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.
List of topics (plus lecture notes):