**Course description:**
This course is an introduction to *p*-adic Hodge theory, the study of *p*-adic Galois representations of *p*-adic fields. This is motivated by the relationship between de Rham cohomology and *p*-adic etale cohomology for algebraic varieties over *p*-adic fields.

After a quick introduction to local fields, we will work closely through the Clay Mathematics Institute Summer School lecture notes of Brinon and Conrad (starting with the 2009 version, plus updates as these become available). Supplemental lecture notes may be provided on certain topics (perfectoid fields and algebras, overconvergence of *p*-adic representations, the de Rham comparison isomorphism).

Note: there are two Math 205 courses running this term. These are deliberately complementary: this course focuses on local fields, while Claus Sorensen's course on class field theory will emphasize global fields.

**Instructor:** Kiran Kedlaya,
kedlaya [at] math [etcetera].

**Lectures:** TTh 10-11:30, APM 5829. Course meeting time subject to change by mutual agreement. (Makeup lectures may be scheduled differently.)

**Office hours:** W 1-2, APM 7402. Appointments can also be made by email.

**Textbook:** No required text. We will be following
Brian Conrad's Clay Summer School lecture notes, starting with the 2009 version (updated versions may appear later).

I will also post some supplemental lecture notes below. For example, at some point we will go through my paper New methods for (phi, Gamma)-modules.

A supplemental reference that may be useful (and the only textbook I am aware of on this subject) is:
Jean-Marc Fontaine and Yi Ouyang, Theory of *p*-adic Galois
representations, Springer, 2013.

**Prerequisites:** Math 200A-C or equivalent (which must be approved by the instructor), plus some background in algebraic number theory.

**Grading:** If you need a meaningful grade (e.g., if you are an undergraduate) then let me know at the start of the course.

**Announcements:**

- I've updated the posted version of "New methods for (phi, Gamma)-modules" based on my lectures. If you find any further typos or confusing points, please let me know!
- Makeup lecture Friday, March 21, 12 PM in APM 2402.

**List of topics (plus lecture notes):**

- Overview of the course.
- Some basics about local fields.
- Hodge-Tate representations.
- Etale phi-modules and Galois representations in positive characteristic.
- Witt vectors and associated constructions.
- Admissible representations, de Rham representations.
- Filtered (phi, N)-modules, crystalline and semistable representations.
- Integral p-adic Hodge theory.
- (phi, Gamma)-modules.
- Overconvergence of (phi, Gamma)-modules.
- p-adic differential equations and p-adic local monodromy.
- Additional topics if time permits.