0 Introduction

In this course, we shall first give an outline of class field theory. We then look at

abelian

L

-functions (Hecke, Tate). We then talk about non-abelian

L

-functions,

and in particular the Weil–Deligne group and local L- and ε-factors.

We then talk about local Langlands for

GL

n

a bit, and do a bit of global

theory and automorphic forms at the end.

The aim is not to prove everything, because that will take 3 courses instead of

one, but we are going to make precise definitions and statements of everything.