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 ...

Introduction 2 1. Harmonic analysis and unitary duals 2 2. Non-discrete locally compact fields, classical groups, reductive groups 4 3. K0-finite vectors 7 4. Smooth representations 8 5. Parabolically induced representations 9 6. Jacquet modules 12 7. Filtrations of Jacquet modules 14 8. Square integrable and tempered representations 15 9. Langlands classification 16 10. Geometric lemma and alg...

Let F be a totally real number field. We define global L-packets for GSp(2) over F which should correspond to the elliptic tempered admissible homomorphisms from the conjectural Langlands group of F to the L-group of GSp(2) which are reducible, or irreducible and induced from a totally real quadratic extension of F . We prove that the elements of these global L-packets occur in the space of cus...

Using the theta correspondence, we extend classification of irreducible representations quasi-split unitary groups (the so-called local Langlands which is due to Mok) non groups. We also prove that our satisfies some good properties, characterize it uniquely. In particular, this paper provides an alternative approach works Kaletha-Mínguez-Shin-White and Mœglin-Renard.

We carry out the first nontrivial cases of the limiting process proposed by Langlands in his manuscript Beyond Endoscopy, with technical variations that enable us to treat the limit unconditionally. This gives an elementary proof, on GL(2), of the classification of forms such that the symmetric square L-function has a pole (including, implicitly, the construction of these forms). The result of ...

Inside the universal deformation space of a local Galois representation one has the set of deformations which are potentially semi-stable of given p-adic Hodge and Galois type. It turns out these points cut out a closed subspace of the deformation space. A deep conjecture due to Breuil-Mézard predicts that part of the structure of this space can be described in terms of the local Langlands corr...

Introduction. These notes are based on a series of lectures given by the author at the Central China Normal University in Wuhan (July 2007). The aim of the lectures was to provide an introduction to the Langlands philosophy. According to a theorem of Chevalley, connected reductive split algebraic groups over a fixed field A are in bijection with certain combinatorial objects called root systems...

We give a short introduction to Beyond Endoscopy, a proposal by Langlands for attacking the general principle of functoriality. We shall try to motivate the proposal by emphasizing its structural similarities with the actual theory of endoscopy. We shall then discuss a few of the many problems that will need to be solved, some of which are suggested by the recent work of A. Altuğ . Mathematics ...