Open Questions: The Langlands Program

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See also: Number theory -- Algebraic number theory

Introduction

Automorphic forms

Reciprocity laws

Nonabelian class field theory


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Introduction



Recommended references: Web sites

Langlands program
Article from Wikipedia.
The work of Robert Langlands
This is an archive of many of Langlands' papers and writings, which indicates the general topics of concern in his "program". Most documents are available in downloadable format.
The Langlands Program
One paragraph summary by Harry Riesmann.


Recommended references: Magazine/journal articles

Harmonic Analysis and Group Representations
James Arthur
Notices of the AMS, January 2000, pp. 26-34
The Langlands program is a way of organizing number theoretic data in terms of analytic objects. This survey paper explains it in terms of Harish-Chandra's fundamental work on representation theory.
[Article in PDF format]
The Nonabelian Reciprocity Law for Local Fields
Jonathan Rogawski
Notices of the AMS, January 2000, pp. 35-41
Reciprocity laws in number theory are far-reaching generalizations of the quadratic reciprocity law in elementary number theory, which specifies when "square roots" modulo a prime exist. This research paper explains how the Langlands program yields a generalized reciprocity law for matrices over a p-adic field.
[Article in PDF format]
On some applications of automorphic forms to number theory
Daniel Bump; Solomon Friedberg; Jeffrey Hoffstein
Bulletin of the AMS, April 1996, pp. 157-175
The basic idea of Dirichlet series is to study an interesting sequence of numbers by means of investingating analytic properties of the Dirichlet series having the numbers as coefficients. Part of the Langlands program is to make general statements about such series. The present paper deals with a generalization in which the coefficients themselves are functions of a complex variable.
[Abstract, references, downloadable text]
Group Representations and Harmonic Analysis from Euler to Langlands, Part II
Anthony W. Knapp
Notices of the AMS, May 1996, pp. 537-549
Group representations provide a mechanism for generalizing harmonic analysis to deal with nonabelian groups. This generalized theory is fundamental to the Langlands program, and is applicable in many other areas as well, such as the theory of Lie groups.
[Article in PDF format]
Group Representations and Harmonic Analysis from Euler to Langlands, Part I
Anthony W. Knapp
Notices of the AMS, April 1996, pp. 410-415
Group representations and harmonic analysis play a central role in the Langlands program. The two topics have applications as diverse as number theory, probability, and mathematical physics. The origins of the main ideas can be found in the work of Euler, Fourier, and Dirichlet.
[Article in PDF format]
Selberg's Conjectures and Artin's L-functions
M. Ram Murty
Bulletin of the AMS, July 1994, pp. 1-14
Nonabelian reciprocity laws can be expressed as an identity between L-functions defined in different ways. In 1989 Atle Selberg stated some conjectures about complex functions having properties typical of L-functions. The relationship of these conjectures to the Langlands program is discussed.
An Elementary Introduction to the Langlands Program
Stephen Gelbart
Bulletin of the AMS, 10(1984), pp. 177-219
The essence of the Langlands conjecture is that the possible number fields of degree n are related to and constrained by the irreducible infinite dimensional representations of GL(n). How automorphic forms and L-functions help provide the connection is explained in this excellent survey paper.


Recommended references: Books


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