Course:    p-adic Analytic Geometry with applications to Representation Theory.

Joseph Bernstein

Fall 2020

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Description of the course.

  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.




  Prerequisites.

What I assume is

(i) Good knowledge of linear algebra.

(ii) Basic Galois theory

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

(iv) Extensions of $p$-adic fields.  Ramified and unramified extensions.
    Elementary facts about Galois groups of $p$-adic field
    
      
   I will mostly use following two souses.

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.

  Probably in addition I will use some parts of other expositions 
of $p$-adic Hodge Theory.

   This is a learning course. I will try not to assume that 
participants know too many things, and will try to give short 
descriptions of notions and theories that we will have to use.
  Probably there will be some some remedial sessions about these 
topics.

   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.

URL: http://www.wisdom.weizmann.ac.il/~dimagur/padicAG.html