Programme de Langlands . Ce programme souhaite relier la th. La correspondance entre ces diff.
De plus, en se donnant un tel groupe G, Langlands construit un groupe de Lie. LG, puis, pour chaque repr. Une de ses conjectures pr. Cette conjecture de fonctorialit. Cette construction, d'une nature analogue .
An Introduction to the Langlands Program. Elementary Theory of L. Informal Introduction to Geometric Langlands. Thomas Hales, Introduction to the Langlands Program and the Fundamental Lemma I.
Gelbart, An Introduction to the Langlands Program. Home ยป Elementary Introduction to the Langlands Program, by Edward Frenkel Workshop Elementary Introduction to the Langlands Program, by Edward Frenkel September 21.
- CiteSeerX - Scientific documents that cite the following paper: An elementary introduction to the Langlands program.
- FIRST STEPS IN THE LANGLANDS PROGRAM The Langlands program is a series of conjectures which aim to build a bridge between two different worlds.
- Stephen Gelbart: An Elementary Introduction to the Langlands Program, Bulletin of the AMS v.10 no. Edward Frenkel: 'Langlands program.
- An elementary introduction to the Langlands program. An elementary introduction to the Langlands program. Introduction aux travaux de Selberg.
- Special Lectures 'Elementary Introduction to the Langlands Program' This is the raw footage of my four lectures, aimed at the beginners, that were filmed at MSRI.
Les tentatives pour obtenir une construction directe n'ont produit que des r. Elle devint beaucoup plus technique pour les groupes de Lie plus grands, parce que les sous- groupes paraboliques (en) sont plus nombreux. Et, du c. Laurent Lafforgue a re. Ce travail prolongeait les recherches men. Seuls des cas particuliers concernant les corps de nombres ont . Laurent Lafforgue (en 2.
Langlands program - Wikipedia, the free encyclopedia. In mathematics, the Langlands program is a web of far- reaching and influential conjectures that relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles.
It was proposed by Robert Langlands (1. Background. Therefore, once the role of some low- dimensional Lie groups such as GL(2) in the theory of modular forms had been recognised, and with hindsight GL(1) in class field theory, the way was open at least to speculation about GL(n) for general n > 2. The cusp form idea came out of the cusps on modular curves but also had a meaning visible in spectral theory as 'discrete spectrum', contrasted with the 'continuous spectrum' from Eisenstein series. It becomes much more technical for bigger Lie groups, because the parabolic subgroups are more numerous.
In all these approaches there was no shortage of technical methods, often inductive in nature and based on Levi decompositions amongst other matters, but the field was and is very demanding. There are many different groups over many different fields for which they can be stated, and for each field there are several different versions of the conjectures. Moreover, the Langlands conjectures have evolved since Langlands first stated them in 1.
There are different types of objects for which the Langlands conjectures can be stated: Representations of reductive groups over local fields (with different subcases corresponding to archimedean local fields, p- adic local fields, and completions of function fields)Automorphic forms on reductive groups over global fields (with subcases corresponding to number fields or function fields). Finite fields. Langlands did not originally consider this case, but his conjectures have analogues for it. More general fields, such as function fields over the complex numbers. Conjectures. The Artin reciprocity law applies to a Galois extension of algebraic number fields whose Galois group is abelian, assigns L- functions to the one- dimensional representations of this Galois group; and states that these L- functions are identical to certain Dirichlet L- series or more general series (that is, certain analogues of the Riemann zeta function) constructed from Hecke characters. The precise correspondence between these different kinds of L- functions constitutes Artin's reciprocity law.
For non- abelian Galois groups and higher- dimensional representations of them, one can still define L- functions in a natural way: Artin L- functions. The insight of Langlands was to find the proper generalization of Dirichlet L- functions, which would allow the formulation of Artin's statement in this more general setting. Automorphic forms.
Langlands then generalized these to automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the general linear group GL(n) over the adele ring of Q. This is known as his . There are numerous variations of this, in part because the definitions of Langlands group and L- group are not fixed. Over local fields this is expected to give a parameterization of L- packets of admissible irreducible representations of a reductive group over the local field. For example, over the real numbers, this correspondence is the Langlands classification of representations of real reductive groups.
Over global fields, it should give a parameterization of automorphic forms. Functoriality. Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial. Generalized functoriality. Furthermore, given such a group G, Langlands constructs the Langlands dual group LG, and then, for every automorphic cuspidal representation of G and every finite- dimensional representation of LG, he defines an L- function. One of his conjectures states that these L- functions satisfy a certain functional equation generalizing those of other known L- functions. He then goes on to formulate a very general . Given two reductive groups and a (well behaved) morphism between their corresponding L- groups, this conjecture relates their automorphic representations in a way that is compatible with their L- functions.
This functoriality conjecture implies all the other conjectures presented so far. It is of the nature of an induced representation construction. Attempts to specify a direct construction have only produced some conditional results. All these conjectures can be formulated for more general fields in place of Q: algebraic number fields (the original and most important case), local fields, and function fields (finite extensions of Fp(t) where p is a prime and Fp(t) is the field of rational functions over the finite field with p elements). Geometric conjectures. In simple cases, it relates l- adic representations of the .
Although Wiles' results have been substantially generalized, in many different directions, the full Langlands conjecture for GL(2, Q) remains unproved. Laurent Lafforgue proved Lafforgue's theorem verifying the Langlands conjectures for the general linear group GL(n, K) for function fields K.
This work continued earlier investigations by Drinfeld, who proved the case GL(2, K)Local Langlands conjectures. Their proof uses a global argument. Richard Taylor and Michael Harris (2. Langlands conjectures for the general linear group GL(n, K) for characteristic 0 local fields K. Guy Henniart (2. 00. Both proofs use a global argument.
Peter Scholze (2. Fundamental lemma.
ISBN 9. 78- 0- 4. All this stuff, as my dad put it, is quite heavy: we've got Hitchin moduli spaces, mirror symmetry, A- branes, B- branes, automorphic sheaves..
One can get a headache just trying to keep track of them all. Believe me, even among specialists, very few people know the nuts and bolts of all elements of this construction. ISBN 9. 78. 04. 65. The Langlands Program is now a vast subject. There is a large community of people working on it in different fields: number theory, harmonic analysis, geometry, representation theory, mathematical physics. Although they work with very different objects, they are all observing similar phenomena.
Le Courrier du Vietnam. New Series, 4. 0 (1): 3.
New Series, 1. 0 (2): 1. Mathematicians (Stockholm, 1.