The Langlands Program: Notes, Day I

These are notes for the first of a two-day series of lectures introducing graduate students to (parts of) the Langlands Program, delivered in the Building Bridges: 2nd EU/US Summer School on Automorphic Forms and Related Topics, July 2014. Introduction: The Big Picture There are two different kinds of number theoretic L-functions that have been studied extensively. The first are Artin L-functions. Let F be a number field, and GF be the Galois group of an algebraic closure F̄ of F over F . A Galois representation is a continuous homomorphism ρ : GF → Aut(V ) where V is a finite dimensional complex vector space. Here continuous means there exists a finite Galois extension K/F such that ρ factors through the finite Galois group Gal(K/F ). For each unramified prime ideal p of F , there is a conjugacy class Frp, the Frobenius class, in Gal(K/F ) that determines how p factors in K. (If p is ramified then one gets a class modulo the inertia subgroup Ip.) Then Artin defined the L-function, given as an infinite product absolutely convergent for <(s) > 1: