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Langlands program

In mathematics, the Langlands program is a web of far-reaching and influential conjectures that connect number theory and the representation theory of certain groups. It was proposed by Robert Langlands beginning in 1967.

Ideas available at the time

In a very broad context, it built on existing ideas: the philosophy of cusp forms formulated a few years earlier by I.M. Gel'fand, the work and approach of Harish-Chandra on semisimple Lie groups, and in technical terms the trace formula of Atle Selberg and others.

For example, in the work of Harish Chandra one finds the principle that what can be done for one semisimple (or reductive) Lie group, should be done for all. Therefore once the role of some low-dimensional Lie groups such as GL(2) in the theory of modular forms had been recognised, the way was open at least to speculation about GL(n).

The cusp form idea came out of the cusps on modular curves but also had a meaning visible in spectral theory as 'discrete spectrum', contrasted with the 'continuous spectrum' from Eisenstein series. It becomes much more technical for bigger Lie groups, because the parabolic subgroups are more numerous.

In all these approaches there was no shortage of technical methods, often inductive in nature, but the field was and is very demanding. And on the side of modular forms, there were examples such as Hilbert modular forms, Siegel modular forms, and theta-series.

Connection with number theory

What initially was very new in Langlands' work, besides technical depth, was the proposed direct connection to number theory, together with the rich organisational structure hypothesised (so-called functoriality).

The starting point of the program is the Artin reciprocity law which generalizes quadratic reciprocity. Artin's law applies to an algebraic number field whose Galois group over Q 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 (that is, the analogues of the Riemann Zeta function constructed from Dirichlet characters). The precise correspondence constitutes the reciprocity law (including quadratic reciprocity as one case).

For non-abelian Galois groups and higher-imensional representations of them, one can still define L-functions in a natural way. 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 setting of automorphic representations

Hecke had earlier related Dirichlet L-functions with automorphic forms, holomorphic functions on the upper half place of C that satisfy certain functional equations. Langlands then generalized these to automorphic cuspidal representations, which are certain infinite dimensional irreducible representations of the general linear group over the adele ring of Q. (This ring simultaneously keeps track of all the completions of Q, see p-adic numbers.)

Langlands attached L-functions to these automorphic cuspidal representations, and conjectured that every L-function arising from finite-dimensional representations of the Galois group is equal to one arising from an automorphic cuspidal representation. This is known as his "Reciprocity Conjecture".

A general principle of functoriality

Langlands then formulated a much more general "Functoriality Principle", which relates automorphic representations of different groups (not just the general linear group) over the adele ring of Q, in a way which is compatible with their L-functions.

All these conjectures can be formulated for more general fields in place of Q: algebraic number fields (the original and most important case), local fields and function fields (finite extensions of Fp(t) where p is a prime and Fp(t) is the field of rational functions over the finite field with p elements).

Parts of the program for local fields were completed in 1998 and for function fields in 1999. Laurent Lafforgue received the Fields Medal in 2002 for his work on the function field case. This work continued earlier investigations by Vladimir Drinfeld, which were honored with the Fields Medal in 1990. Only special cases of the number field case have been proven, some by Langlands himself.

Langlands received the Wolf Prize in 1996 for his work on these conjectures.