‎In the spirit of the Langlands proposal on  Beyond Endoscopy ‎we discuss the explicit relation between the Langlands functorial transfers and automorphic $L$-functions‎. ‎It is well-known that the poles of the $L$-functions have deep impact to the Langlands functoriality‎. ‎Our discussion also‎ ‎includes the meaning of the central value of the tensor product $L$-functions in terms of the Langl...

Introduction 1 1. The unramified global Langlands correspondence 5 2. Classical local Langlands correspondence 9 3. Geometric local Langlands correspondence over C 12 4. Center and opers 18 5. Opers vs. local systems 23 6. Harish–Chandra categories 26 7. Local Langlands correspondence: unramified case 30 8. Local Langlands correspondence: tamely ramified case 41 9. Ramified global Langlands cor...

This article is an introduction to the early life and work of Robert P. Langlands, creator and founder of the Langlands program. The story is, to a large extent, told by Langlands himself, in his own words. Our focus is on two of Langlands’ major discoveries: automorphic L-functions and the principle of functoriality. It was Langlands’ desire to communicate his excitement about his newly discov...

The Langlands program of number theory, or what we might call Langlands duality, was proposed in more or less its present form by Robert Langlands, in the late 1960s. It is a kind of unified scheme for many results in number theory ranging from quadratic reciprocity, which is hundreds of years old, to modern results such as Andrew Wiles’ proof of Fermat’s last theorem, which involved a sort of ...

The Langlands Program relates Galois representations and automorphic representations of reductive algebraic groups. The trace formula is a powerful tool in the study of this connection and the Langlands Functoriality Conjecture. After giving an introduction to the Langlands Program and its geometric version, which applies to curves over finite fields and over the complex field, I give a survey ...

‎We illustrate a method of computing Langlands-Shahidi local coefficients via the theory of types and covers‎. ‎The purpose of this paper is to illustrate a method of computing the Langlands-Shahidi local coefficients using the theory of types and covers.

This text gives an introduction to the Langlands correspondence for function fields and in particular to some recent works in this subject. We begin with a short historical account (all notions used below are recalled in the text). The Langlands correspondence [Lan70] is a conjecture of utmost importance, concerning global fields, i.e. number fields and function fields. Many excellent surveys a...

In the late 1960s Robert Langlands launched what has become known as the Langlands Program with the ambitious goal of relating deep questions in Number Theory to Harmonic Analysis [L]. In particular, Langlands conjectured that Galois representations and motives can be described in terms of the more tangible data of automorphic representations. A striking application of this general principle is...

In this essay I hope to explain the classification of characters of algebraic tori defined over R, particularly Langlands’ parametrization in terms of homomorphisms from the Weil group into the associated L-group. I’ll follow more or less the treatment in §§2 and 3 of [Langlands:1974/1989], avoiding most subsequent ‘improvements’. The only slightly new thing is how I relate Langlands’ formula f...

Part I. The origins of the Langlands Program 9 1. The Langlands correspondence over number fields 9 1.1. Galois group 9 1.2. Abelian class field theory 10 1.3. Frobenius automorphisms 13 1.4. Rigidifying ACFT 14 1.5. Non-abelian generalization? 15 1.6. Automorphic representations of GL2(AQ) and modular forms 18 1.7. Elliptic curves and Galois representations 22 2. From number fields to function...