# 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.

### 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, 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.

See here

### Logistics (To be continuously updated thorughout the semester)

• When: Wednesdays, 4:20-5:50 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)
Jean-Marc 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 p-divisible 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 Hodge-Tate decomposition via perfectoid spaces, link

Fargues-Fontaine Curve
[JL01] Jacob Lurie - Lectures on the Fargues-Fontine Curve, link
[Far16] Laurent Fargues - Geometrization of the local Langlands correspondence: an overview, link
[FF12] Laurent Fargues and Jean-Marc Fontaine - Vector bundles and p-adic 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 Jean-Marc Fontaine - Vector bundles on curves and p-adic 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 Jean-Marc 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 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
Week 3 (02/08)
TBD
More on finite flat group schemes
This week, we will unpack more on finite flat group schemes. Finishing up where we have left last week on basic definitions and properties, we will talk more about Cartier duality, then we will introduce finite étale group schemes. After that, we will briefly talk about the connected-étale sequence and the Frobenius morphism
notes(to be updated by 02/05)
Week 4 (02/15)
Xiaorun Wu
TBD
TBD