Newsgroups: sci.math
From: dg@dgupta.hpl.hp.com (Dipankar Gupta)
Subject: Re: Langlands program
Date: Thu, 9 Feb 1995 12:23:42 GMT
sal> [...] that the real importance of Wiles' work was that it proved
sal> some of the] Tanimaya conjectures; but that the *really real*
sal> importance of Wiles's work was that it was a step towards
sal> proving the Langlands program. So what's the Langlands program?
sal> The article talked about it meaning that analysis and number
sal> theory were identical, but that doesn't really explain it.
From what little I know of it, the Langlands program concerns itself
with identities between automorphic Lfunctions and ``motivic'' ones,
which arise in contexts of representation theory and number theory
respectively. It is essentially a series of conjectures to understand
nonabelian reciprocity laws. The Artin reciprocity law and
Shimura/Taniyama conjecture are examples of such reciprocities.
I had filed the following message written by Tal Kubo on the
subject. In addition, you might wish to refer to:
M. Ram Murty _A motivated introduction to the Langlands program_
pp 3766 in ``Advances in Number Theory'' (F. Gouvea and
N. Yui, eds)
Clarendon Press, Oxford, 1993
Hope this was useful.
Dipankar

From: kubo@brauer.harvard.edu (Tal Kubo)
Subject: Re: What is "Langlands Philosophy"?
Date: 26 Mar 94 16:05:43 EST
The Langlands program is a system of conjectures connecting
number theory and the representation theory of Lie groups.
It predicts that a large class of the zeta and Lfunctions coming from
number theory and algebraic geometry coincide with Lfunctions coming
from representation theory. Lfunctions have been used for over a century
in number theory; Langlands isolated the correct analogue from
representation theory (socalled "automorphic" Lfunctions) and was
the first to understand the general picture.
As an example of why one might expect some sort of connection between
Lie groups and number theory, consider the Galois group G = Gal(K/L)
where K is an algebraic closure of a number field L. Number theorists
are very interested in representations of K. G is a profinite group
(projective limit of finite groups). Representation theory of profinite
groups is not so welldeveloped but there is at least one situation where
there is some hope: algebraic groups over padic rings. Lie groups over
padic fields turn out to be prominent actors in Langlands' conjectures.
Standard references for the Langlands program are:
Stephen Gelbart, "An Elementary Introduction to the Langlands Program",
Bulletin of the AMS v.10 no. 2 April 1984.
Proc. Sympos. Pure Math., v. 33, parts 1 & 2. This is the proceedings
of a conference on the Langlands conjectures, including expository
articles on the Langlands program and some of the background.

==============================================================================
From: Bill Dubuque
Newsgroups: sci.math
Subject: Re: Langlands Conjecture
Date: 20 Aug 1998 07:51:47 0400
Cliff writes:

 When I asked many people what is the most difficult areas of math to
 understand, and also what is the most important unsolved mathematical
 problem, they always responded with two words: "Langlands Conjecture"
 (or "Langlands philosophy").

 Can anyone explain what this is so that a general audience could
 understand it? If this is not possible, could anyone give a flavor
 of what this means?
Alas, to appreciate the ideas in the Langlands program requires at
least a PhDlevel math education. It would be virtually impossible
to attempt to convey these ideas to an audience lesseducated.
Below are references to works of expository character which touch on
topics related to the Langlands program. I'd suggest starting with
Shafarevich, Gelbart (1984) and Murty  some of which should be
accessible to bright math undergrads.
Bill Dubuque
Kapranov, M. M. Analogies between the Langlands correspondence and
topological quantum field theory. Functional analysis on the eve of
the 21st century, Vol. 1 (New Brunswick, NJ, 1993), 119151,
Progr. Math., 131, Birkhauser Boston, Boston, MA, 1995.
MR 97c:11069 (Reviewer: I. Dolgachev) 11G45 (11R39 14A20 19F05 58D29)
Murty, M. Ram. A motivated introduction to the Langlands program.
Advances in number theory (Kingston, ON, 1991), 3766,
Oxford Sci. Publ., Oxford Univ. Press, New York, 1993.
MR 96j:11157 (Reviewer: Alexey A. Panchishkin) 11R39 (11F11 11F70 11G05 11R56)
Gelbart, Stephen. Automorphic forms and Artin's conjecture. II.
Mathematische Wissenschaften gestern und heute. 300 Jahre Mathematische
Gesellschaft in Hamburg, Teil 4 (Hamburg, 1990).
Mitt. Math. Ges. Hamburg 12 (1991), no. 4, 907947 (1992).
MR 94g:11103 11R39 (11F70 11F80)
Murty, Ram. Theta functions: from the classical to the modern. Preface,
viix, CRM Proc. Lecture Notes, 1, Amer. Math. Soc., Providence, RI, 1993.
MR 94c:11038 (Reviewer: Antonia Wilson Bluher) 11F27 (1102 11F32 11R39)
Varadarajan, V. S. Symmetry in mathematics.
Comput. Math. Appl. 24 (1992), no. 3, 3744. MR 93f:20001 2001
Langlands, Robert P. Representation theory: its rise and its role in
number theory. Proceedings of the Gibbs Symposium (New Haven, CT, 1989),
181210, Amer. Math. Soc., Providence, RI, 1990.
MR 92d:11053 (Reviewer: Stephen Gelbart) 11F70 (11G40 11R39 22E50 22E55)
Shafarevich, I. R. Abelian and nonabelian mathematics.
Math. Intelligencer 13 (1991), no. 1, 6775.
MR 92b:01048 (Reviewer: G. Eisenreich) 01A60 (1101)
Neukirch, Jurgen. Algebraische Zahlentheorie. (German)
[Algebraic number theory] Ein Jahrhundert Mathematik 18901990,
587628, Dokumente Gesch. Math., 6, Vieweg, Braunschweig, 1990.
MR 92a:01057 (Reviewer: J. S. Joel) 01A60 (1103 1403)
Patterson, S. J. Erich Hecke und die Rolle der LReihen in der
Zahlentheorie. (German) [Erich Hecke and the role of Lseries in
number theory] Ein Jahrhundert Mathematik 18901990, 629655,
Dokumente Gesch. Math., 6, Vieweg, Braunschweig, 1990.
MR 91m:01020 (Reviewer: J. Dieudonne) 01A60 (01A55 1103)
Manin, Yu. I.; Panchishkin, A. A. Introduction to number theory. (Russian)
Current problems in mathematics. Fundamental directions, Vol. 49 (Russian),
5348, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i
Tekhn. Inform., Moscow, 1990.
MR 91j:11001b (Reviewer: Yurey A. Drakokhrust) 1102
Langlands, R. P. Eisenstein series, the trace formula, and the modern
theory of automorphic forms. Number theory, trace formulas and discrete
groups (Oslo, 1987), 125155, Academic Press, Boston, MA, 1989.
MR 90e:11077 (Reviewer: Stephen Gelbart) 11F70 (01A65 11R39 22E55)
Langlands, R. P. Representation theory and arithmetic. The mathematical
heritage of Hermann Weyl (Durham, NC, 1987), 2533, Proc. Sympos. Pure
Math., 48, Amer. Math. Soc., Providence, RI, 1988.
MR 90e:11076 (Reviewer: Stephen Gelbart) 11F70 (22E55)
Koch, Helmut. Die Rolle der Zetafunktionen in der Zahlentheorie von Euler
bis zur Gegenwart. (German) [The role of zeta functions in number theory,
from Euler to the present] Ceremony and scientific conference on the
occasion of the 200th anniversary of the death of Leonhard Euler (Berlin,
1983), 120124, Abh. Akad. Wiss. DDR, Abt. Math. Naturwiss. Tech., 851,
AkademieVerlag, Berlin, 1985.
MR 87f:01009 (Reviewer: E. J. Barbeau) 01A50 (01A60 1103)
Gelbart, Stephen. An elementary introduction to the Langlands program.
Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177219.
MR 85e:11094 (Reviewer: Joe Repka) 11R39 (1102 11F70)
Langlands, R. P. Some contemporary problems with origins in the Jugendtraum.
Mathematical developments arising from Hilbert problems (Proc. Sympos. Pure
Math., Vol. XXVIII, Northern Illinois Univ., De Kalb, Ill., 1974),
pp. 401418. Amer. Math. Soc., Providence, R. I., 1976.
MR 55#10426 (Reviewer: Stephen Gelbart) 12A65 (10D15)
Tate, J. Problem 9: The general reciprocity law. Mathematical developments
arising from Hilbert problems (Proc. Sympos. Pure Math., Northern Illinois
Univ., De Kalb, Ill., 1974), pp. 311322. Proc. Sympos. Pure Math.,
Vol. XXVIII, Amer. Math. Soc., Providence, R. I., 1976.
MR 55#2849 (Reviewer: Jacques Martinet) 12A65