60:
proved strong approximation for some classical groups. Strong approximation was established in the 1960s and 1970s, for semisimple simply-connected algebraic groups over
605:
682:
525:
386:
490:"Allgemeine Kongruenzklasseneinteilungen der Ideale einfacher Algebren über algebraischen Zahlkörpern und ihre L-Reihen."
113:
568:
Platonov, V. P. (1969), "The problem of strong approximation and the Kneser–Tits hypothesis for algebraic groups",
677:
475:
32:
624:
17:
649:
598:
Algebraic groups and number theory. (Translated from the 1991 Russian original by Rachel Rowen.)
641:
601:
577:
549:
509:
77:
522:
Algebraic Groups and
Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965)
489:
633:
501:
93:
661:
615:
589:
561:
533:
657:
622:
Prasad, Gopal (1977), "Strong approximation for semi-simple groups over function fields",
611:
585:
557:
529:
463:
441:
36:
671:
540:
Margulis, G. A. (1977), "Cobounded subgroups in algebraic groups over local fields",
455:
69:
358:
101:
89:
85:
65:
61:
43:
600:, Pure and Applied Mathematics, vol. 139, Boston, MA: Academic Press, Inc.,
109:
505:
459:
645:
581:
553:
513:
393:
251:-rational, then it satisfies weak approximation with respect to any set
653:
108:). In the number field case Platonov also proved a related result over
637:
454:
Weak approximation holds for a broader class of groups, including
466:, showing that the strong approximation property is restrictive.
542:
Akademija Nauk SSSR. Funkcional'nyi Analiz i ego Priloženija
373:, p.188) states that a non-solvable linear algebraic group
404:
is simply connected, and each almost simple component
291:
is an algebraic number field then any connected group
570:
Izvestiya
Akademii Nauk SSSR. Seriya Matematicheskaya
295:
satisfies weak approximation with respect to the set
275:
satisfies weak approximation with respect to any set
440:The proofs of strong approximation depended on the
329:) has dense image, or equivalently whether the set
259:, p.402). More generally, for any connected group
284:
256:
128:be a linear algebraic group over a global field
520:Kneser, Martin (1966), "Strong approximation",
494:Journal für die Reine und Angewandte Mathematik
444:for algebraic groups, which for groups of type
596:Platonov, Vladimir; Rapinchuk, Andrei (1994),
369:). The main theorem of strong approximation (
8:
381:has strong approximation for the finite set
313:approximation is whether the embedding of
222:approximation is whether the embedding of
97:
81:
18:Strong approximation in algebraic groups
144:is a non-empty finite set of places of
57:
370:
105:
73:
451:was only proved several years later.
7:
165:for the product of the completions
25:
120:Formal definitions and properties
243:) has dense image. If the group
1:
526:American Mathematical Society
285:Platonov & Rapinchuk 1994
257:Platonov & Rapinchuk 1994
416:has a non-compact component
287:, p.415). In particular, if
27:In algebraic group theory,
699:
506:10.1515/crll.1938.179.227
476:Superstrong approximation
33:Chinese remainder theorem
488:Eichler, Martin (1938),
263:, there is a finite set
31:are an extension of the
309:The question asked in
279:that is disjoint with
218:The question asked in
114:Kneser–Tits conjecture
29:approximation theorems
625:Annals of Mathematics
683:Diophantine geometry
528:, pp. 187–196,
524:, Providence, R.I.:
377:over a global field
306:of infinite places.
267:of finite places of
182:. For any choice of
385:if and only if its
178:in the finite set
136:the adele ring of
64:. The results for
628:, Second Series,
247:is connected and
16:(Redirected from
690:
678:Algebraic groups
664:
618:
592:
564:
548:(2): 45–57, 95,
536:
516:
464:Chevalley groups
152:for the ring of
148:, then we write
37:algebraic groups
21:
698:
697:
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667:
638:10.2307/1970924
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487:
484:
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442:Hasse principle
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352:
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55:
23:
22:
15:
12:
11:
5:
696:
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632:(3): 553–572,
619:
606:
593:
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537:
517:
483:
480:
479:
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471:
468:
456:adjoint groups
448:
433:(depending on
420:
355:
354:
348:
303:
238:
202:
169:
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121:
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86:function field
58:Eichler (1938)
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14:
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2:
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607:0-12-558180-7
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583:
579:
576:: 1211–1219,
575:
571:
566:
563:
559:
555:
551:
547:
543:
538:
535:
531:
527:
523:
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499:
496:(in German),
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491:
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139:
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90:finite fields
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83:
79:
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67:
66:number fields
63:
62:global fields
59:
52:
50:
48:
45:
44:global fields
41:
38:
34:
30:
19:
629:
623:
597:
573:
569:
545:
541:
521:
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401:
397:
389:
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378:
374:
366:
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359:dense subset
356:
349:
345:
341:
337:
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314:
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300:
296:
292:
288:
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264:
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252:
248:
244:
239:
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231:
227:
223:
219:
217:
212:
208:
203:
199:
195:
194:) embeds in
191:
187:
183:
179:
175:
170:
166:
161:
157:
156:-adeles and
153:
149:
145:
141:
137:
133:
129:
125:
123:
110:local fields
92:, is due to
56:
46:
39:
28:
26:
500:: 227–251,
460:inner forms
371:Kneser 1966
112:called the
88:case, over
68:are due to
672:Categories
482:References
271:such that
646:0003-486X
582:0373-2436
554:0374-1990
514:0075-4102
425:for some
394:unipotent
470:See also
94:Margulis
78:Platonov
662:0444571
654:1970924
616:1278263
590:0258839
562:0442107
534:0213361
387:radical
104: (
96: (
84:); the
80: (
72: (
53:History
660:
652:
644:
614:
604:
588:
580:
560:
552:
532:
512:
311:strong
207:) and
174:, for
132:, and
102:Prasad
100:) and
76:) and
70:Kneser
650:JSTOR
357:is a
321:) in
230:) in
140:. If
42:over
642:ISSN
602:ISBN
578:ISSN
550:ISSN
510:ISSN
458:and
220:weak
124:Let
106:1977
98:1977
82:1969
74:1966
634:doi
630:105
502:doi
498:179
462:of
437:).
429:in
408:of
392:is
361:in
215:).
35:to
674::
658:MR
656:,
648:,
640:,
612:MR
610:,
586:MR
584:,
574:33
572:,
558:MR
556:,
546:11
544:,
530:MR
508:,
492:,
396:,
299:=
186:,
116:.
49:.
636::
504::
449:8
446:E
435:H
431:S
427:s
422:s
418:H
414:N
412:/
410:G
406:H
402:N
400:/
398:G
390:N
383:S
379:k
375:G
367:A
365:(
363:G
353:)
350:S
346:A
344:(
342:G
340:)
338:k
336:(
334:G
327:A
325:(
323:G
319:k
317:(
315:G
304:∞
301:S
297:S
293:G
289:k
283:(
281:T
277:S
273:G
269:k
265:T
261:G
255:(
253:S
249:k
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240:S
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234:(
232:G
228:k
226:(
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198:(
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192:k
190:(
188:G
184:S
180:S
176:s
171:s
167:k
162:S
158:A
154:S
150:A
146:k
142:S
138:k
134:A
130:k
126:G
47:k
40:G
20:)
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