Fundamental lemma (Langlands program)

In the mathematical theory of automorphic forms, the fundamental lemma relates orbital integrals on a reductive group over a local field to stable orbital integrals on its endoscopic groups.^{[clarification needed]} It was conjectured by Robert Langlands (1983) in the course of developing the Langlands program. The fundamental lemma was proved by Gérard Laumon and Ngô Bảo Châu in the case of unitary groups and then by Ngô (2010) for general reductive groups, building on a series of important reductions made by Jean-Loup Waldspurger to the case of Lie algebras. Time magazine placed Ngô's proof on the list of the "Top 10 scientific discoveries of 2009".^[1] In 2010, Ngô was awarded the Fields Medal for this proof.

Motivation and history

Langlands outlined a strategy for proving local and global Langlands conjectures using the Arthur–Selberg trace formula, but in order for this approach to work, the geometric sides of the trace formula for different groups must be related in a particular way. This relationship takes the form of identities between orbital integrals on reductive groups G and H over a nonarchimedean local field F, where the group H, called an endoscopic group of G, is constructed from G and some additional data.

The first case considered was $G={\rm {SL}}_{2}$ (Labesse & Langlands 1979). Langlands and Diana Shelstad (1987) then developed the general framework for the theory of endoscopic transfer and formulated specific conjectures. However, during the next two decades only partial progress was made towards proving the fundamental lemma.^[2]^[3] Harris called it a "bottleneck limiting progress on a host of arithmetic questions".^[4] Langlands himself, writing on the origins of endoscopy, commented:

... it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of Shimura varieties; it is the stabilized (or stable) trace formula, the reduction of the trace formula itself to the stable trace formula for a group and its endoscopic groups, and the stabilization of the Grothendieck–Lefschetz formula. None of these are possible without the fundamental lemma and its absence rendered progress almost impossible for more than twenty years.^[5]

Statement

The fundamental lemma states that an orbital integral O for a group G is equal to a stable orbital integral SO for an endoscopic group H, up to a transfer factor Δ (Nadler 2012):

SO_{\gamma _{H}}(1_{K_{H}})=\Delta (\gamma _{H},\gamma _{G})O_{\gamma _{G}}^{\kappa }(1_{K_{G}})

where

F is a local field,
G is an unramified group defined over F, in other words a quasi-split reductive group defined over F that splits over an unramified extension of F,
H is an unramified endoscopic group of G associated to κ,
K_G and K_H are hyperspecial maximal compact subgroups of G and H, which means roughly that they are the subgroups of points with coefficients in the ring of integers of F,
1_{K_G} and 1_{K_H} are the characteristic functions of K_G and K_H,
Δ(γ_H,γ_G) is a transfer factor, a certain elementary expression depending on γ_H and γ_G,
γ_H and γ_G are elements of G and H representing stable conjugacy classes, such that the stable conjugacy class of G is the transfer of the stable conjugacy class of H,
κ is a character of the group of conjugacy classes in the stable conjugacy class of γ_G,
SO and O are stable orbital integrals and orbital integrals depending on their parameters.

Approaches

Shelstad (1982) proved the fundamental lemma for Archimedean fields.

Waldspurger (1991) verified the fundamental lemma for general linear groups.

Kottwitz (1992) and Blasius & Rogawski (1992) verified some cases of the fundamental lemma for 3-dimensional unitary groups.

Hales (1997) and Weissauer (2009) verified the fundamental lemma for the symplectic and general symplectic groups Sp₄, GSp₄.

A paper of George Lusztig and David Kazhdan pointed out that orbital integrals could be interpreted as counting points on certain algebraic varieties over finite fields. Further, the integrals in question can be computed in a way that depends only on the residue field of F; and the issue can be reduced to the Lie algebra version of the orbital integrals. Then the problem was restated in terms of the Springer fiber of algebraic groups.^[6] The circle of ideas was connected to a purity conjecture; Laumon gave a conditional proof based on such a conjecture, for unitary groups. Laumon and Ngô (2008) then proved the fundamental lemma for unitary groups, using Hitchin fibration introduced by Ngô (2006), which is an abstract geometric analogue of the Hitchin system of complex algebraic geometry. Waldspurger (2006) showed for Lie algebras that the function field case implies the fundamental lemma over all local fields, and Waldspurger (2008) showed that the fundamental lemma for Lie algebras implies the fundamental lemma for groups.

Notes

^ "Top 10 Scientific Discoveries of 2009". Time. Archived from the original on December 13, 2009. Retrieved December 14, 2009.
^ Kottwitz and Rogawski for ${\rm {U}}_{3}$ , Wadspurger for ${\rm {SL}}_{n}$ , Hales and Weissauer for ${\rm {Sp}}_{4}$ .
^ Fundamental Lemma and Hitchin Fibration Archived 2011-07-17 at the Wayback Machine, Gérard Laumon, May 13, 2009
^ INTRODUCTION TO “THE STABLE TRACE FORMULA, SHIMURA VARIETIES, AND ARITHMETIC APPLICATIONS” Archived 2009-07-31 at the Wayback Machine, p. 1., Michael Harris
^ publications.ias.edu
^ The Fundamental Lemma for Unitary Groups Archived 2010-06-12 at the Wayback Machine, at p. 12., Gérard Laumon

References

Blasius, Don; Rogawski, Jonathan D. (1992), "Fundamental lemmas for U(3) and related groups", in Langlands, Robert P.; Ramakrishnan, Dinakar (eds.), The zeta functions of Picard modular surfaces, Montreal, QC: Univ. Montréal, pp. 363–394, ISBN 978-2-921120-08-1, MR 1155234
Casselman, W. (2009), Langlands' Fundamental Lemma for SL(2) (PDF)
Dat, Jean-François (November 2004), Lemme fondamental et endoscopie, une approche géométrique, d'après Gérard Laumon et Ngô Bao Châu (PDF), Séminaire Bourbaki, no 940
Hales, Thomas C. (1997), "The fundamental lemma for Sp(4)", Proceedings of the American Mathematical Society, 125 (1): 301–308, doi:10.1090/S0002-9939-97-03546-6, ISSN 0002-9939, MR 1346977
Harris, M. (ed.), Stabilisation de la formule des traces, variétés de Shimura, et applications arithmétiques, archived from the original on 2012-04-20, retrieved 2012-01-04
Kazhdan, David; Lusztig, George (1988), "Fixed point varieties on affine flag manifolds", Israel Journal of Mathematics, 62 (2): 129–168, doi:10.1007/BF02787119, ISSN 0021-2172, MR 0947819
Kottwitz, Robert E. (1992), "Calculation of some orbital integrals", in Langlands, Robert P.; Ramakrishnan, Dinakar (eds.), The zeta functions of Picard modular surfaces, Montreal, QC: Univ. Montréal, pp. 349–362, ISBN 978-2-921120-08-1, MR 1155233
Labesse, Jean-Pierre; Langlands, R. P. (1979), "L-indistinguishability for SL(2)", Canadian Journal of Mathematics, 31 (4): 726–785, doi:10.4153/CJM-1979-070-3, ISSN 0008-414X, MR 0540902, S2CID 17447242
Langlands, Robert P. (1983), 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, Paris: Université de Paris VII U.E.R. de Mathématiques, MR 0697567
Langlands, Robert P.; Shelstad, Diana (1987), "On the definition of transfer factors", Mathematische Annalen, 278 (1): 219–271, doi:10.1007/BF01458070, ISSN 0025-5831, MR 0909227, S2CID 14141632
Laumon, Gérard (2006), "Aspects géométriques du Lemme Fondamental de Langlands-Shelstad", International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, pp. 401–419, MR 2275603, archived from the original on 2012-03-15, retrieved 2012-01-09
Laumon, Gérard; Ngô, Bao Châu (2008), "Le lemme fondamental pour les groupes unitaires", Annals of Mathematics, Second Series, 168 (2): 477–573, arXiv:math/0404454, doi:10.4007/annals.2008.168.477, ISSN 0003-486X, MR 2434884, S2CID 119606388
Nadler, David (2012), "The geometric nature of the fundamental lemma", Bulletin of the American Mathematical Society, 49: 1–50, arXiv:1009.1862, doi:10.1090/S0273-0979-2011-01342-8, ISSN 0002-9904, S2CID 30785271
Ngô, Bao Châu (2006), "Fibration de Hitchin et endoscopie", Inventiones Mathematicae, 164 (2): 399–453, arXiv:math/0406599, Bibcode:2006InMat.164..399N, doi:10.1007/s00222-005-0483-7, ISSN 0020-9910, MR 2218781, S2CID 52064585
Ngô, Bao Châu (2010), "Le lemme fondamental pour les algèbres de Lie", Publications Mathématiques de l'Institut des Hautes Études Scientifiques, 111: 1–169, arXiv:0801.0446, doi:10.1007/s10240-010-0026-7, ISSN 0073-8301, MR 2653248
Shelstad, Diana (1982), "L-indistinguishability for real groups", Mathematische Annalen, 259 (3): 385–430, doi:10.1007/BF01456950, ISSN 0025-5831, MR 0661206, S2CID 121385109
Waldspurger, Jean-Loup (1991), "Sur les intégrales orbitales tordues pour les groupes linéaires: un lemme fondamental", Canadian Journal of Mathematics, 43 (4): 852–896, doi:10.4153/CJM-1991-049-5, ISSN 0008-414X, MR 1127034
Waldspurger, Jean-Loup (2006), "Endoscopie et changement de caractéristique", Journal of the Institute of Mathematics of Jussieu, 5 (3): 423–525, doi:10.1017/S1474748006000041, ISSN 1474-7480, MR 2241929, S2CID 122919302
Waldspurger, Jean-Loup (2008), "L'endoscopie tordue n'est pas si tordue" [Twisted endoscopy is not so twisted] (PDF), Memoirs of the American Mathematical Society (in French), 194 (908), Providence, R.I.: American Mathematical Society: 261, doi:10.1090/memo/0908, ISBN 978-0-8218-4469-4, ISSN 0065-9266, MR 2418405
Weissauer, Rainer (2009), Endoscopy for GSP(4) and the Cohomology of Siegel Modular Threefolds, Lecture Notes in Mathematics, vol. 1968, Berlin, New York: Springer-Verlag, doi:10.1007/978-3-540-89306-6, ISBN 978-3-540-89305-9, MR 2498783

External links

Gerard Laumon lecture on the fundamental lemma for unitary groups
Basken, Paul (September 12, 2010). "Understanding the Langlands Fundamental Lemma". The Chronicle of Higher Education.

[1] "Top 10 Scientific Discoveries of 2009". Time. Archived from the original on December 13, 2009. Retrieved December 14, 2009.

[2] Kottwitz and Rogawski for ${\rm {U}}_{3}$ , Wadspurger for ${\rm {SL}}_{n}$ , Hales and Weissauer for ${\rm {Sp}}_{4}$ .

[3] Fundamental Lemma and Hitchin Fibration Archived 2011-07-17 at the Wayback Machine, Gérard Laumon, May 13, 2009

[4] INTRODUCTION TO “THE STABLE TRACE FORMULA, SHIMURA VARIETIES, AND ARITHMETIC APPLICATIONS” Archived 2009-07-31 at the Wayback Machine, p. 1., Michael Harris

[5] ublications.ias.edu

[6] The Fundamental Lemma for Unitary Groups Archived 2010-06-12 at the Wayback Machine, at p. 12., Gérard Laumon

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