Sommersemester 2008 - Galois Representations

• Dates: Monday 10-12 am at the Institut fuer Experimentelle Mathematik.
• This course will be taught in English.
• It will be accompanied by exercises.
• Current version of the lecture notes.

Course Description

This lecture gives an introduction to the theory of Galois representations. If time allows, it will consist of the following four main parts.

• The first part will be on general representation theory. Among other things, we will prove the Brauer-Nesbitt theorem and discuss fields of definition of Galois representations.
• A long chapter will be devoted to the local theory of Galois representations. For an l-adic and a mod l Galois representation there are huge differences between the local representation at p different from l and the one at l. We will define and discuss conductors for Artin representations and l-adic and mod l representations away from l. We will also introduce Weil-Deligne representations that serve to classify these. Moreover, also the local representation at l will be discussed and fundamental characters will be introduced. The goal of this chapter will be the precise formulation of Serre's conjecture.
• Another chapter will treat complex Galois representations. A great deal of it will focus on the 1-dimensional case. We will use those to give or sketch a proof of Chebotarev's density theorem. The remainder of the chapter will discuss how the Mellin transformation and its inverse allows to move between modular forms and their L-functions. Consequences for Artin's conjecture will be mentioned.
• If the time allows, in a final chapter we will sketch the construction of the Galois representation attached to a newform (at least in certain cases).

Prerequisites:

Basic knowledge of Algebraic number theory and class field theory. For the applications, knowledge about elliptic curves and modular forms is desirable but not strictly necessary.

Leistungsnachweis/certificate

A certificate (Leistungsnachweis) can be obtained by regularly and successfully solving the exercises and by passing an oral exam at the end of the term.

Perspectives

If you intend to write your Diplomarbeit, Master's thesis or Staatsarbeit in number theory, this is the lecture to attend. At the end of the lecture, subjects can be obtained.

Last modification: 4 April 2008.