Langlands program

The Langlands program of mathematics consists in a series of far-reaching conjectures that relate the number theory and the representation theory of groups with each other. They were erected by Robert Langlands since 1967.

Connection to number theory

As a starting point of the program, you can view the reciprocity law of Artin, which generalizes the quadratic reciprocity law. Artin reciprocity law assigns a algebraic number field whose Galois group over commutative ( abelian ) is an L- function of the one-dimensional representations of the Galois group, and states that this L- function coincides with a certain Dirichlet L- series.

For non- abelian Galois groups and higher dimensional representations, one can also define L-functions in a natural way.

Automorphic representations

The idea of Langlands was to find a suitable generalization of the Dirichlet L-functions, which makes it possible to formulate the statement of Artin in this more general framework.

Hedge had brought earlier Dirichlet L-functions with automorphic forms, ie, with holomorphic functions of the upper half-plane of complex numbers that satisfy certain functional equations in conjunction. Langlands generalized this to automorphic representations kuspidale. It is infinite-dimensional irreducible representations of the general linear group GLn over the ring of Adele, this ring all completions of account, see p- adic numbers.

Langlands rejected this automorphic representations for certain L-functions and assumed that each L- function of a finite-dimensional representation of the Galois group coincides with the L- function of an automorphic representation kuspidalen. This is the so-called " Reziprozitätsvermutung ".

A general Funktorialitätsprinzip

Langlands generalized this further: Instead of the general linear group GLn can consider other reductive groups. For such a group G Langlands constructed a complex Lie group LG, and for each kuspidale automorphic representation of G and every finite-dimensional representation of LG, he defined an L- function. One of his conjecture then states that these L-functions satisfy certain functional equations, which generalize those of known L-functions.

In this framework, Langlands formulated a general " Funktorialitätsprinzip ": If two reductive groups and a morphism is given between their L- groups, so this supposed principle are connected in a manner according to their automorphic representations with each other, which is compatible with their L-functions. This functoriality implies all other assumptions. It is in type, the construction of an induced representation of what was called in the traditional theory of automorphic forms a ' foot lifter '. Attempts to provide such a structure directly have led to only limited results.

All of these assumptions may also be formulated for other body. Instead of you can algebraic number, local bodies and body function, ie finite field extensions of Fp (t ) consider, where p is a prime number and Fp (t ) denotes the field of rational functions over the finite field with p elements.

Ideas that led to the Langlands program

In the program, the following ideas were received: the philosophy of cusp forms, which had been a few years earlier by Israel Gelfand, the access of Harish - Chandra on semisimple Lie groups and in the technical sense, the trace formula of Selberg and others. What was new in Langlands work, in addition to the technical depth, the presumed direct connection with number theory and the functorial structure of the whole.

In the work of Harish - Chandra can be found, for example, the principle that one what you can do with a semisimple (or reductive ) Lie group, should make for all. Thus, if the role of low-dimensional Lie groups such as the GL ( 2) had been recognized in the theory of modular forms, so the way was open for speculation on GL ( n ) for any n > 2

The idea of ​​the tip shape stemmed from the tips on modular curves, but she was also visible in the spectral theory of a discrete spectrum, in contrast to the continuous spectrum of Eisenstein series. This relationship is technically far more complicated for larger Lie groups, since the parabolic subgroups are numerous.

Results and Prices

Parts of the program for local bodies were finished in 1998 and the function body for 1999. Laurent Lafforgue was awarded the Fields Medal in 2002 for his work in the case of function fields. This continued earlier studies of Vladimir Drinfeld, which was also awarded the Fields Medal in 1990. For a number field the program is proved only in a few special cases, some of Langlands itself for local function body the Langland presumption of Gérard Laumon, Michael Rapoport, Ulrich Stuhler, - elliptic sheaves and the Langlands correspondence, Invent was. Math 113 (1993) 217-338 proven. The local Langlands conjecture ( for local p- adic body ) was proved in 1998 by Michael Harris, Richard Taylor and regardless of Guy Henniart.

Langlands received the 1996 Wolf Prize in Mathematics for his work on these assumptions. For the proof of Ngo Bao Chau Fundamentallemmas 2010 received the Fields Medal.

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