106:
found examples where the
Tamagawa numbers are not integers. The second observation, that the Tamagawa numbers of simply connected semisimple groups seem to be 1, became known as the Weil conjecture.
102:
and observed that it is an integer in all considered cases and that it was equal to 1 in the cases when the group is simply connected. The first observation does not hold for all groups:
57:
190:
149:
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292:
165:
615:
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625:
511:
246:
168:
160:
announced a proof of the conjecture for algebraic groups over function fields over finite fields, formally published in
407:
241:
378:
Langlands, R. P. (1966), "The volume of the fundamental domain for some arithmetical subgroups of
Chevalley groups",
284:
121:
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141:
factors. V. I. Chernousov (1989) removed this restriction, by proving the Hasse principle for the resistant
80:
462:
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610:
236:
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487:
437:
334:
182:
used the Weil conjecture to calculate the
Tamagawa numbers of all semisimple algebraic groups.
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479:
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113:
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588:"Yang-Mills theory and Tamagawa Numbers: the fascination of unexpected links in mathematics"
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382:, Proc. Sympos. Pure Math., Providence, R.I.: Amer. Math. Soc., pp. 143–148,
553:
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20:
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133:
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The Siegel Mass
Formula, Tamagawa Numbers, and Nonabelian Poincaré Duality
558:, Progress in Mathematics, vol. 23, Boston, MA: Birkhäuser Boston,
491:
441:
338:
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Chernousov, V. I. (1989), "The Hasse principle for groups of type E8",
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510:, Proc. Sympos. Pure Math., vol. IX, Providence, R.I.:
152:), thus completing the proof of Weil's conjecture. In 2011,
283:, Annals of Mathematics Studies, vol. 199, Princeton:
120:. K. F. Lai (1980) extended the class of known cases to
132:, which at the time was known for all groups without
36:
410:(1963), "On the Tamagawa number of algebraic tori",
544:, Séminaire Bourbaki, vol. 5, pp. 249–257
220:
218:
98:) calculated the Tamagawa number in many cases of
51:
528:Algebraic Groups and their Birational Invariants
400:Tamagawa Numbers via Nonabelian Poincaré Duality
317:Kottwitz, Robert E. (1988), "Tamagawa numbers",
280:Weil's Conjecture for Function Fields (Volume I)
67:defined over a number field is 1. In this case,
356:"Tamagawa number of reductive algebraic groups"
586:Aravind Asok, Brent Doran and Frances Kirwan,
161:
8:
508:Algebraic Groups and Discontinuous Subgroups
458:"On the relative theory of Tamagawa numbers"
380:Algebraic Groups and Discontinuous Subgroups
164:, and a future proof using a version of the
541:Exp. No. 186, Adèles et groupes algébriques
35:
150:strong approximation in algebraic groups
128:proved it for all groups satisfying the
125:
214:
171:will be published in a second volume.
325:(3), Annals of Mathematics: 629–646,
224:
7:
95:
506:Tamagawa, Tsuneo (1966), "Adèles",
191:Smith–Minkowski–Siegel mass formula
189:, the conjecture implies the known
179:
103:
25:Weil conjecture on Tamagawa numbers
14:
16:Conjecture in algebraic geometry
79:sense, which is not always the
46:
40:
1:
526:Voskresenskii, V. E. (1991),
512:American Mathematical Society
162:Gaitsgory & Lurie (2019)
555:Adeles and algebraic groups
242:Encyclopedia of Mathematics
122:quasisplit reductive groups
75:covering" in the algebraic
71:means "not having a proper
642:
285:Princeton University Press
27:is the statement that the
616:Theorems in group theory
52:{\displaystyle \tau (G)}
277:; Lurie, Jacob (2019),
169:Lefschetz trace formula
116:methods to show it for
360:Compositio Mathematica
287:, pp. viii, 311,
53:
463:Annals of Mathematics
456:Ono, Takashi (1965),
413:Annals of Mathematics
54:
626:Diophantine geometry
597:posted June 8, 2012.
514:, pp. 113–121,
81:topologists' meaning
34:
590:, February 22, 2013
354:Lai, K. F. (1980),
256:Soviet Math. Dokl.
112:(1966) introduced
49:
565:978-3-7643-3092-7
530:, AMS translation
466:, Second Series,
416:, Second Series,
294:978-0-691-18213-1
275:Gaitsgory, Dennis
237:"Tamagawa number"
114:harmonic analysis
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621:Algebraic groups
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158:Dennis Gaitsgory
118:Chevalley groups
110:Robert Langlands
100:classical groups
69:simply connected
61:simply connected
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581:Further reading
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203:Tamagawa number
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130:Hasse principle
126:Kottwitz (1988)
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65:algebraic group
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29:Tamagawa number
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175:Applications
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77:group theory
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611:Conjectures
550:Weil, André
536:Weil, André
262:: 592–596,
187:spin groups
154:Jacob Lurie
21:mathematics
605:Categories
593:J. Lurie,
311:1439.14006
225:Lurie 2014
209:References
180:Ono (1965)
148:case (see
104:Ono (1963)
552:(1982) ,
484:0003-486X
434:0003-486X
247:EMS Press
73:algebraic
38:τ
538:(1959),
397:(2014),
197:See also
574:0670072
520:0212025
500:0177991
492:1970563
450:0156851
442:1970502
388:0213362
372:0581580
347:0942522
339:2007007
303:3887650
268:1014762
249:, 2001
94: (
87:History
63:simple
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23:, the
488:JSTOR
438:JSTOR
335:JSTOR
321:, 2,
59:of a
560:ISBN
480:ISSN
430:ISSN
289:ISBN
185:For
156:and
96:1959
92:Weil
472:doi
422:doi
327:doi
323:127
307:Zbl
19:In
607::
570:MR
568:,
516:MR
496:MR
494:,
486:,
478:,
468:82
460:,
446:MR
444:,
436:,
428:,
418:78
384:MR
368:MR
364:41
362:,
358:,
343:MR
341:,
333:,
305:,
299:MR
297:,
264:MR
260:39
258:,
245:,
239:,
217:^
193:.
124:.
83:.
474::
424::
350:.
329::
227:.
146:8
143:E
138:8
135:E
47:)
44:G
41:(
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