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 L-functions 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 37--66 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 L-functions coming from number theory and algebraic geometry coincide with L-functions coming from representation theory. L-functions have been used for over a century in number theory; Langlands isolated the correct analogue from representation theory (so-called "automorphic" L-functions) 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 well-developed but there is at least one situation where there is some hope: algebraic groups over p-adic rings. Lie groups over p-adic 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 PhD-level math education. It would be virtually impossible to attempt to convey these ideas to an audience less-educated. 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), 119--151, 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), 37--66, 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, 907--947 (1992). MR 94g:11103 11R39 (11F70 11F80) Murty, Ram. Theta functions: from the classical to the modern. Preface, vii--x, CRM Proc. Lecture Notes, 1, Amer. Math. Soc., Providence, RI, 1993. MR 94c:11038 (Reviewer: Antonia Wilson Bluher) 11F27 (11-02 11F32 11R39) Varadarajan, V. S. Symmetry in mathematics. Comput. Math. Appl. 24 (1992), no. 3, 37--44. MR 93f:20001 20-01 Langlands, Robert P. Representation theory: its rise and its role in number theory. Proceedings of the Gibbs Symposium (New Haven, CT, 1989), 181--210, 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, 67--75. MR 92b:01048 (Reviewer: G. Eisenreich) 01A60 (11-01) Neukirch, Jurgen. Algebraische Zahlentheorie. (German) [Algebraic number theory] Ein Jahrhundert Mathematik 1890--1990, 587--628, Dokumente Gesch. Math., 6, Vieweg, Braunschweig, 1990. MR 92a:01057 (Reviewer: J. S. Joel) 01A60 (11-03 14-03) Patterson, S. J. Erich Hecke und die Rolle der L-Reihen in der Zahlentheorie. (German) [Erich Hecke and the role of L-series in number theory] Ein Jahrhundert Mathematik 1890--1990, 629--655, Dokumente Gesch. Math., 6, Vieweg, Braunschweig, 1990. MR 91m:01020 (Reviewer: J. Dieudonne) 01A60 (01A55 11-03) Manin, Yu. I.; Panchishkin, A. A. Introduction to number theory. (Russian) Current problems in mathematics. Fundamental directions, Vol. 49 (Russian), 5--348, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1990. MR 91j:11001b (Reviewer: Yurey A. Drakokhrust) 11-02 Langlands, R. P. Eisenstein series, the trace formula, and the modern theory of automorphic forms. Number theory, trace formulas and discrete groups (Oslo, 1987), 125--155, 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), 25--33, 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), 120--124, Abh. Akad. Wiss. DDR, Abt. Math. Naturwiss. Tech., 85-1, Akademie-Verlag, Berlin, 1985. MR 87f:01009 (Reviewer: E. J. Barbeau) 01A50 (01A60 11-03) Gelbart, Stephen. An elementary introduction to the Langlands program. Bull. Amer. Math. Soc. (N.S.) 10 (1984), no. 2, 177--219. MR 85e:11094 (Reviewer: Joe Repka) 11R39 (11-02 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. 401--418. 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. 311--322. Proc. Sympos. Pure Math., Vol. XXVIII, Amer. Math. Soc., Providence, R. I., 1976. MR 55#2849 (Reviewer: Jacques Martinet) 12A65