Galois representations   TCC Fall 2016

Assessment

8 exercises set as course assignments - one each week. Please hand them in by emailing TCCGalRep@gmail.com. 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:

• Riemann ζ-function and L-functions of Dirichlet characters
• Number fields and Artin representations
• L-functions of Artin representations as pieces of Dedekind ζ-functions
• Artin formalism
• Conductor
• l-adic representations coming from elliptic curves
• Good reduction and bad reduction
• Weil and Weil-Deligne representations (local case)
• Compatible systems of l-adic representations (global case)
• Links, big conjectures, what is known

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

• Galois theory (essential)
• Number fields (essential; though I will not need unit groups or class groups)
• Having seen elliptic curves would help
• p-adic numbers and their extensions would also help
• A bit of representation theory would be good (though not crucial)
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.