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Langlands proved the Langlands conjectures for groups over the archimedean local fields R and C by giving the Langlands classification of their irreducible representations.
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Langlands and proved
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.
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.
Ngô Bảo Châu proved an auxiliary but difficult statement, the so-called " Fundamental Lemma ", originally conjectured by Langlands.
In subsequent work, Taylor ( along with Michael Harris ) proved the local Langlands conjectures for GL ( n ) over a number field.
Langlands and conjectures
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.
There are a number of related Langlands conjectures.
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.
Moreover, the Langlands conjectures have evolved since Langlands first stated them in 1967.
There are different types of objects for which the Langlands conjectures can be stated:
Langlands did not originally consider this case, but his conjectures have analogues for it.
There are several different ways of stating Langlands conjectures, which are closely related but not obviously equivalent.
The Langlands conjectures for GL ( 1, K ) follow from ( and are essentially equivalent to ) class field theory.
Lusztig's classification of the irreducible representations of groups of Lie type over finite fields can be considered an analogue of the Langlands conjectures for finite fields.
Andrew Wiles ' proof of modularity of semi-stable elliptic curves over rationals can be viewed as an exercise in the Langlands conjectures.
* Local Langlands conjectures
The modularity theorem is a special case of more general conjectures due to Robert Langlands.
Langlands and for
To this day, ℓ-adic cohomology remains a fundamental tool for number theorists, with important applications to the Langlands program.
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.
The Langlands conjecture for GL ( 2, Q ) still remains unproved.
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.
According to Rebecca Langlands, Cordus's story "... is a tale which vividly demonstrates the possibility that a text might be received in a way which the author had not intended or anticipated, and be received in a way which might have dire consequences for author and text.
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 >.
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.
Langlands and groups
For example, over the real numbers, this is the Langlands classification of representations of real reductive groups.
Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial.
Langlands generalized the idea of functoriality: instead of using the general linear group GL ( n ), other connected reductive groups can be used.
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 ).
Ramanujan bounds for groups other than can be obtained as an application of known cases of Langlands functoriality.
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.
The name Langlands can refer to one of several individuals or groups:
Langlands and over
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.
The Langlands program seeks to attach an automorphic form or automorphic representation ( a suitable generalization of a modular form ) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field.
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 1974, at the age of twenty, Drinfeld announced a proof of the Langlands conjectures for GL < sub > 2 </ sub > over a global field of positive characteristic.
The largest crowd at Langlands Park was just over 8, 000 for a game between Wests and Easts in 1977, however 5, 000 people attended the ground on 11 September 2004 to watch Easts defeat the Wynnum-Manly Seagulls to make the 2004 Queensland Cup Grand Final.
Langlands and local
There are several other nonabelian theories, local and global, which provide alternative to the Langlands correspondence point of view.
* Langlands – Deligne local constant
Langlands Park is named for Langlands which was the name of the residence that previous existed on the property ( owned by the local Nicklin family ).
Firstly Langlands and Deligne established a factorisation into Langlands – Deligne local constants ; this is significant in relation to conjectural relationships to automorphic representations.
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