# Course Identification

## Lecturers and Teaching Assistants

## Course Schedule and Location

## Field of Study, Course Type and Credit Points

## Comments

## Prerequisites

(i) Good knowledge of linear algebra.

(ii) Basic Galois theory

(ii) $p$-adic fields and their topology.

(iii) Extensions of $p$-adic fields. Ramified and unramified extensions.

Elementary facts about Galois groups of $p$-adic field

## Restrictions

## Language of Instruction

## Attendance and participation

## Grade Type

## Grade Breakdown (in %)

## Evaluation Type

**Take-home exam**

## Scheduled date 1

## Estimated Weekly Independent Workload (in hours)

## Syllabus

During the last decade it became clear that methods of p-adic Analytic Geometry

play more and more important role in Representation Theory, including Langlands’ program.

These methods produce very powerful new tools to construct and study the representations,

and I have a filling that we have to learn these tools to continue to work in Representation Theory.

The p-adic analytic theory by now is very rich and highly developed theory.

In this course I will try to describe basic notions and results of this theory. In my exposition I will

try to emphasize how this theory is related to the Representation Theory, but I am not sure

that I will have time to describe this relation in some details.

There are two main directions in the development of this theory.

1. Rigid Analytic Geometry over p-adic numbers.

This is a p-adic analogue of the theory of complex manifolds.

2. p-adic Hodge Theory.

This is an analogue of complex Hodge theory.

Of course these two directions are highly intertwined.

In my course I will mostly discuss the Hodge theory. It is closely related to the theory of representations of Galois groups.

I think that proper understanding of this relation gives a new insight into the standard theory of representations of p-adic groups.

In my lectures I will try to formulate most of concepts and results that I need in the lectures.

However, since the material of the course is rather advanced, some preliminary knowledge of many of these topics will be very helpful.

## Learning Outcomes

I hope that as a result of this course the participants will be able to read the literature on $p$-adic Hodge Theory and apply it to some problems in Representation Theory.

## Reading List

1. Laurent Berger, An introduction to the theory of p-adic representations.

2. OLIVIER __BRINON__ AND BRIAN CONRAD,

__CMI__ SUMMER SCHOOL NOTES ON p-__ADIC__ HODGE THEORY.