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Bulletin of the American Mathematical Society

The Bulletin publishes expository articles on contemporary mathematical research, written in a way that gives insight to mathematicians who may not be experts in the particular topic. The Bulletin also publishes reviews of selected books in mathematics and short articles in the Mathematical Perspectives section, both by invitation only.

ISSN 1088-9485 (online) ISSN 0273-0979 (print)

The 2020 MCQ for Bulletin of the American Mathematical Society is 0.84.

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The geometric nature of the fundamental lemma
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by David Nadler PDF
Bull. Amer. Math. Soc. 49 (2012), 1-50 Request permission


The Fundamental Lemma is a somewhat obscure combinatorial identity introduced by Robert P. Langlands in 1979 as an ingredient in the theory of automorphic representations. After many years of deep contributions by mathematicians working in representation theory, number theory, algebraic geometry, and algebraic topology, a proof of the Fundamental Lemma was recently completed by Ngô Bao Châu in 2008, for which he was awarded a Fields Medal. Our aim here is to touch on some of the beautiful ideas contributing to the Fundamental Lemma and its proof. We highlight the geometric nature of the problem which allows one to attack a question in $p$-adic analysis with the tools of algebraic geometry.
  • James Arthur, The problem of classifying automorphic representations of classical groups, Advances in mathematical sciences: CRM’s 25 years (Montreal, PQ, 1994) CRM Proc. Lecture Notes, vol. 11, Amer. Math. Soc., Providence, RI, 1997, pp. 1–12. MR 1479667, DOI 10.1090/crmp/011/01
  • James Arthur, An introduction to the trace formula, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 1–263. MR 2192011
  • James Arthur, Report on the trace formula, Automorphic forms and $L$-functions I. Global aspects, Contemp. Math., vol. 488, Amer. Math. Soc., Providence, RI, 2009, pp. 1–12. MR 2522025, DOI 10.1090/conm/488/09562
  • D. Ben-Zvi, D. Nadler, The character theory of a complex group, arXiv:0904.1247.
  • P.-H. Chaudouard, G. Laumon, Sur l’homologie des fibres de Springer affines tronquées, arXiv:math/0702586.
  • P.-H. Chaudouard, G. Laumon, Le lemme fondamental pondéré I : constructions géométriques, arXiv:math/0902.2684.
  • P.-H. Chaudouard, G. Laumon, Le lemme fondamental pondéré. II. Énoncés cohomologiques, arXiv:math/0702586.
  • R. Cluckers, T. Hales, F. Loeser, Transfer principle for the Fundamental Lemma, arXiv:0712.0708.
  • Raf Cluckers and François Loeser, Ax-Kochen-ErÅ¡ov theorems for $p$-adic integrals and motivic integration, Geometric methods in algebra and number theory, Progr. Math., vol. 235, Birkhäuser Boston, Boston, MA, 2005, pp. 109–137. MR 2159379, DOI 10.1007/0-8176-4417-2_{5}
  • Raf Cluckers and François Loeser, Constructible exponential functions, motivic Fourier transform and transfer principle, Ann. of Math. (2) 171 (2010), no. 2, 1011–1065. MR 2630060, DOI 10.4007/annals.2010.171.1011
  • Stephen DeBacker, The fundamental lemma: what is it and what do we know?, Current developments in mathematics, 2005, Int. Press, Somerville, MA, 2007, pp. 151–171. MR 2459300
  • Jan Denef and François Loeser, Definable sets, motives and $p$-adic integrals, J. Amer. Math. Soc. 14 (2001), no. 2, 429–469. MR 1815218, DOI 10.1090/S0894-0347-00-00360-X
  • V. Drinfeld, Informal notes available at
  • E. Frenkel, R. Langlands, B. C. Ngô, Formule des Traces et Fonctorialité: le Début d’un Programme, arXiv:1003.4578.
  • E. Frenkel, B. C. Ngô, Geometrization of trace formulas, arXiv:1004.5323.
  • William Fulton and Joe Harris, Representation theory, Graduate Texts in Mathematics, vol. 129, Springer-Verlag, New York, 1991. A first course; Readings in Mathematics. MR 1153249, DOI 10.1007/978-1-4612-0979-9
  • Victor Ginsburg, Intégrales sur les orbites nilpotentes et représentations des groupes de Weyl, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 5, 249–252 (French, with English summary). MR 693785
  • Mark Goresky, Robert Kottwitz, and Robert MacPherson, Equivariant cohomology, Koszul duality, and the localization theorem, Invent. Math. 131 (1998), no. 1, 25–83. MR 1489894, DOI 10.1007/s002220050197
  • Mark Goresky, Robert Kottwitz, and Robert Macpherson, Homology of affine Springer fibers in the unramified case, Duke Math. J. 121 (2004), no. 3, 509–561. MR 2040285, DOI 10.1215/S0012-7094-04-12135-9
  • Mark Goresky, Robert Kottwitz, and Robert MacPherson, Purity of equivalued affine Springer fibers, Represent. Theory 10 (2006), 130–146. MR 2209851, DOI 10.1090/S1088-4165-06-00200-7
  • Mikhail Grinberg, A generalization of Springer theory using nearby cycles, Represent. Theory 2 (1998), 410–431. MR 1657203, DOI 10.1090/S1088-4165-98-00053-3
  • Thomas C. Hales, On the fundamental lemma for standard endoscopy: reduction to unit elements, Canad. J. Math. 47 (1995), no. 5, 974–994. MR 1350645, DOI 10.4153/CJM-1995-051-5
  • Thomas C. Hales, A statement of the fundamental lemma, Harmonic analysis, the trace formula, and Shimura varieties, Clay Math. Proc., vol. 4, Amer. Math. Soc., Providence, RI, 2005, pp. 643–658. MR 2192018
  • M. Harris et. al., The stable trace formula, Shimura varieties, and arithmetic applications, book project available at
  • M. A. de Cataldo, T. Hausel, L. Migliorini, Topology of Hitchin systems and Hodge theory of character varieties, arXiv:1004.1420.
  • Nigel Hitchin, Stable bundles and integrable systems, Duke Math. J. 54 (1987), no. 1, 91–114. MR 885778, DOI 10.1215/S0012-7094-87-05408-1
  • R. Hotta and M. Kashiwara, The invariant holonomic system on a semisimple Lie algebra, Invent. Math. 75 (1984), no. 2, 327–358. MR 732550, DOI 10.1007/BF01388568
  • D. Kazhdan and G. Lusztig, Fixed point varieties on affine flag manifolds, Israel J. Math. 62 (1988), no. 2, 129–168. MR 947819, DOI 10.1007/BF02787119
  • Robert E. Kottwitz, Stable trace formula: cuspidal tempered terms, Duke Math. J. 51 (1984), no. 3, 611–650. MR 757954, DOI 10.1215/S0012-7094-84-05129-9
  • Robert E. Kottwitz, Stable trace formula: elliptic singular terms, Math. Ann. 275 (1986), no. 3, 365–399. MR 858284, DOI 10.1007/BF01458611
  • R. P. Langlands, Les débuts d’une formule des traces stable, Publications Mathématiques de l’Université Paris VII [Mathematical Publications of the University of Paris VII], vol. 13, Université de Paris VII, U.E.R. de Mathématiques, Paris, 1983 (French). MR 697567
  • Robert P. Langlands, Base change for $\textrm {GL}(2)$, Annals of Mathematics Studies, No. 96, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1980. MR 574808
  • R. P. Langlands, Informal remarks available at series.php?series=54.
  • R. P. Langlands, Informal remarks available at series.php?series=56.
  • R. P. Langlands and D. Shelstad, On the definition of transfer factors, Math. Ann. 278 (1987), no. 1-4, 219–271. MR 909227, DOI 10.1007/BF01458070
  • E. Lapid, The relative trace formula and its applications, Automorphic Forms and Automorphic L-Functions (Kyoto, 2005), Surikaisekikenkyusho Kokyuroku No. 1468 (2006), 76-87.
  • G. Laumon, The Fundamental Lemma for Unitary Groups, lecture at Clay Math. Inst., available at
  • G. Laumon, Fundamental Lemma and Hitchin Fibration, lecture at Newton Inst., available at
  • Gérard Laumon, Fibres de Springer et jacobiennes compactifiées, Algebraic geometry and number theory, Progr. Math., vol. 253, Birkhäuser Boston, Boston, MA, 2006, pp. 515–563 (French, with French summary). MR 2263199, DOI 10.1007/978-0-8176-4532-8_{9}
  • G. Laumon, Sur le lemme fondamental pour les groupes unitaires, arXiv:math/0212245.
  • G. Laumon and B. C. Ngô, Le lemme fondamental pour les groupes unitaires, arXiv:math/0404454v2.
  • S. Morel, Étude de la cohomologie de certaines varietes de Shimura non compactes, arXiv:0802.4451.
  • Ngô Báo Châu, Le lemme fondamental de Jacquet et Ye en caractéristique positive, Duke Math. J. 96 (1999), no. 3, 473–520 (French). MR 1671212, DOI 10.1215/S0012-7094-99-09615-1
  • Bao Châu Ngô, Le lemme fondamental pour les algèbres de Lie, Publ. Math. Inst. Hautes Études Sci. 111 (2010), 1–169 (French). MR 2653248, DOI 10.1007/s10240-010-0026-7
  • Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, vol. 123, Princeton University Press, Princeton, NJ, 1990. MR 1081540, DOI 10.1515/9781400882441
  • Jean-Pierre Serre, Linear representations of finite groups, Graduate Texts in Mathematics, Vol. 42, Springer-Verlag, New York-Heidelberg, 1977. Translated from the second French edition by Leonard L. Scott. MR 0450380
  • D. Shelstad, $L$-indistinguishability for real groups, Math. Ann. 259 (1982), no. 3, 385–430. MR 661206, DOI 10.1007/BF01456950
  • S.-W. Shin, Galois representations arising from some compact Shimura varieties, to appear in Annals of Math.
  • J.-L. Waldspurger, Sur les intégrales orbitales tordues pour les groupes linéaires: un lemme fondamental, Canad. J. Math. 43 (1991), no. 4, 852–896 (French). MR 1127034, DOI 10.4153/CJM-1991-049-5
  • J.-L. Waldspurger, Le lemme fondamental implique le transfert, Compositio Math. 105 (1997), no. 2, 153–236 (French). MR 1440722, DOI 10.1023/A:1000103112268
  • J.-L. Waldspurger, Endoscopie et changement de caractéristique, J. Inst. Math. Jussieu 5 (2006), no. 3, 423–525 (French, with English and French summaries). MR 2241929, DOI 10.1017/S1474748006000041
  • J.-L. Waldspurger, L’endoscopie tordue n’est pas si tordue, Mem. Amer. Math. Soc. 194 (2008), no. 908, x+261 (French, with English summary). MR 2418405, DOI 10.1090/memo/0908
  • J.-L. Waldspurger, À propos du lemme fondamental pondéré tordu, Math. Ann. 343 (2009), no. 1, 103–174 (French, with English summary). MR 2448443, DOI 10.1007/s00208-008-0267-7
  • Z. Yun, Towards a Global Springer Theory I: The affine Weyl group action, arXiv:0810.2146.
  • Z. Yun, The fundamental lemma of Jacquet-Rallis in positive characteristics, arXiv:0901.0900.
  • Z. Yun, Towards a Global Springer Theory II: the double affine action, arXiv:0904.3371.
  • Z. Yun, Towards a Global Springer Theory III: Endoscopy and Langlands duality, arXiv:0904.3372.
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Additional Information
  • David Nadler
  • Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208-2370
  • MR Author ID: 620327
  • Email:
  • Received by editor(s): January 30, 2001
  • Received by editor(s) in revised form: April 18, 2011
  • Published electronically: July 26, 2011
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Bull. Amer. Math. Soc. 49 (2012), 1-50
  • MSC (2010): Primary 11R39, 14D24
  • DOI:
  • MathSciNet review: 2869006