Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Bulletin of the American Mathematical Society
Bulletin of the American Mathematical Society
ISSN 1088-9485(online) ISSN 0273-0979(print)

Langlands Program, trace formulas, and their geometrization


Author: Edward Frenkel
Journal: Bull. Amer. Math. Soc. 50 (2013), 1-55
MSC (2010): Primary 11R39, 14D24, 22E57
Published electronically: October 12, 2012
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The Langlands Program relates Galois representations and automorphic representations of reductive algebraic groups. The trace formula is a powerful tool in the study of this connection and the Langlands Functoriality Conjecture. After giving an introduction to the Langlands Program and its geometric version, which applies to curves over finite fields and over the complex field, I give a survey of my recent joint work with Robert Langlands and Ngô Bao Châu on a new approach to proving the Functoriality Conjecture using the trace formulas, and on the geometrization of the trace formulas. In particular, I discuss the connection of the latter to the categorification of the Langlands correspondence.


References [Enhancements On Off] (What's this?)

  • [AG] D. Arinkin and D. Gaitsgory, Singular support of coherent sheaves, and the geometric Langlands conjecture, Preprint arXiv:1201.6343.
  • [Art1] J. Arthur, The principle of functoriality, Bull. Amer. Math. Soc. 40 (2002) 39-53.
  • [Art2] J. Arthur, An Introduction to the trace formula, Clay Mathematics Proceedings 4, American Mathematical Society, Providence, RI, 2005.
  • [AB] M. Atiyah and R. Bott, A Lefschetz fixed point formula for elliptic complexes, I. Ann. of Math. (2) 86 (1967) 374-407; and II. Ann. of Math. (2) 88 1968 451-491.
  • [BCDT] C. Breuil, B. Conrad, F. Diamond and R. Taylor, On the modularity of elliptic curves over $ {\mathbf Q}$: wild $ 3$-adic exercises, J. Amer. Math. Soc. 14 (2001) 843-939.
  • [BL] A. Beauville and Y. Laszlo, Un lemme de descente, C.R. Acad. Sci. Paris, Sér. I Math. 320 (1995) 335-340.
  • [Be] K. Behrend, Derived $ l$-adic categories for algebraic stacks, Mem. Amer. Math. Soc. 163 (2003), no. 774.
  • [BeDh] K. Behrend and A. Dhillon, Connected components of moduli stacks of torsors via Tamagawa numbers, Canad. J. Math. 61 (2009) 3-28.
  • [BFN] D. Ben-Zvi, J. Francis, and D. Nadler, Integral transforms and Drinfeld centers in derived algebraic geometry, Preprint arXiv:0805.0157, to appear in Journal of AMS.
  • [BN] D. Ben-Zvi and D. Nadler, Loop spaces and connections, Preprint arXiv:1002.3636.
  • [BD] A. Beilinson and V. Drinfeld, Quantization of Hitchin's integrable system and Hecke eigensheaves, Preprint, available at www.math.uchicago.edu/$ \sim $mitya/langlands
  • [Bou] N. Bourbaki, Groupes et algèbres de Lie Chapitres IV, V, VI, Hermann, Paris 1968.
  • [CL] P.-H. Chaudouard and G. Laumon, Le lemme fondamental pondéré I : constructions géométriques, Preprint arXiv:0902.2684.
  • [D1] V.G. Drinfeld, Two-dimensional $ \ell $-adic representations of the fundamental group of a curve over a finite field and automorphic forms on $ GL(2)$, Amer. J. Math. 105 (1983) 85-114.
  • [D2] V.G. Drinfeld, Langlands conjecture for $ GL(2)$ over function field, Proc. Int. Congress of Math. (Helsinki, 1978), pp. 565-574; Moduli varieties of $ F$-sheaves, Funct. Anal. Appl. 21 (1987) 107-122; The proof of Petersson's conjecture for $ GL(2)$ over a global field of characteristic $ p$, Funct. Anal. Appl. 22 (1988) 28-43.
  • [F1] E. Frenkel, ``Lectures on the Langlands Program and Conformal Field Theory'', in Frontiers in Number Theory, Physics and Geometry II, eds. P. Cartier, et al., pp. 387-536, Springer Verlag, 2007 (hep-th/0512172).
  • [F2] E. Frenkel, Gauge theory and Langlands duality, Séminaire Bourbaki, Juin 2009 (arXiv:0906.2747).
  • [FGV1] E. Frenkel, D. Gaitsgory and K. Vilonen, Whittaker patterns in the geometry of moduli spaces of bundles on curves, Ann. of Math. 153 (2001) 699-748, (arXiv:math/9907133).
  • [FGV2] E. Frenkel, D. Gaitsgory and K. Vilonen, On the geometric Langlands conjecture, Jour. Amer. Math. Soc. 15 (2001) 367-417 (arXiv:math/0012255).
  • [FLN] E. Frenkel, R. Langlands and B.C. Ngô, La formule des traces et la functorialité. Le début d'un Programme, Ann. Sci. Math. Québec 34 (2010) 199-243 (arXiv:1003.4578).
  • [FN] Edward Frenkel and Bao Châu Ngô, Geometrization of trace formulas, Bull. Math. Sci. 1 (2011), no. 1, 129–199. MR 2823791 (2012k:22026), http://dx.doi.org/10.1007/s13373-011-0009-0
  • [FW] E. Frenkel and E. Witten, Geometric endoscopy and mirror symmetry, Communications in Number Theory and Physics, 2 (2008) 113-283 (arXiv:0710.5939).
  • [FK] E. Freitag, R. Kiehl, Etale Cohomology and the Weil conjecture, Springer, 1988.
  • [G] D. Gaitsgory, On a vanishing conjecture appearing in the geometric Langlands correspondence, Ann. Math. 160 (2004) 617-682.
  • [GM] Edward Frenkel and Edward Witten, Geometric endoscopy and mirror symmetry, Commun. Number Theory Phys. 2 (2008), no. 1, 113–283. MR 2417848 (2009e:14017)
  • [GP] B.H. Gross and D. Prasad, On the decomposition of a representation of $ SO_n$ when restricted to $ SO_{n-1}$, Canad. J. Math. 44 (1992) 974-1002.
  • [Ha] G. Harder. Über die Galoiskohomologie halbenfacher algebraischer Gruppen. III, J. Reine Angew. Math. 274/275 (1975) 125-138.
  • [H1] N. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987) 59-126.
  • [H2] N. Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987) 91-114.
  • [Il] L. Illusie, Formule de Lefschetz, SGA 5, Lect. Notes in Math. 589, pp. 73-137, Springer Verlag, 1977.
  • [Ic] A. Ichino, On critical values of adjoint $ L$-functions for $ GSp(4)$, Preprint, available at http://www.math.ias.edu/$ \sim $ichino/ad.pdf
  • [IcIk] A. Ichino and T. Ikeda, On the periods of automorphic forms on special orthogonal groups and the Gross-Prasad conjecture, Geom. Func. Anal. 19 (2010) 1378-1425.
  • [J] H. Jacquet, ``A guide to the relative trace formula'', in Automorphic Representations, L-functions and Applications: Progress and Prospects, Ohio State University Mathematical Research Institute Publications, Volume 11, pp. 257-272, De Gruyter, Berlin, 2005.
  • [JL] H. Jacquet and R. Langlands, Automorphic forms on $ GL(2)$, Lect. Notes in Math. 114, Springer, 1970. X
  • [KW] A. Kapustin and E. Witten, Electric-magnetic duality and the geometric Langlands Program, Preprint hep-th/0604151.
  • [KS] M. Kashiwara and P. Schapira, Sheaves on Manifolds, Springer, 1990.
  • [K] M. Kontsevich, Notes on motives in finite characteristic, Preprint arXiv:math/0702206.
  • [LLaf] L. Lafforgue, Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002) 1-241.
  • [VLaf] V. Lafforgue, Quelques calculs reliés à la correspondance de Langlands géométrique pour $ {\mathbb{P}}^1$, available at http://people.math.jussieu.fr/$ \sim $vlafforg/geom.pdf
  • [LL] J.-P. Labesse and R.P. Langlands, L-indistinguishability for SL(2), Canad. J. Math. 31 (1979) 726-785.
  • [LafL] V. Lafforgue and S. Lysenko, Compatibility of the theta correspondence with the Whittaker functors, Preprint arXiv:0902.0051.
  • [L1] R. Langlands, Problems in the theory of automorphic forms, in Lect. Notes in Math. 170, pp. 18-61, Springer Verlag, 1970.
  • [L2] R. Langlands, ``Beyond endoscopy'', in Contributions to automorphic forms, geometry, and number theory, pp. 611-697, Johns Hopkins Univ. Press, Baltimore, MD, 2004.
  • [L3] R. Langlands, Un nouveau point de repère dans la théorie des formes automorphes, Canad. Math. Bull. 50 (2007) no. 2, 243-267.
  • [L4] E. Langlands, Singularités et Transfert, 2010, available at http://publications.ias.edu/rpl
  • [L5] E. Langlands, Functoriality and Reciprocity (in Russian), 2011, http://publica-
    tions.ias.edu/rpl
  • [LO] Y. Laszlo and M. Olsson, The six operations for sheaves on Artin stacks. I. Finite coefficients, Publ. Math. IHES 107 (2008) 109-168.
  • [Lau1] G. Laumon, Transformation de Fourier, constantes d'équations fonctionelles et conjecture de Weil, Publ. IHES 65 (1987) 131-210.
  • [Lau2] G. Laumon, Transformation de Fourier généralisée, Preprint alg-geom/9603004.
  • [Lau3] G. Laumon, Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math. J. 54 (1987) 309-359.
  • [Ly1] S. Lysenko, Geometric theta-lifting for the dual pair $ SO_{2m}, Sp_{2n}$, Ann. Sci. École Norm. Sup. 44 (2011) 427-493.
  • [Ly2] S. Lysenko, Geometric theta-lifting for the dual pair $ GSp_{2n}, GO_{2m}$, Preprint arXiv:0802.0457.
  • [M] J.S. Milne, Étale cohomology, Princeton University Press, 1980.
  • [MV] I. Mirković and K. Vilonen, Geometric Langlands duality and representations of algebraic groups over commutative rings, Preprint math.RT/0401222.
  • [N1] B.C. Ngô, Fibration de Hitchin et endoscopie, Invent. Math. 164 (2006) 399-453.
  • [N2] B.C. Ngô, Le lemme fondamental pour les algebres de Lie, Preprint arXiv:0801.0446.
  • [Ni1] Ye. Nisnevich, Étale cohomology and arithmetic of semisimple groups, Ph.D. Thesis. Harvard University, 1982. Available at http://proquest.umi.com/pqdlink?RQT=306
  • [Ni2] Ye. Nisnevich, Espaces homogènes principaux rationnellement triviaux et arithmétique des schémas en groupes réductifs sur les anneaux de Dedekind, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984) 5-8.
  • [P] A. Polishchuk, Lefschetz type formulas for dg-categories, Preprint arXiv:1111.0728.
  • [R] M. Rothstein, Connections on the total Picard sheaf and the KP hierarchy, Acta Applicandae Mathematicae 42 (1996) 297-308.
  • [S] P. Sarnak, Comments on Robert Langland's Lecture: "Endoscopy and Beyond", available at http://www.math.princeton.edu/sarnak/SarnakLectureNotes-1.pdf
  • [TW] R. Taylor and A. Wiles, Ring-theoretic properties of certain Hecke algebras, Ann. of Math. (2) 141 (1995) 553-572.
  • [Ve] A. Venkatesh, ``Beyond endoscopy'' and special forms on $ GL(2)$, J. Reine Angew. Math. 577 (2004) 23-80.
  • [Vi] E. B. Vinberg, On reductive algebraic semigroups, Lie Groups and Lie Algebras, in E. B. Dynkin Seminar (S. Gindikin, E. Vinberg, eds.), AMS Transl. (2) 169 (1995), 145-182.
  • [W] A. Wiles, Modular elliptic curves and Fermat's Last Theorem, Ann. of Math. (2) 141 (1995) 443-551.

Similar Articles

Retrieve articles in Bulletin of the American Mathematical Society with MSC (2010): 11R39, 14D24, 22E57

Retrieve articles in all journals with MSC (2010): 11R39, 14D24, 22E57


Additional Information

Edward Frenkel
Affiliation: Department of Mathematics, University of California, Berkeley, California 94720

DOI: http://dx.doi.org/10.1090/S0273-0979-2012-01387-3
PII: S 0273-0979(2012)01387-3
Received by editor(s): February 10, 2012
Received by editor(s) in revised form: June 17, 2012
Published electronically: October 12, 2012
Additional Notes: Supported by DARPA under the grant HR0011-09-1-0015
Dedicated: Notes for the AMS Colloquium Lectures at the Joint Mathematics Meetings in Boston, January 4–6, 2012
Article copyright: © Copyright 2012 American Mathematical Society