Galois representations   TCC Spring 2025

Course overview

This is 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 (Samir Siksek's course), elliptic curves, étale cohomology, proof of Fermat's Last Theorem, algebraic groups, and the Langlands program. They are quite central to modern number theory, but this also makes them difficult to study - they rely on numerous neighbouring areas and the full theory requires a lot of background. Here is what the course will use:

• Galois theory
• Number fields (though I will not need unit groups or class groups)
• Representation theory of finite groups would be helpful
• Having seen elliptic curves (torsion and Tate module) would be good for the second part of the course
• p-adic numbers and their extensions would also help