Introduction to p-adic Hodge Theory

Course Description

In this course, we will give an introduction to p-adic Hodge theory and we will focus on the geometric aspect of this theory.

Possible topics includes:

1 p-divisible groups and abelian varieties, Hodge-Tate decomposition for p-divisible groups, deformations of p-divisible groups.

2 p-adic representations, Fontaine’s period rings.

3 p-adic comparison theorems.


The choice of material will be based on the interests and prerequisites of the participants.


Berger, Laurent, An introduction to the theory of p-adic representations, Geometric aspects of Dwork theory, I, Berlin: Walter de Gruyter GmbH & Co. KG,

Brinon, Olivier; Conrad, Brian, CMI Summer School notes on p-adic Hodge theory

J-M. Fontaine, Groupes p-divisibles sur les corps locaux, Société Mathématique de France, Astérisque, No. 47-48, 1977.

J-M. Fontaine, Sur certains types de représentations p-adiques du groupe de Galois d’un corps local ; construction d’un anneau de Barsotti–Tate, Annals of Math. (2) 115 (1982), no. 3, pp. 529–577.

J-M. Fontaine, "Arithmétique des représentations galoisiennes p-adiques" in Cohomologies p-adiques et applications arithmétiques (III), Astérisque 295 (2004), pp. 1–115.

A. Grothendieck, Groupes de Barsotti-Tate et cristaux de Dieudonné, Sém. Math. Sup. 45, Presses de l'Université de Montréal, 1974.

L. Illusie, Déformations de groupes de Barsotti-Tate, d'après A. Grothendieck, in Séminaire sur les pinceaux arithmétiques : la conjecture de Mordell, Lucien Szpiro, Astérisque 127, SMF, 1985, 151-198.

J. Tate, "p-divisible groups” in Proc. Conf. on Local Fields (Driebergen), Springer-Verlag, 1967, pp. 148–183.

Course Meeting Time and Location
Tuesday and Thursday
2:30 - 3:55 pm
122 Math Building (Building 15)

Course Instructor Contact Information and Office Hours
Dr. Daxin Xu
210-4 Math Building (Building 15)

Course Schedule and Textbook

Course Policies

 Date PostedAssignment Due Date 

Midterm and Final Exam

Collaboration Table
You may consult:  
Course textbook (including answers in the back)YESYES
Other booksYESNO
Solution manualsNONO
Your notes (taken in class)YESYES
Class notes of othersYESNO
Your hand copies of class notes of othersYESYES
Photocopies of class notes of othersYESNO
Electronic copies of class notes of othersYESNO
Course handoutsYESYES
Your returned homework / examsYESYES
Solutions to homework / exams (posted on webpage)YESYES
Homework / exams of previous yearsNONO
Solutions to homework / exams of previous yearsNONO
Emails from TAsYESNO
You may:

Discuss problems with othersYESNO
Look at communal materials while writing up solutionsYESNO
Look at individual written work of othersNONO
Post about problems onlineNONO
For computational aids, you may use:


* You may use a computer or calculator while doing the homework, but may not refer to this as justification for your work.  For example, "by Mathematica" is not an acceptable justification for deriving one equation from another.  Also, since computers and calculators will not be allowed on the exams, it's best not to get too dependent on them.