Langlands program (nonfiction)

In mathematics, the Langlands program is a web of far-reaching and influential conjectures about connections between number theory and geometry.

Proposed by Robert Langlands (1967, 1970), it seeks to relate Galois groups in algebraic number theory to automorphic forms and representation theory of algebraic groups over local fields and adeles.

Widely seen as the single biggest project in modern mathematics, Edward Frenkel described the Langlands program as "a kind of grand unified theory of mathematics."

There are several different ways of stating Langlands conjectures, which are closely related but not obviously equivalent.

The starting point of the program may be seen as Emil Artin's reciprocity law, which generalizes quadratic reciprocity. 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.

The functoriality conjecture states that a suitable homomorphism of L-groups is expected to give a correspondence between automorphic forms (in the global case) or representations (in the local case). Roughly speaking, the Langlands reciprocity conjecture is the special case of the functoriality conjecture when one of the reductive groups is trivial.

The so-called geometric Langlands program, suggested by Gérard Laumon following ideas of Vladimir Drinfeld, arises from a geometric reformulation of the usual Langlands program that attempts to relate more than just irreducible representations. In simple cases, it relates l-adic representations of the étale fundamental group of an algebraic curve to objects of the derived category of l-adic sheaves on the moduli stack of vector bundles over the curve.

In the News

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  Robert Phelan Langlands is an American-Canadian mathematician, founder of the Langlands program, a web of conjectures and results connecting representation theory and automorphic forms to the study of Galois groups in number theory.

Fiction cross-reference

• Mathematics

Nonfiction cross-reference

• Emil Artin (nonfiction)
• Fundamental lemma (Langlands program) (nonfiction)
• Robert Langlands (nonfiction)
• Mathematics (nonfiction)

External links:

• Langlands program @ Wikipedia