Seminaire Fontaine

Hi! Welcome to the webpage for the Spring 2025 Seminar on p-adic Hodge Theory.

Since this is top secret, feel free to share this webpage with anyone who might be interested!

We’ll meet every Tuesday from 2:00–3:15 pm in 4N49. The main purpose of this seminar is to force me to study p-adic Hodge theory, so I plan to give most of the talks myself. However, if you’d like to present on any of the listed topics or related subjects, your contribution would be greatly appreciated! 

This semester, we’ll cover the following topics:

• Motivation and overview of p-adic Hodge theory

• Finite group schemes and p-divisible groups

• Proof of the Hodge-Tate decomposition for p-divisible groups

• Dieudonné modules

• Construction of period rings

• Basics of $(\phi, \Gamma)$-modules

• Proof of comparison theorems for p-divisible groups

• Additional topics (time permitting): the Fargues-Fontaine curve, crystalline cohomology, etc.

Week 1 and Week 2: Overview of p-adic Hodge theory, introduction to finite flat group schemes.
Notes for week 1 and 2

Week 3: Finite flat group schemes

Week 4: p-divisible groups and formal Lie groups

Week 5: Serre-Tate Theorem

Week 6: Logarithm and pairing

Week 7: Proof of the Hodge-Tate decomposition theorem for p-divisible groups

Notes for week 3 to 7

Week 8: Spring Break

Week 9: (\phi, \Gamma)-modules

Week 10: Perfectoid fields and the tilting equivalence 

Week 11: Construction & properties of B_dR

Week 12: de Rham representations

Week 13: Crystalline representations

Week 14: Semistable representations

Notes for week 9 to 14

Week 15: The Fargue-Fontaine Curve

Notes for week 15

End of the semester!

References

Overview

[Be01] An introduction to the theory of p-adic Galois representations, Laurent Berger

[Ni19] Hodge theory of p-adic varieties: a survey, Wieslawa Niziol

Comprehensive references

[BC09] CMI summer school notes on p adic Hodge theory, Olivier Brinon and Brian Conrad 

[Be22] An introduction to p adic Hodge theory, Denis Benois

Finite flat group schemes and basics of p-divisivle groups

[Se68] Groupes p-divisibles (d'apres J. Tate), Jean-Pierre Serre

[St09] A course on finite flat group schemes and p-divisible groups, Jakob Stix

[Ta66] p-divisible groups, John Tate

[Yo18] p-divisible groups, formal groups, and the Serre-Tate theorem, Alex Youcis

Dieudonné Modules

[De72] Lectures on p-divisible groups, Michel Demazure

[Pi05] Finite group schemes, Richard Pink 

Comparison Theorems, Grothendieck-Messing Theory

[Gr70] Groupes de Barsotti–Tate et cristaux de Dieudonne, Alexander Grothendieck. English Translation by Eric Peterson

[Me72] The crystals associated to Barsotti-Tate groups, William Messing

[Pa21] Crystalline comparison theorem for p-divisible groups, Andrea Panonti  

[Zh17] Comparison theorems for p-divisible groups, Luming Zhao

Fargue-Fontaine Curve

[dS22] The Fargue-Fontaine curve and p-adic Hodge Theory, Ehud de Shalit

[Hong] Notes on p-adic Hodge Theory, Serin Hong

[Mo18] The Fargue-Fontaine curve and diamonds, Matthew Morrow