Vector bundles and padic Galois representations, Fifth International Congress of Chinese Mathematicians, AMS/IP Studies in Advanced Mathematics, vol. 51, Cambridge Univ. Press, Cambridge, 2012, pp. 77–114.

[FF14] Laurent Fargues and JeanMarc Fontaine  Vector bundles on curves and padic Hodge theory, , Automorphic forms and Galois representations. Vol. 2, London Math. Soc. Lecture Note Ser., vol. 415, Cambridge Univ. Press, Cambridge, 2014, pp. 17–104

[FF18] Laurent Fargues and JeanMarc Fontaine  , Courbes et fibrés vectoriels en théorie de Hodge \(p\)adique, , Astérisque 406 (2018).
Schedule

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 FarguesFontaine 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 0202)

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 Connectedétale 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 SerreTate equivalence for connected \(p\)divisible groups. If time allows, we will discuss about DieudonnéManin classification.

notes

Week 5 (02/22)

Xiaorun Wu

HodgeTate Decomposition (Part I)

This week, we will finish DieudonnéManin classification. Then we will talk about HodgeTate decomposition. We will talk about the completed algebraic closure of a \(p\)adic field.

notes

Week 6 (03/01)

Xiaorun Wu

HodgeTate 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 DieudonnéManin classification. Then we will talk about HodgeTate decomposition.

notes

Week 7 (03/08)

Xiaorun Wu

HodgeTate Decomposition (Part III)

This week, we will finish our discussion on HodgeTate 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.

notes

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 sectionwe 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.

notes

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 \(B_{dR}\).
 notes

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.
 notes

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.
 notes

Week 12 (04/19)

Xiaorun Wu

Crystalline representations (III) & FarguesFontaine 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 FarguesFontaine curve.
 notes

Week 13 (04/26)

Xiaorun Wu

FarguesFontaine curveintroduction

This week we will continue on discussion of FarguesFontaine curve. First we will talk about untilts of a perfectoid field, then we will briefly discuss about the schematic FarguesFontaine curve. If time allows, we will talk about the adic FarguesFontaine curve.
 notes

Week 14 (05/01)

Xiaorun Wu

FarguesFontaine curve

We conclude this semester of seminar by talking about the adic FarguesFontaine 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)