205:) 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. Harris called it a "bottleneck limiting progress on a host of arithmetic questions". Langlands himself, writing on the origins of endoscopy, commented:
387:
636:
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
112:, 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
213:; 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
252:
191:
818:
784:
750:
690:
961:
1288:
902:
Blasius, Don; Rogawski, Jonathan D. (1992), "Fundamental lemmas for U(3) and related groups", in
Langlands, Robert P.; Ramakrishnan, Dinakar (eds.),
851:
1718:
1655:
1609:
1072:
911:
832:
217:. None of these are possible without the fundamental lemma and its absence rendered progress almost impossible for more than twenty years.
214:
881:
1000:
1690:
209:... it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of
109:
641:; and the issue can be reduced to the Lie algebra version of the orbital integrals. Then the problem was restated in terms of the
1723:
1494:
1143:, Publications Mathématiques de l'Université Paris VII , vol. 13, Paris: Université de Paris VII U.E.R. de Mathématiques,
1091:
700:
1593:
1022:
1492:
Waldspurger, Jean-Loup (1991), "Sur les intégrales orbitales tordues pour les groupes linéaires: un lemme fondamental",
92:
magazine placed NgĂ´'s proof on the list of the "Top 10 scientific discoveries of 2009". In 2010, NgĂ´ was awarded the
1728:
1713:
1708:
1202:
382:{\displaystyle SO_{\gamma _{H}}(1_{K_{H}})=\Delta (\gamma _{H},\gamma _{G})O_{\gamma _{G}}^{\kappa }(1_{K_{G}})}
1336:
1138:
649:; Laumon gave a conditional proof based on such a conjecture, for unitary groups. Laumon and NgĂ´ (
669:
showed for Lie algebras that the function field case implies the fundamental lemma over all local fields, and
67:
17:
848:
455:, which means roughly that they are the subgroups of points with coefficients in the ring of integers of
1448:
1232:
1157:
1063:(1992), "Calculation of some orbital integrals", in Langlands, Robert P.; Ramakrishnan, Dinakar (eds.),
105:
79:
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Lemme fondamental et endoscopie, une approche géométrique, d'après Gérard Laumon et Ngô Bao Châu
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59:
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showed that the fundamental lemma for Lie algebras implies the fundamental lemma for groups.
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Laumon, Gérard (2006), "Aspects géométriques du Lemme
Fondamental de Langlands-Shelstad",
1182:
1144:
1118:
1078:
1060:
1049:
1020:
Kazhdan, David; Lusztig, George (1988), "Fixed point varieties on affine flag manifolds",
1006:
1002:
Stabilisation de la formule des traces, variétés de
Shimura, et applications arithmétiques
988:
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885:
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INTRODUCTION TO “THE STABLE TRACE FORMULA, SHIMURA VARIETIES, AND ARITHMETIC APPLICATIONS”
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117:
88:
39:
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Laumon, Gérard; Ngô, Bao Châu (2008), "Le lemme fondamental pour les groupes unitaires",
1359:
1155:
Langlands, Robert P.; Shelstad, Diana (1987), "On the definition of transfer factors",
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662:
642:
629:
198:
1702:
1569:
1485:
1279:
942:
633:
563:Îş is a character of the group of conjugacy classes in the stable conjugacy class of Îł
71:
1391:
1327:
1311:
1194:
1130:
929:
1089:
Labesse, Jean-Pierre; Langlands, R. P. (1979), "L-indistinguishability for SL(2)",
93:
83:
78:
for general reductive groups, building on a series of important reductions made by
975:
617:
verified the fundamental lemma for the symplectic and general symplectic groups Sp
579:
are stable orbital integrals and orbital integrals depending on their parameters.
1255:
128:
43:
607:
verified some cases of the fundamental lemma for 3-dimensional unitary groups.
552:
representing stable conjugacy classes, such that the stable conjugacy class of
1647:
1545:
1532:
Waldspurger, Jean-Loup (2006), "Endoscopie et changement de caractéristique",
1422:
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1431:
1375:
1319:
1263:
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1114:
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984:
1508:
1105:
1601:
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Publications Mathématiques de l'Institut des Hautes Études
Scientifiques
1461:
1398:
Ngô, Bao Châu (2010), "Le lemme fondamental pour les algèbres de Lie",
1286:
Nadler, David (2012), "The geometric nature of the fundamental lemma",
1170:
1036:
1216:
517:) is a transfer factor, a certain elementary expression depending on Îł
1350:
1246:
1636:
Endoscopy for GSP(4) and the
Cohomology of Siegel Modular Threefolds
1638:, Lecture Notes in Mathematics, vol. 1968, Berlin, New York:
1446:
Shelstad, Diana (1982), "L-indistinguishability for real groups",
1412:
1302:
1679:
Gerard Laumon lecture on the fundamental lemma for unitary groups
653:) then proved the fundamental lemma for unitary groups, using
1334:
Ngô, Bao Châu (2006), "Fibration de
Hitchin et endoscopie",
959:
Hales, Thomas C. (1997), "The fundamental lemma for Sp(4)",
645:
of algebraic groups. The circle of ideas was connected to a
409:, in other words a quasi-split reductive group defined over
104:
Langlands outlined a strategy for proving local and global
597:
verified the fundamental lemma for general linear groups.
226:
The fundamental lemma states that an orbital integral
792:
758:
727:
591:
255:
159:
1534:Journal of the Institute of Mathematics of Jussieu
1067:, Montreal, QC: Univ. Montréal, pp. 349–362,
906:, Montreal, QC: Univ. Montréal, pp. 363–394,
812:
778:
744:
661:), which is an abstract geometric analogue of the
381:
185:
1204:International Congress of Mathematicians. Vol. II
556:is the transfer of the stable conjugacy class of
202:
962:Proceedings of the American Mathematical Society
194:
1686:"Understanding the Langlands Fundamental Lemma"
604:
207:
1580:[Twisted endoscopy is not so twisted]
447:are hyperspecial maximal compact subgroups of
1289:Bulletin of the American Mathematical Society
1207:, Eur. Math. Soc., Zürich, pp. 401–419,
1065:The zeta functions of Picard modular surfaces
904:The zeta functions of Picard modular surfaces
8:
1586:Memoirs of the American Mathematical Society
413:that splits over an unramified extension of
670:
666:
594:
18:Fundamental lemma of Langlands and Shelstad
1140:Les débuts d'une formule des traces stable
1578:"L'endoscopie tordue n'est pas si tordue"
1507:
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55:
879:The Fundamental Lemma for Unitary Groups
600:
588:
830:Fundamental Lemma and Hitchin Fibration
691:"Top 10 Scientific Discoveries of 2009"
682:
62:. The fundamental lemma was proved by
931:Langlands' Fundamental Lemma for SL(2)
243:
234:is equal to a stable orbital integral
610:
423:is an unramified endoscopic group of
7:
941:Dat, Jean-François (November 2004),
484:are the characteristic functions of
405:is an unramified group defined over
1684:Basken, Paul (September 12, 2010).
658:
75:
46:to stable orbital integrals on its
799:
796:
765:
762:
731:
302:
172:
169:
58:) in the course of developing the
25:
1691:The Chronicle of Higher Education
186:{\displaystyle G={\rm {SL}}_{2}}
1576:Waldspurger, Jean-Loup (2008),
1495:Canadian Journal of Mathematics
1312:10.1090/S0273-0979-2011-01342-8
1092:Canadian Journal of Mathematics
665:of complex algebraic geometry.
38:relates orbital integrals on a
813:{\displaystyle {\rm {Sp}}_{4}}
779:{\displaystyle {\rm {SL}}_{n}}
376:
356:
331:
305:
296:
276:
215:Grothendieck–Lefschetz formula
153:The first case considered was
30:In the mathematical theory of
1:
1594:American Mathematical Society
1137:Langlands, Robert P. (1983),
1023:Israel Journal of Mathematics
976:10.1090/S0002-9939-97-03546-6
839:, GĂ©rard Laumon, May 13, 2009
745:{\displaystyle {\rm {U}}_{3}}
605:Blasius & Rogawski (1992)
242:, up to a transfer factor Δ (
1719:Theorems in abstract algebra
195:Labesse & Langlands 1979
110:Arthur–Selberg trace formula
1256:10.4007/annals.2008.168.477
27:Theorem in abstract algebra
1745:
1634:Weissauer, Rainer (2009),
888:, at p. 12., GĂ©rard Laumon
786:, Hales and Weissauer for
721:Kottwitz and Rogawski for
150:and some additional data.
1648:10.1007/978-3-540-89306-6
1592:(908), Providence, R.I.:
1546:10.1017/S1474748006000041
1423:10.1007/s10240-010-0026-7
1368:10.1007/s00222-005-0483-7
197:). Langlands and
1337:Inventiones Mathematicae
657:introduced by NgĂ´ (
238:for an endoscopic group
50:. It was conjectured by
1724:Lemmas in number theory
858:, p. 1., Michael Harris
1509:10.4153/CJM-1991-049-5
1106:10.4153/CJM-1979-070-3
928:Casselman, W. (2009),
814:
780:
746:
383:
219:
187:
146:, is constructed from
127:over a nonarchimedean
100:Motivation and history
1449:Mathematische Annalen
1233:Annals of Mathematics
1158:Mathematische Annalen
815:
781:
747:
384:
188:
106:Langlands conjectures
80:Jean-Loup Waldspurger
868:publications.ias.edu
790:
756:
725:
703:on December 13, 2009
253:
157:
52:Robert Langlands
1360:2006InMat.164..399N
1061:Kottwitz, Robert E.
355:
1462:10.1007/BF01456950
1171:10.1007/BF01458070
1037:10.1007/BF02787119
999:Harris, M. (ed.),
952:SĂ©minaire Bourbaki
884:2010-06-12 at the
854:2009-07-31 at the
835:2011-07-17 at the
810:
776:
742:
671:Waldspurger (2008)
667:Waldspurger (2006)
595:Waldspurger (1991)
379:
334:
183:
134:, where the group
1729:Langlands program
1714:Automorphic forms
1657:978-3-540-89305-9
1611:978-0-8218-4469-4
1602:10.1090/memo/0908
1236:, Second Series,
1074:978-2-921120-08-1
913:978-2-921120-08-1
752:, Wadspurger for
655:Hitchin fibration
647:purity conjecture
399:is a local field,
211:Shimura varieties
114:orbital integrals
60:Langlands program
48:endoscopic groups
36:fundamental lemma
32:automorphic forms
16:(Redirected from
1736:
1709:Algebraic groups
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1005:, archived from
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615:Weissauer (2009)
544:are elements of
427:associated to Îş,
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140:endoscopic group
118:reductive groups
96:for this proof.
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601:Kottwitz (1992)
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82:to the case of
70:in the case of
40:reductive group
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15:
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5:
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1673:External links
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1610:
1573:
1540:(3): 423–525,
1529:
1502:(4): 852–896,
1489:
1456:(3): 385–430,
1443:
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1344:(2): 399–453,
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663:Hitchin system
643:Springer fiber
630:George Lusztig
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1399:
1351:math/0406599
1341:
1335:
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1247:math/0404454
1237:
1231:
1221:, retrieved
1217:the original
1203:
1162:
1156:
1139:
1096:
1090:
1064:
1027:
1021:
1011:, retrieved
1007:the original
1001:
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930:
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84:Lie algebras
74:and then by
68:Ngô Bảo Châu
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628:A paper of
244:Nadler 2012
129:local field
44:local field
1703:Categories
1223:2012-01-09
1013:2012-01-04
896:References
584:Approaches
108:using the
76:NgĂ´ (2010)
1620:0065-9266
1570:122919302
1554:1474-7480
1518:0008-414X
1486:121385109
1470:0025-5831
1432:0073-8301
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1376:0020-9910
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310:γ
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266:γ
222:Statement
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1123:0540902
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