To this day, ℓ-adic cohomology remains a fundamental tool for number theorists, with important applications to the Langlands program.

The Langlands program is a far-reaching web of these ideas of ' unifying conjectures ' that link different subfields of mathematics, e. g. number theory and representation theory of Lie groups ; some of these conjectures have since been proved.

He has made outstanding contributions to Langlands ' program in the fields of number theory and analysis, and in particular proved the Langlands conjectures for the automorphism group of a function field.

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.

The so-called geometric Langlands program, suggested by Gérard Laumon following ideas of Vladimir Drinfel'd, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations.

Another field, known as the Langlands program, likewise starts with apparently haphazard similarities ( in this case, between number-theoretical results and representations of certain groups ) and looks for constructions from which both sets of results would be corollaries.

* Langlands program

In fact the Langlands program ( or philosophy ) is much more like a web of unifying conjectures ; it really does postulate that the general theory of automorphic forms is regulated by the L-groups introduced by Robert Langlands.

This development preceded the Langlands program by a few years, and can be regarded as complementary to it: Langlands ' work relates largely to Artin L-functions, which, like Hecke's L-functions, were defined several decades earlier, and to L-functions attached to general automorphic representations.

# REDIRECT Langlands program

He facilitated the now-celebrated visit of Robert Langlands to Turkey ( now famous for the Langlands program, among many other things ); during which Langlands worked out some arduous calculations on the epsilon factors of Artin L-functions.

Taylor received the 2007 Shaw Prize in Mathematical Sciences for his work on the Langlands program with Robert Langlands.

The associated reductive Lie groups are of significant interest: the Langlands program is based on the premise that what is done for one reductive Lie group should be done for all.

Drinfeld has also collaborated with Alexander Beilinson to rebuild the theory of vertex algebras, which have become increasingly important to conformal field theory, string theory, and the geometric Langlands program.

** Langlands program

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.

Langlands attached automorphic L-functions to these automorphic representations, and conjectured that every Artin L-function arising from a finite-dimensional representation of the Galois group of a number field is equal to one arising from an automorphic cuspidal representation.

Roughly speaking, the reciprocity conjecture gives a correspondence between automorphic representations of a reductive group and homomorphisms from a Langlands group to an L-group.

Furthermore, given such a group G, Langlands constructs the Langlands dual group < sup > L </ sup > G, and then, for every automorphic cuspidal representation of G and every finite-dimensional representation of < sup > L </ sup > G, he defines an L-function.

Robert Langlands showed how ( in generality, many particular cases being known ) the Riemann-Roch theorem could be applied to the calculation of dimensions of automorphic forms ; this is a kind of post hoc check on the validity of the notion.

Drinfeld's work connected algebraic geometry over finite fields with number theory, especially the theory of automorphic forms, through the notions of elliptic module and the theory of the geometric Langlands correspondence.

In fact, Murty showed that Artin L-functions corresponding to irreducible representations of the Galois group of a solvable extension of the rationals are automorphic as predicted by the Langlands conjectures.

One of the aims of proposed non-abelian class field theory is to incorporate the complex-analytic nature of Artin L-functions into a larger framework, such as is provided by automorphic forms and Langlands ' philosophy.

Firstly Langlands and Deligne established a factorisation into Langlands – Deligne local constants ; this is significant in relation to conjectural relationships to automorphic representations.

pointed out that the Artin conjecture follows from strong enough results from the Langlands philosophy, relating to the L-functions associated to automorphic representations for GL ( n ) for all.

More precisely, the Langlands conjectures associate an automorphic representation of the adelic group GL < sub > n </ sub >( A < sub > Q </ sub >) to every n-dimensional irreducible representation of the Galois group, which is a cuspidal representation if the Galois representation is irreducible, such that the Artin L-function of the Galois representation is the same as the automorphic L-function of the automorphic representation.

All irreducible unitary representations are admissible ( or rather their Harish-Chandra modules are ), and the admissible representations are given by the Langlands classification, and it is easy to tell which of them have a non-trivial invariant sesquilinear form.

The general Langlands philosophy includes a comparable explanation of the connection of polynomials of degree m, and the representation theory for GL < sub > m </ sub >.

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.

Another significant related point is that the Langlands approach stands apart from the whole development triggered by monstrous moonshine ( connections between elliptic modular functions as Fourier series, and the group representations of the Monster group and other sporadic groups ).

Sometimes Hilbert's statements were not precise enough to specify a particular problem but were suggestive enough so that certain problems of more contemporary origin seem to apply, e. g. most modern number theorists would probably see the 9th problem as referring to the ( conjectural ) Langlands correspondence on representations of the absolute Galois group of a number field.

The modularity theorem is a special case of more general conjectures due to Robert Langlands.

There had been proposed extensions along Portland Road, up John Finnie Street, West Langlands Street and eventually towards Crosshouse, but by this time, increasing costs and the far more flexible motor bus had made inroads and the trams ceased operation in 1926 during the General Strike.

The burn increases in size around Langlands Moss and more properly becomes the Calder Water.

Langlands generalized the idea of functoriality: instead of using the general linear group GL ( n ), other connected reductive groups can be used.

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 Vladimir Drinfel'd, who proved the case GL ( 2, K )

proved the local Langlands conjectures for the general linear group GL ( 2, K ) over local fields.

proved the local Langlands conjectures for the general linear group GL ( n, K ) for positive characteristic local fields K. Their proof uses a global argument.

proved the local Langlands conjectures for the general linear group GL ( n, K ) for characteristic 0 local fields K. gave another proof.

Contemporary successors of the theory are the Arthur-Selberg trace formula applying to the case of general semisimple G, and the many studies of the trace formula in the Langlands philosophy ( dealing with technical issues such as endoscopy ).

Some versions of the Langlands conjectures are vague, or depend on objects such as the Langlands groups, whose existence in unproven, or on the L-group that has several inequivalent definitions.

There are different types of objects for which the Langlands conjectures can be stated: