482:
has shown that the dimension bounds the transcendence degree; so that the
Mumford–Tate group catches sufficiently many algebraic relations between the periods. This is a special case of the full Grothendieck period conjecture.
209:
236:
consisting of the integral cohomology classes. Not quite so much is needed for the definition of the
Mumford–Tate group, but it does assume that the vector space
450:
of the Galois image. This conjecture is known only in particular cases. Through generalisations of this conjecture, the
Mumford–Tate group has been connected to the
129:
used to describe Hodge structures has a concrete matrix representation, as the 2×2 invertible matrices of the shape that is given by the action of
478:, in the sense of the field generated by its entries, predicted by the dimension of its Mumford–Tate group, as in the previous section. Work of
155:
595:
351:(1), from the complex field to the real field, an algebraic torus whose character group consists of the two homomorphisms to
564:
647:
527:
240:
underlying the Hodge structure has a given rational structure, i.e. is given over the rational numbers
670:
106:
582:(1967), "Sur les groupes de Galois attachés aux groupes p-divisibles", in Springer, Tonny A. (ed.),
455:
58:
233:
665:
561:
Algebraic Groups and
Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965)
408:
102:
499:
625:
605:
572:
475:
451:
345:
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The original context for the formulation of the group in question was the question of the
8:
638:
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Mumford-Tate groups, families of Calabi-Yau varieties and analogue André-Oort problems I
20:
229:
642:
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225:
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511:
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62:
47:
40:
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305:. Thinking of the matrix in terms of the complex number λ it represents,
659:
556:
467:
215:
82:
74:
612:
Tate, John T. (1967), "p-divisible groups.", in
Springer, Tonny A. (ed.),
447:
412:
422:. Conjecturally, the image of such a Galois representation, which is an
272:
of the Hodge structure describes the action of the diagonal matrices of
358:
Once formulated in this fashion, the rational representation ρ of
515:
539:
528:
http://math.cts.nthu.edu.tw/Mathematics/preprints/prep2005-6-002.pdf
204:{\displaystyle {\begin{bmatrix}a&b\\-b&a\end{bmatrix}}.}
423:
584:
Proceedings of a
Conference on Local Fields (Driebergen, 1966)
395:) is by definition the smallest algebraic group defined over
466:
A related conjecture on abelian varieties states that the
244:. For the purposes of the theory the complex vector space
500:
http://www.math.columbia.edu/~thaddeus/seattle/voisin.pdf
224:
Hodge structures arising in geometry, for example on the
430:, is determined by the corresponding Mumford–Tate group
284:, under that action. Under the action of the full group
454:, and, for example, the general issue of extending the
313:
th power and of the complex conjugate of λ by the
214:
The circle group inside this group of matrices is the
164:
158:
280:is supposed therefore to be homogeneous of weight
203:
657:
512:http://math.berkeley.edu/~ribet/Articles/mg.pdf
297:, complex conjugate in pairs under switching
101:-adic analogue of Mumford's construction for
614:Proc. Conf. Local Fields( Driebergen, 1966)
402:
355:(1), interchanged by complex conjugation.
89:) introduced Mumford–Tate groups over the
73:of the image in the representation of the
559:(1966), "Families of abelian varieties",
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253:, obtained by extending the scalars of
86:
658:
117:, and named them Mumford–Tate groups.
578:
94:
611:
461:
442:)), to the extent that knowledge of
434:(coming from the Hodge structure on
110:
426:Lie group for a given prime number
340:underlying the matrix group is the
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336:In more abstract terms, the torus
14:
682:
632:
540:https://arxiv.org/abs/0805.2569v1
309:has the action of λ by the
93:under the name of Hodge groups.
366:setting up the Hodge structure
533:
521:
505:
493:
120:
1:
565:American Mathematical Society
549:
370:determines the image ρ(
7:
317:th power. Here necessarily
10:
687:
456:Sato–Tate conjecture
293:breaks up into subspaces
141:} of the complex numbers
486:
403:Mumford–Tate conjecture
399:containing this image.
59:rational representation
474:over number field has
205:
409:Galois representation
206:
39:) constructed from a
639:Lecture slides (PDF)
616:, Berlin, New York:
590:, pp. 118–131,
586:, Berlin, New York:
567:, pp. 347–351,
563:, Providence, R.I.:
476:transcendence degree
452:motivic Galois group
346:multiplicative group
156:
125:The algebraic torus
105:, using the work of
65:, the definition of
16:Mathematics concept
580:Serre, Jean-Pierre
201:
192:
115:p-divisible groups
103:Hodge–Tate modules
25:Mumford–Tate group
21:algebraic geometry
643:Phillip Griffiths
597:978-3-540-03953-2
462:Period conjecture
458:(now a theorem).
226:cohomology groups
678:
671:Algebraic groups
651:, preprint (PDF)
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608:
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543:
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525:
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502:, pp. 7–9.
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342:Weil restriction
230:Kähler manifolds
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137:on the basis {1,
79:rational numbers
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618:Springer-Verlag
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588:Springer-Verlag
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446:determines the
417:abelian variety
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97:introduced the
91:complex numbers
71:Zariski closure
63:algebraic torus
48:algebraic group
41:Hodge structure
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633:External links
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557:Mumford, David
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491:
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480:Pierre Deligne
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666:Hodge theory
648:
613:
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514:, survey by
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98:
95:Serre (1967)
75:circle group
66:
54:
50:
43:
36:
32:
28:
24:
18:
448:Lie algebra
413:Tate module
268:The weight
265:, is used.
121:Formulation
77:, over the
29:Hodge group
660:Categories
550:References
69:is as the
516:Ken Ribet
232:, have a
180:−
374:(1)) in
626:0231827
606:0242839
573:0206003
542:, p. 7.
530:, p. 3.
411:on the
387:); and
344:of the
234:lattice
109: (
85: (
83:Mumford
53:. When
624:
604:
594:
571:
424:l-adic
415:of an
276:, and
61:of an
23:, the
487:Notes
257:from
221:(1).
145:over
113:) on
592:ISBN
301:and
111:1967
107:Tate
87:1966
27:(or
641:by
470:of
362:on
261:to
228:of
81:.
19:In
662::
622:MR
620:,
602:MR
600:,
569:MR
389:MT
376:GL
353:GL
349:GL
328:=
324:+
149::
135:bi
33:MT
31:)
518:.
472:A
444:G
440:A
438:(
436:H
432:G
428:l
420:A
397:Q
393:F
391:(
384:C
380:V
378:(
372:U
368:F
364:V
360:T
338:T
332:.
330:k
326:q
322:p
315:q
311:p
307:V
303:q
299:p
295:V
290:C
286:V
282:k
278:V
274:T
270:k
263:C
259:Q
255:V
250:C
246:V
242:Q
238:V
219:U
199:.
194:]
188:a
183:b
173:b
168:a
162:[
147:R
143:C
139:i
133:+
131:a
127:T
99:p
67:G
55:F
51:G
44:F
37:F
35:(
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