Galois representations   TCC Fall 2016

Thursdays 10-12 from October 20 to December 1 (8 weeks)

Lecture notes

Lecture 1 - Riemann ζ, L-functions and Dedekind ζ and TCC boards
Lecture 2 - Dirichlet L-functions and cyclotomic fields and TCC boards
Lecture 3 - Artin representations and their L-functions and TCC boards
Lecture 4 - Dirichlet and Dedekind are Artin and TCC boards
Lecture 5 - Characters, induction, Artin formalism and TCC boards
Lecture 6 - Γ-factors, ε-factors, conductors and TCC boards
Lecture 7 - Compatible systems of l-adic representations and TCC boards
Lecture 8 - l-adic representations of elliptic curves and TCC boards

Full course notes in LaTeX by Emma Bailey


8 exercises set as course assignments - one each week. Please hand them in by emailing The seventh one is due by Thursday December 1 and the last one by Thursday December 8.

Course overview

In this course I would like to give an introduction to the theory of Galois representations and L-functions. My current plan, subject to reality, is to cover the following topics:

Galois representations and L-functions are a big subject, with links to modular forms (and Ariel Pacetti's course will touch on this), elliptic curves, étale cohomology, proof of Fermat's Last Theorem, algebraic groups, the Langlands program and what not. They are absolutely central to modern number theory, but this also makes them difficult to absorb - they rely on numerous neighbouring areas and the full theory requires a lot of background.

I will try my best to navigate through the area avoiding technicalities if possible, and concentrate on the main topics and how they link to one another. Nevertheless, just to be able to to talk about them and to present the most important examples, I will need

It is unreasonable to expect that you have seen all of this, and I hope you can enjoy the course without some of these topics. However, if you are willing to read about p-adic numbers and elliptic curves in advance, you will probably enjoy it more.


Questions/comments to Tim Dokchitser