mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, Lafforgue's theorem, due to

Laurent Lafforgue Laurent Lafforgue (; born 6 November 1966) is a French mathematician. 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 ...

, completes the

Langlands program In representation theory and algebraic number theory, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry. Proposed by , it seeks to relate Galois groups in algebraic num ...

for

general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...

s over

algebraic function field In mathematics, an algebraic function field (often abbreviated as function field) of ''n'' variables over a field ''k'' is a finitely generated field extension ''K''/''k'' which has transcendence degree ''n'' over ''k''. Equivalently, an algebraic ...

s, by giving a correspondence between

automorphic form In harmonic analysis and number theory, an automorphic form is a well-behaved function from a topological group ''G'' to the complex numbers (or complex vector space) which is invariant under the action of a discrete subgroup \Gamma \subset G of ...

s on these groups and representations of

Galois group In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the pol ...

s. The Langlands conjectures were introduced by and describe a correspondence between representations of the

Weil group In mathematics, a Weil group, introduced by , is a modification of the absolute Galois group of a local or global field, used in class field theory. For such a field ''F'', its Weil group is generally denoted ''WF''. There also exists "finite lev ...

of an

and representations of

algebraic group In mathematics, an algebraic group is an algebraic variety endowed with a group structure which is compatible with its structure as an algebraic variety. Thus the study of algebraic groups belongs both to algebraic geometry and group theory. Man ...

s over the function field, generalizing

class field theory In mathematics, class field theory (CFT) is the fundamental branch of algebraic number theory whose goal is to describe all the abelian Galois extensions of local and global fields using objects associated to the ground field. Hilbert is credit ...

of function fields from abelian Galois groups to non-abelian Galois groups.

Langlands conjectures for GL₁

The Langlands conjectures for GL₁(''K'') follow from (and are essentially equivalent to)

. More precisely the

Artin map The Artin reciprocity law, which was established by Emil Artin in a series of papers (1924; 1927; 1930), is a general theorem in number theory that forms a central part of global class field theory. The term "reciprocity law" refers to a long line ...

gives a map from the idele class group to the abelianization of the Weil group.

Automorphic representations of GL_''n''(''F'')

The representations of GL_''n''(''F'') appearing in the Langlands correspondence are automorphic representations.

Lafforgue's theorem for GL_''n''(''F'')

Here ''F'' is a global field of some positive characteristic ''p'', and ℓ is some prime not equal to ''p''. Lafforgue's theorem states that there is a bijection σ between: *Equivalence classes of cuspidal representations π of GL_''n''(''F''), and *Equivalence classes of irreducible ℓ-adic representations σ(π) of dimension ''n'' of the absolute Galois group of ''F'' that preserves the ''L''-function at every place of ''F''. The proof of Lafforgue's theorem involves constructing a representation σ(π) of the absolute Galois group for each cuspidal representation π. The idea of doing this is to look in the ℓ-adic cohomology of the moduli stack of

shtuka In mathematics, a Drinfeld module (or elliptic module) is roughly a special kind of module over a ring of functions on a curve over a finite field, generalizing the Carlitz module. Loosely speaking, they provide a function field analogue of comple ...

s of rank ''n'' that have compatible level ''N'' structures for all ''N''. The cohomology contains subquotients of the form :π⊗σ(π)⊗σ(π)^∨ which can be used to construct σ(π) from π. A major problem is that the moduli stack is not of finite type, which means that there are formidable technical difficulties in studying its cohomology.

Applications

Lafforgue's theorem implies the

Ramanujan–Petersson conjecture In mathematics, the Ramanujan conjecture, due to , states that Ramanujan's tau function given by the Fourier coefficients of the cusp form of weight :\Delta(z)= \sum_\tau(n)q^n=q\prod_\left (1-q^n \right)^ = q-24q^2+252q^3- 1472q^4 + 4830q^5-\ ...

that if an automorphic form for GL_''n''(''F'') has central character of finite order, then the corresponding Hecke eigenvalues at every unramified place have absolute value 1. Lafforgue's theorem implies the conjecture of that an irreducible finite-dimensional ''l''-adic representation of the absolute Galois group with determinant character of finite order is pure of weight 0.

References

* * * * * Lafforgue, Laurent (2002)
"Chtoucas de Drinfeld, formule des traces d'Arthur-Selberg et correspondance de Langlands."
(Drinfeld shtukas, Arthur-Selberg trace formula and Langlands correspondence) Proceedings of the International Congress of Mathematicians, Vol. I (Beijing, 2002), 383–400, Higher Ed. Press, Beijing, 2002. * * * *Gérard Laumon (2002)
"The work of Laurent Lafforgue"
Proceedings of the ICM, Beijing 2002, vol. 1, 91–97, *G. Laumon (2000)
"La correspondance de Langlands sur les corps de fonctions (d'après Laurent Lafforgue)"
(The Langlands correspondence over function fields (according to Laurent Lafforgue)), Séminaire Bourbaki, 52e année, 1999–2000, no. 873.

External links

The work of Robert Langlands
*{{citation, last=Rapoport, title=The work of Laurent Lafforgue, url=http://www.math.uni-bonn.de/ag/alggeom/preprints/lafforgue.pdf Theorems in algebraic number theory Representation theory of Lie groups Automorphic forms Conjectures Class field theory Langlands program Theorems in representation theory

Langlands conjectures for GL1

Automorphic representations of GL''n''(''F'')