# Course Identification

p-adic Analytic Geometry with applications to Representation Theory
20214111

## Lecturers and Teaching Assistants

Prof. Dmitry Gourevitch, Prof. Joseph Bernstein
N/A

## Course Schedule and Location

2021
First Semester
Monday, 14:15 - 16:00
26/10/2020

## Field of Study, Course Type and Credit Points

Mathematics and Computer Science: Lecture; Elective; Regular; 3.00 points

N/A

## 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.

30

English

## Attendance and participation

Expected and Recommended

Pass / Fail

100%

Take-home exam

N/A
N/A
-
N/A

3

## 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.

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.