6475:
3788:. This classification is equivalent to that of connected algebraic groups over the complex field, and Cartan subalgebras correspond to maximal tori in these. In fact the classification carries over to the case of an arbitrary base field under the assumption that there exists a split maximal torus (which is automatically satisfied over an algebraically closed field). Without the splitness assumption things become much more complicated and a more detailed theory has to be developed, which is still based in part on the study of adjoint actions of tori.
5848:. These provide representations of fundamental groupoids of the base with respect the fpqc topology. If the torus is locally trivializable with respect to a weaker topology such as the etale topology, then the sheaves of groups descend to the same topologies and these representations factor through the respective quotient groupoids. In particular, an etale sheaf gives rise to a quasi-isotrivial torus, and if
2530:
6462:
The notion of invariant given above generalizes naturally to tori over arbitrary base schemes, with functions taking values in more general rings. While the order of the extension group is a general invariant, the other two invariants above do not seem to have interesting analogues outside the realm
3147:
th power map on the multiplicative group. The functor given by taking weights is an antiequivalence of categories between tori and free abelian groups, and the coweight functor is an equivalence. In particular, maps of tori are characterized by linear transformations on weights or coweights, and
6038:
Since taking a weight lattice is an equivalence of categories, short exact sequences of tori correspond to short exact sequences of the corresponding weight lattices. In particular, extensions of tori are classified by Ext sheaves. These are naturally isomorphic to the flat cohomology groups
3447:
is the multiplicative group, then this gives the restriction of scalars a permutation module structure. Tori whose weight lattices are permutation modules for the Galois group are called quasi-split, and all quasi-split tori are finite products of restrictions of scalars.
2368:
3438:
6157:
1323:
6434:
3914:
577:
4936:. It is in fact a maximal flat subspace and all maximal such are obtained as orbits of split tori in this way. Thus there is a geometric definition of the real rank, as the maximal dimension of a flat subspace in
3254:
3081:
1032:
2791:-torus of rank one is either split or isomorphic to the kernel of the norm of a quadratic extension. The two examples above are special cases of this: the compact real torus is the kernel of the field norm of
4113:
2271:
295:
6328:
T. Ono showed that the
Tamagawa number of a torus over a number field is such an invariant. Furthermore, he showed that it is a quotient of two cohomological invariants, namely the order of the group
5711:
2026:
1779:
1738:
3953:
1495:
2903:
3263:) and arbitrary abelian groups, and such a generalization can be convenient if one wants to work in a well-behaved category, since the category of tori doesn't have kernels or filtered colimits.
6022:
5810:
4657:
4700:
4160:
5944:
5883:
for which their base extensions are isomorphic, i.e., it is a torus of the same rank. Isomorphism classes of twisted forms of a split torus are parametrized by nonabelian flat cohomology
436:
3137:
729:
363:
219:
150:
56:
932:
5749:
2821:
332:
2227:
2089:
5169:
4766:
4394:
This reduces the classification problem to anisotropic groups, and to determining which Tits indices can occur for a given Dynkin diagram. The latter problem has been solved in
4308:
4273:
1082:
4199:
1851:
5486:
5373:
2850:
2311:
2194:
2162:
1805:
85:
3004:
2957:
2125:
1207:
1137:
3319:
6607:
5508:
5437:
5395:
5344:
5308:
5283:
5241:
5219:
5137:
5115:
5070:
5048:
5026:
5004:
4569:
4547:
4330:
4015:
3833:
3811:
3712:
3553:
3499:
2056:
1633:
1611:
1563:
1537:
1403:
1232:
188:
107:
4359:
4044:
804:
4874:
4801:
6024:
with trivial Galois action on the coefficients. In the one-dimensional case, the coefficients form a group of order two, and isomorphism classes of twisted forms of
5093:
2739:
1349:
6042:
4239:
4219:
3853:
833:
392:
4726:
2769:
2560:
2363:
2337:
1438:
879:
757:
2525:{\displaystyle \left\{{\begin{pmatrix}t&du\\u&t\end{pmatrix}}:t,u\in \mathbb {F} _{q},t^{2}-du^{2}=1\right\}\subset \mathrm {SL} _{2}(\mathbb {F} _{q}).}
5830:
5457:
5415:
5261:
5193:
4982:
4954:
4934:
4914:
4894:
4848:
4824:
4609:
4589:
4521:
4488:
4464:
4444:
4420:
4387:
3993:
3973:
3756:
3732:
3686:
3662:
3642:
3620:
3597:
3573:
3519:
2789:
2704:
2684:
2664:
2644:
2624:
2604:
2580:
1891:
1871:
1680:
1653:
1587:
1515:
1373:
1252:
1180:
1158:
1110:
853:
777:
698:
674:
637:
617:
597:
499:
479:
504:
441:
Tori are of fundamental importance in the theory of algebraic groups and Lie groups and in the study of the geometric objects associated to them such as
2706:-rank 1 (note that restriction of scalars over an inseparable field extension will yield a commutative algebraic group that is not a torus). The kernel
1257:
3152:. The quasi-inverse of the weights functor is given by a dualization functor from free abelian groups to tori, defined by its functor of points as:
6331:
3286:
with algebraic homomorphisms and the category of finitely generated torsion free abelian groups with an action of the absolute Galois group of
3858:
3274:
are defined as the respective lattices over the separable closure. This induces canonical continuous actions of the absolute Galois group of
3158:
3020:
4332:(the latter is only a reductive group). As its name indicates it is an anisotropic group, and its absolute type is uniquely determined by
5754:
461:(for example finite or characteristic zero). This hypothesis is required to have a smooth group scheme, since for an algebraic group
4049:
2232:
227:
6534:
5664:
1896:
1699:
if there exists an isogeny from the first to the second. Isogenies between tori are particularly well-behaved: for any isogeny
6648:
3919:
3278:
on the lattices. The weights and coweights that are fixed by this action are precisely the maps that are defined over
1443:
2855:
6168:
1743:
1702:
5957:
4614:
3017:. These are both free abelian groups whose rank is that of the torus, and they have a canonical nondegenerate pairing
937:
4662:
4389:-algebraic groups are isomorphic if and only if they have the same Tits indices and isomorphic anisotropic kernels.
4118:
6667:
6628:
Tits, Jacques (1966). "Classification of algebraic semisimple groups". In Borel, Armand; Mostow, George D. (eds.).
5886:
397:
3086:
703:
337:
193:
124:
30:
884:
5719:
3461:
As seen in the examples above tori can be represented as linear groups. An alternative definition for tori is:
2918:
2794:
300:
3259:
This equivalence can be generalized to pass between groups of multiplicative type (a distinguished class of
2203:
2065:
6159:. Over a field, the extensions are parametrized by elements of the corresponding Galois cohomology group.
5640:
5142:
6172:
4731:
4282:
4247:
1037:
6456:
4169:
3433:{\displaystyle X^{\bullet }(\mathrm {Res} _{L/K}T)\cong \mathrm {Ind} _{G_{L}}^{G_{K}}X^{\bullet }(T).}
1823:
6175:
introduced a type of functorial invariants of tori over finite separable extensions of a chosen field
5462:
5349:
2826:
2287:
2170:
2133:
1784:
806:. To define it properly as an algebraic group one can take the affine variety defined by the equation
61:
5946:, where the coefficient group forms a constant sheaf. In particular, twisted forms of a split torus
2973:
2926:
2099:
1185:
1115:
6498:
3466:
A linear algebraic group is a torus if and only if it is diagonalisable over an algebraic closure.
5639:. In general, the multiplicity of this product (i.e., the dimension of the scheme) is called the
5491:
5420:
5378:
5327:
5291:
5266:
5224:
5202:
5120:
5098:
5053:
5031:
5009:
4987:
4552:
4530:
4313:
3998:
3816:
3794:
3695:
3536:
3482:
2039:
1616:
1594:
1546:
1520:
1386:
1215:
171:
90:
4335:
4020:
782:
4853:
4279:: this is the semisimple algebraic group obtained as the derived subgroup of the centraliser of
6152:{\displaystyle H^{1}(S,\mathrm {Hom} _{\mathbb {Z} }(X^{\bullet }(T_{1}),X^{\bullet }(T_{2})))}
5860:
asserts that any torus of finite type is quasi-isotrivial, i.e., split by an etale surjection.
5319:
4771:
3773:
446:
1695:
between algebraic groups is a surjective morphism with finite kernel; two tori are said to be
5078:
2709:
1328:
5853:
5524:
4224:
4204:
3838:
651:
809:
368:
8:
6672:
5310:-rank of any lattice in a semisimple Lie group, as the dimension of its asymptotic cone.
4705:
3282:. The functor of taking weights is an antiequivalence between the category of tori over
2748:
2537:
2342:
2316:
2165:
1415:
858:
734:
153:
6632:. Proceedings of symposia in pure math. Vol. 9. American math. soc. pp. 33–62.
4221:. The Tits index is a "relative" version of the "absolute" Dynkin diagram associated to
6480:
5815:
5442:
5400:
5246:
5196:
5178:
4967:
4939:
4919:
4899:
4879:
4833:
4809:
4594:
4574:
4506:
4473:
4449:
4429:
4405:
4372:
4241:; obviously, only finitely many Tits indices can correspond to a given Dynkin diagram.
4166:
is a way to encode the properties of this map and of the action of the Galois group of
3978:
3958:
3741:
3717:
3671:
3647:
3627:
3605:
3582:
3558:
3504:
2774:
2689:
2669:
2649:
2629:
2609:
2589:
2565:
1876:
1856:
1807:
is a power map. In particular being isogenous is an equivalence relation between tori.
1665:
1638:
1572:
1500:
1358:
1237:
1165:
1143:
1095:
838:
762:
683:
659:
622:
602:
582:
484:
464:
6644:
6527:
6474:
5714:
4423:
4399:
3777:
3471:
The torus is split over a field if and only if it is diagonalisable over this field.
2200:
Any real torus is isogenous to a finite sum of those two; for example the real torus
121:. Higher dimensional algebraic tori can be modelled as a product of algebraic groups
114:
5073:
2583:
2128:
1318:{\displaystyle \mathbf {T} ({\overline {F}})\cong ({\overline {F}}^{\times })^{r}}
5172:
4827:
2921:
442:
165:
110:
4470:
of this quasi-split group, and those are classified by the Galois cohomology of
6488:
3785:
118:
5844:, weights and coweights are defined as fpqc sheaves of free abelian groups on
5650:
Most notions defined for tori over fields carry to this more general setting.
6661:
6429:{\displaystyle H^{1}(G_{k},X^{\bullet }(T))\cong Ext^{1}(T,\mathbb {G} _{m})}
5543:
3310:
2164:. It is an anisotropic torus. As a Lie group, it is also isomorphic to the 1-
2094:
458:
5546:
isomorphic to a finite product of copies of the multiplicative group scheme
642:
In general one has to use separable closures instead of algebraic closures.
6503:
6437:
5857:
5535:
3260:
2282:
881:. The multiplication is then given by restricting the regular rational map
222:
3270:
is not separably closed, the weight and coweight lattices of a torus over
5659:
3781:
20:
5072:
has also a geometric significance. To get to it one has to introduce an
3909:{\displaystyle V=X^{*}(\mathbf {T} )\otimes _{\mathbb {Z} }\mathbb {R} }
2196:, which explains the picture of diagonalisable algebraic groups as tori.
1853:
there is up to isomorphism a unique torus of any given rank. For a rank
6179:. Such an invariant is a collection of positive real-valued functions
4467:
4163:
3767:
2742:
572:{\displaystyle (\cdot )^{p^{r}}:{\mathcal {O}}(G)\to {\mathcal {O}}(G)}
5622:
an algebraic group whose extension to some finite separable extension
4364:
The first step towards a classification is then the following theorem
3148:
the automorphism group of a torus is a general linear group over
6463:
of fraction fields of one-dimensional domains and their completions.
6031:
are in natural bijection with separable quadratic extensions of
3249:{\displaystyle D(M)_{S}(X):=\mathrm {Hom} (M,\mathbb {G} _{m,S}(X)).}
3076:{\displaystyle X^{\bullet }(T)\times X_{\bullet }(T)\to \mathbb {Z} }
161:
5171:, which is a Riemannian orbifold and hence a metric space. Then any
3692:
if and only if equality holds (that is, there is a maximal torus in
5954:
are parametrized by elements of the Galois cohomology pointed set
5856:), the torus is isotrivial. As a partial converse, a theorem of
4422:. More precisely to each Tits index there is associated a unique
1692:
5513:
3835:
then over the algebraic closure it gives rise to a root system
1034:
and the inverse is the restriction of the regular rational map
6555:. Translations of mathematical monographs. American Math. Soc.
6528:"Algebraic Groups: The Theory of Group Schemes of Finite Type"
4498:
4108:{\displaystyle X^{*}(\mathbf {T} )\to X^{*}(_{F}\mathbf {T} )}
6493:
4876:
is a maximal split torus then there exists a unique orbit of
4571:-algebraic group whose group of real points is isomorphic to
2266:{\displaystyle \mathbb {R} ^{\times }\times \mathbb {T} ^{1}}
6638:
6559:
5751:. Then, with the origin removed, the induced projection map
5658:
One common example of an algebraic torus is to consider the
290:{\displaystyle \mathbb {C} ^{*}={\text{Spec}}(\mathbb {C} )}
2273:. This gives an example of isogenous, non-isomorphic tori.
1209:
to a finite product of copies of the multiplicative group.
2058:
there are exactly (up to isomorphism) two tori of rank 1:
1351:. The basic terminology associated to tori is as follows.
365:-action on a complex vector space can be pulled back to a
297:, which is the scheme theoretic analogue of the Lie group
6583:
6571:
6313:
whose weight lattice is a projective Galois module, then
5706:{\displaystyle {\text{Aff}}(X)\subset \mathbb {A} ^{n+1}}
5397:
correspond to the apartments of the Bruhat-Tits building
5199:
with top-dimensional simplices of dimension equal to the
2313:
there are two rank-1 tori: the split one, of cardinality
2021:{\displaystyle \mathbf {G} _{m}^{n}={\text{Spec}}_{k}(k)}
27:, where a one dimensional torus is typically denoted by
5643:
of the torus, and it is a locally constant function on
5563:. In other words, there exists a faithfully flat map
5155:
3948:{\displaystyle {}_{F}\mathbf {T} \subset \mathbf {T} }
3761:
2382:
1490:{\displaystyle \mathbf {T} (E)\cong (E^{\times })^{r}}
6334:
6045:
5960:
5889:
5818:
5757:
5722:
5667:
5494:
5465:
5445:
5423:
5403:
5381:
5352:
5330:
5294:
5269:
5249:
5227:
5205:
5181:
5145:
5123:
5101:
5081:
5056:
5034:
5012:
4990:
4984:
is obtained as the real points of an algebraic group
4970:
4942:
4922:
4902:
4882:
4856:
4836:
4812:
4774:
4734:
4708:
4665:
4617:
4597:
4577:
4555:
4533:
4509:
4476:
4452:
4432:
4408:
4375:
4338:
4316:
4285:
4250:
4227:
4207:
4172:
4121:
4052:
4023:
4001:
3981:
3961:
3922:
3861:
3841:
3819:
3797:
3744:
3720:
3698:
3674:
3650:
3630:
3608:
3585:
3561:
3539:
3507:
3485:
3322:
3161:
3089:
3023:
2976:
2929:
2898:{\displaystyle \mathbb {F} _{q^{2}}/\mathbb {F} _{q}}
2858:
2829:
2797:
2777:
2751:
2712:
2692:
2672:
2652:
2632:
2612:
2592:
2568:
2540:
2371:
2345:
2319:
2290:
2235:
2206:
2173:
2136:
2102:
2068:
2042:
1899:
1879:
1859:
1826:
1815:
1787:
1746:
1705:
1668:
1641:
1619:
1597:
1575:
1549:
1523:
1503:
1446:
1418:
1389:
1361:
1331:
1260:
1240:
1218:
1188:
1168:
1146:
1118:
1098:
1040:
940:
887:
861:
841:
812:
785:
765:
737:
706:
686:
662:
625:
605:
585:
507:
487:
467:
400:
371:
340:
303:
230:
196:
174:
127:
93:
64:
33:
6470:
3813:is a maximal torus in a semisimple algebraic group
3474:
1774:{\displaystyle \psi :\mathbf {T} '\to \mathbf {T} }
1733:{\displaystyle \phi :\mathbf {T} \to \mathbf {T} '}
6428:
6151:
6016:
5938:
5852:is locally noetherian and normal (more generally,
5824:
5804:
5743:
5705:
5502:
5480:
5451:
5431:
5409:
5389:
5367:
5338:
5302:
5277:
5255:
5235:
5213:
5187:
5163:
5131:
5117:, which roughly is the group of integer points of
5109:
5087:
5064:
5042:
5020:
4998:
4976:
4948:
4928:
4908:
4888:
4868:
4842:
4818:
4795:
4760:
4720:
4694:
4651:
4603:
4583:
4563:
4541:
4515:
4482:
4458:
4438:
4414:
4381:
4353:
4324:
4302:
4267:
4233:
4213:
4193:
4154:
4107:
4038:
4009:
3987:
3967:
3947:
3908:
3847:
3827:
3805:
3780:play a fundamental rĂ´le in the classification via
3750:
3726:
3706:
3680:
3656:
3636:
3614:
3591:
3567:
3547:
3513:
3493:
3432:
3248:
3131:
3075:
2998:
2951:
2897:
2844:
2815:
2783:
2763:
2745:is also a torus, which is anisotropic and of rank
2733:
2698:
2678:
2658:
2638:
2618:
2598:
2574:
2554:
2524:
2357:
2331:
2305:
2265:
2221:
2188:
2156:
2119:
2083:
2050:
2020:
1885:
1865:
1845:
1799:
1773:
1732:
1674:
1647:
1627:
1605:
1581:
1557:
1531:
1509:
1489:
1432:
1397:
1367:
1343:
1317:
1246:
1226:
1201:
1174:
1152:
1131:
1104:
1076:
1026:
926:
873:
847:
827:
798:
771:
751:
723:
692:
668:
645:
631:
611:
591:
571:
493:
473:
430:
386:
357:
326:
289:
213:
182:
144:
101:
79:
50:
6017:{\displaystyle H^{1}(G_{K},GL_{n}(\mathbb {Z} ))}
5805:{\displaystyle \pi :({\text{Aff}}(X)-\{0\})\to X}
4652:{\displaystyle (\mathbb {R} ^{\times })^{r}\to G}
3456:
2365:. The latter can be realised as the matrix group
457:In most places we suppose that the base field is
6659:
6553:Algebraic groups and their birational invariants
4775:
4244:Another invariant associated to the split torus
3668:Obviously the rank is greater than or equal the
3555:(note that all maximal tori are conjugated over
1497:. There is a unique minimal finite extension of
1027:{\displaystyle ((x,y),(x',y'))\mapsto (xx',yy')}
599:, meaning the image of the corresponding map on
5812:gives the structure of an algebraic torus over
4695:{\displaystyle \mathrm {SL} _{n}(\mathbb {R} )}
2093:the compact form, which can be realised as the
579:must be geometrically reduced for large enough
6272:Restriction: For a finite separable extension
4155:{\displaystyle \Phi \to {}_{F}\Phi \cup \{0\}}
6550:
5939:{\displaystyle H^{1}(S,GL_{n}(\mathbb {Z} ))}
4916:which is a totally geodesic flat subspace in
3624:) is the maximal rank of a torus subgroup in
3533:) is the rank of a maximal torus subgroup in
3501:is a semisimple algebraic group over a field
3006:is the group of algebraic homomorphisms
2229:is doubly covered by (but not isomorphic to)
452:
6636:
6589:
6577:
5790:
5784:
5514:Algebraic tori over an arbitrary base scheme
4149:
4143:
3451:
1613:is the maximal rank of a split sub-torus of
431:{\displaystyle U(1)\subset \mathbb {C} ^{*}}
6298:evaluated on its restriction of scalars to
5610:. One particularly important case is when
5579:whose image is an open affine subscheme of
4499:Flat subspaces and rank of symmetric spaces
3132:{\displaystyle (f,g)\mapsto \deg(f\circ g)}
724:{\displaystyle \mathbf {G} _{\mathbf {m} }}
358:{\displaystyle \mathbf {G} _{\mathbf {m} }}
214:{\displaystyle \mathbf {G} _{\mathbf {m} }}
145:{\displaystyle \mathbf {G} _{\mathbf {m} }}
51:{\displaystyle \mathbf {G} _{\mathbf {m} }}
5879:for which there exists a fpqc covering of
3143:such that the composition is equal to the
927:{\displaystyle F^{2}\times F^{2}\to F^{2}}
6630:Algebraic groups and discontinuous groups
6413:
6194:runs over finite separable extensions of
6079:
6004:
5926:
5744:{\displaystyle X\subset \mathbb {P} ^{n}}
5731:
5687:
5468:
5355:
5207:
5036:
5014:
4685:
4623:
4557:
4535:
3902:
3895:
3293:Given a finite separable field extension
3215:
3069:
2885:
2861:
2832:
2816:{\displaystyle \mathbb {C} /\mathbb {R} }
2809:
2799:
2506:
2433:
2339:, and the anisotropic one of cardinality
2293:
2253:
2238:
2209:
2071:
2044:
418:
320:
255:
233:
176:
168:). For example, over the complex numbers
156:were named by analogy with the theory of
95:
67:
6622:On the Tamagawa number of algebraic tori
6162:
4490:with coefficients in the adjoint group.
2959:is the group of algebraic homomorphisms
2908:
2031:
327:{\displaystyle U(1)\subset \mathbb {C} }
3738:if it contains no split tori (i.e. its
2913:Over a separably closed field, a torus
6660:
5575:has a quasi-compact open neighborhood
2562:is a finite field extension of degree
2276:
2222:{\displaystyle \mathbb {C} ^{\times }}
2084:{\displaystyle \mathbb {R} ^{\times }}
1635:. A torus is split if and only if its
5587:yields a finite product of copies of
5164:{\displaystyle M=\Gamma \backslash X}
4959:
6627:
6565:
6521:
6519:
6186:on isomorphism classes of tori over
5288:Note that this allows to define the
4611:such that there exists an embedding
4493:
4395:
1820:Over any algebraically closed field
1254:-group it is a torus if and only if
779:-points are isomorphic to the group
4523:is a semisimple Lie group then its
3762:Classification of semisimple groups
2917:admits two primary invariants. The
2852:is the kernel of the field norm of
1162:is an algebraic group defined over
13:
6624:Annals of Mathematics 78 (1) 1963.
6073:
6070:
6067:
5152:
5082:
4761:{\displaystyle \mathrm {SO} (p,q)}
4739:
4736:
4671:
4668:
4348:
4303:{\displaystyle {}_{F}\mathbf {T} }
4268:{\displaystyle {}_{F}\mathbf {T} }
4228:
4208:
4137:
4122:
4033:
4017:gives rise to another root system
3842:
3379:
3376:
3373:
3344:
3341:
3338:
3200:
3197:
3194:
2491:
2488:
2141:
2138:
1893:this is given by the group scheme
1816:Over an algebraically closed field
1391:
1112:be a field with algebraic closure
1077:{\displaystyle (x,y)\mapsto (y,x)}
731:such that for any field extension
555:
536:
109:, is a type of commutative affine
14:
6684:
6640:Introduction to Arithmetic Groups
6525:
6516:
6436:(sometimes mistakenly called the
6202:Multiplicativity: Given two tori
5875:, a twisted form is a torus over
5626:is a finite product of copies of
5439:. In particular the dimension of
4466:-group with the same index is an
4194:{\displaystyle {\overline {F}}/F}
1846:{\displaystyle k={\overline {k}}}
481:to be smooth over characteristic
6643:. Deductive Press. p. 492.
6540:from the original on 2016-03-07.
6473:
5496:
5481:{\displaystyle \mathbb {Q} _{p}}
5425:
5383:
5368:{\displaystyle \mathbb {Q} _{p}}
5332:
5296:
5271:
5229:
5125:
5103:
5058:
4992:
4659:. For example, the real rank of
4549:-rank as defined above (for any
4318:
4296:
4261:
4098:
4067:
4003:
3941:
3933:
3882:
3821:
3799:
3700:
3541:
3487:
3475:Split rank of a semisimple group
2845:{\displaystyle \mathbb {F} _{q}}
2306:{\displaystyle \mathbb {F} _{q}}
2189:{\displaystyle \mathbf {T} ^{1}}
2176:
2157:{\displaystyle \mathrm {SO} (2)}
2104:
1902:
1800:{\displaystyle \psi \circ \phi }
1767:
1755:
1722:
1713:
1621:
1599:
1551:
1525:
1448:
1262:
1220:
715:
709:
349:
343:
205:
199:
136:
130:
80:{\displaystyle \mathbb {G} _{m}}
42:
36:
6444:, although it doesn't classify
6198:, satisfying three properties:
5518:
4398:. The former is related to the
3975:-split torus its action on the
2999:{\displaystyle X_{\bullet }(T)}
2952:{\displaystyle X^{\bullet }(T)}
2823:and the anisotropic torus over
2626:of the multiplicative group of
2120:{\displaystyle \mathbf {U} (1)}
2036:Over the field of real numbers
1655:-rank equals its absolute rank.
1202:{\displaystyle {\overline {F}}}
1132:{\displaystyle {\overline {F}}}
646:Multiplicative group of a field
6544:
6423:
6402:
6380:
6377:
6371:
6345:
6146:
6143:
6140:
6127:
6111:
6098:
6085:
6056:
6011:
6008:
6000:
5971:
5933:
5930:
5922:
5900:
5796:
5793:
5778:
5772:
5764:
5679:
5673:
4790:
4778:
4755:
4743:
4689:
4681:
4643:
4634:
4618:
4591:), in other words the maximal
4125:
4102:
4088:
4074:
4071:
4063:
3886:
3878:
3457:Linear representations of tori
3424:
3418:
3365:
3333:
3240:
3237:
3231:
3204:
3187:
3181:
3172:
3165:
3126:
3114:
3105:
3102:
3090:
3065:
3062:
3056:
3040:
3034:
2993:
2987:
2946:
2940:
2516:
2501:
2151:
2145:
2114:
2108:
2015:
2012:
1938:
1932:
1763:
1740:there exists a "dual" isogeny
1717:
1539:is split, which is called the
1478:
1464:
1458:
1452:
1306:
1285:
1279:
1266:
1071:
1059:
1056:
1053:
1041:
1021:
993:
990:
987:
984:
962:
956:
944:
941:
911:
566:
560:
550:
547:
541:
515:
508:
410:
404:
381:
375:
313:
307:
284:
281:
259:
251:
1:
6599:
6551:Voskresenskii, V. S. (1998).
5653:
3575:so the rank is well-defined);
3139:, where degree is the number
1087:
115:projective algebraic geometry
6457:Tate–Shafarevich group
5503:{\displaystyle \mathbf {G} }
5432:{\displaystyle \mathbf {G} }
5390:{\displaystyle \mathbf {G} }
5339:{\displaystyle \mathbf {G} }
5313:
5303:{\displaystyle \mathbf {Q} }
5278:{\displaystyle \mathbf {G} }
5236:{\displaystyle \mathbf {G} }
5214:{\displaystyle \mathbb {Q} }
5195:is homeomorphic to a finite
5132:{\displaystyle \mathbf {G} }
5110:{\displaystyle \mathbf {G} }
5065:{\displaystyle \mathbf {G} }
5043:{\displaystyle \mathbb {Q} }
5021:{\displaystyle \mathbb {Q} }
4999:{\displaystyle \mathbf {G} }
4564:{\displaystyle \mathbb {R} }
4542:{\displaystyle \mathbb {R} }
4325:{\displaystyle \mathbf {G} }
4178:
4010:{\displaystyle \mathbf {G} }
3828:{\displaystyle \mathbf {G} }
3806:{\displaystyle \mathbf {T} }
3707:{\displaystyle \mathbf {G} }
3548:{\displaystyle \mathbf {G} }
3494:{\displaystyle \mathbf {G} }
2051:{\displaystyle \mathbb {R} }
1838:
1686:
1628:{\displaystyle \mathbf {T} }
1606:{\displaystyle \mathbf {T} }
1558:{\displaystyle \mathbf {T} }
1532:{\displaystyle \mathbf {T} }
1398:{\displaystyle \mathrm {T} }
1294:
1274:
1227:{\displaystyle \mathbf {T} }
1194:
1124:
183:{\displaystyle \mathbb {C} }
102:{\displaystyle \mathbb {T} }
7:
6637:Witte-Morris, Dave (2015).
6466:
5614:is the spectrum of a field
5583:, such that base change to
5346:is a semisimple group over
4354:{\displaystyle {}_{F}\Phi }
4039:{\displaystyle {}_{F}\Phi }
3776:over the complex field the
3772:In the classical theory of
3688:-rank; the group is called
2970:, and the coweight lattice
1810:
799:{\displaystyle E^{\times }}
619:is smooth for large enough
394:-action from the inclusion
10:
6689:
6305:Projective triviality: If
5840:For a general base scheme
5835:
5530:, an algebraic torus over
5375:the maximal split tori in
5317:
5263:is compact if and only if
4869:{\displaystyle T\subset G}
3765:
649:
453:Algebraic tori over fields
5854:geometrically unibranched
5139:, and the quotient space
4796:{\displaystyle \min(p,q)}
3452:Tori in semisimple groups
1182:which is isomorphic over
835:in the affine plane over
6509:
6499:Torus based cryptography
6455:), and the order of the
5006:over the rational field
3916:. On the other hand, if
1408:The torus is said to be
16:Specific algebraic group
6668:Linear algebraic groups
5571:such that any point in
5088:{\displaystyle \Gamma }
4728:, and the real rank of
3774:semisimple Lie algebras
3734:). The group is called
2734:{\displaystyle N_{E/F}}
1412:over a field extension
1344:{\displaystyle r\geq 1}
700:is the algebraic group
6430:
6153:
6018:
5940:
5826:
5806:
5745:
5707:
5618:, making a torus over
5504:
5482:
5453:
5433:
5411:
5391:
5369:
5340:
5320:Building (mathematics)
5304:
5279:
5257:
5237:
5215:
5189:
5165:
5133:
5111:
5089:
5066:
5044:
5022:
5000:
4978:
4950:
4930:
4910:
4890:
4870:
4844:
4820:
4797:
4762:
4722:
4696:
4653:
4605:
4585:
4565:
4543:
4517:
4484:
4460:
4440:
4416:
4383:
4355:
4326:
4304:
4269:
4235:
4215:
4195:
4156:
4109:
4046:. The restriction map
4040:
4011:
3989:
3969:
3949:
3910:
3849:
3829:
3807:
3752:
3728:
3708:
3682:
3658:
3638:
3616:
3593:
3569:
3549:
3515:
3495:
3434:
3250:
3133:
3077:
3000:
2953:
2899:
2846:
2817:
2785:
2765:
2735:
2700:
2680:
2660:
2640:
2620:
2600:
2576:
2556:
2526:
2359:
2333:
2307:
2267:
2223:
2190:
2158:
2121:
2085:
2052:
2022:
1887:
1867:
1847:
1801:
1775:
1734:
1676:
1658:A torus is said to be
1649:
1629:
1607:
1583:
1559:
1533:
1511:
1491:
1434:
1399:
1369:
1345:
1319:
1248:
1228:
1203:
1176:
1154:
1133:
1106:
1078:
1028:
928:
875:
849:
829:
800:
773:
753:
725:
694:
670:
633:
613:
593:
573:
495:
475:
432:
388:
359:
328:
291:
215:
184:
146:
103:
81:
52:
6431:
6163:Arithmetic invariants
6154:
6019:
5941:
5827:
5807:
5746:
5708:
5505:
5483:
5454:
5434:
5412:
5392:
5370:
5341:
5305:
5280:
5258:
5238:
5216:
5190:
5166:
5134:
5112:
5090:
5067:
5045:
5023:
5001:
4979:
4951:
4931:
4911:
4891:
4871:
4845:
4821:
4798:
4763:
4723:
4697:
4654:
4606:
4586:
4566:
4544:
4518:
4485:
4461:
4441:
4417:
4384:
4356:
4327:
4305:
4270:
4236:
4234:{\displaystyle \Phi }
4216:
4214:{\displaystyle \Phi }
4196:
4157:
4110:
4041:
4012:
3990:
3970:
3950:
3911:
3850:
3848:{\displaystyle \Phi }
3830:
3808:
3753:
3729:
3709:
3683:
3659:
3639:
3617:
3594:
3570:
3550:
3516:
3496:
3435:
3251:
3134:
3078:
3001:
2954:
2909:Weights and coweights
2900:
2847:
2818:
2786:
2766:
2736:
2701:
2681:
2661:
2641:
2621:
2601:
2577:
2557:
2527:
2360:
2334:
2308:
2268:
2224:
2191:
2159:
2122:
2086:
2053:
2032:Over the real numbers
2023:
1888:
1873:algebraic torus over
1868:
1848:
1802:
1776:
1735:
1677:
1650:
1630:
1608:
1584:
1560:
1534:
1512:
1492:
1435:
1400:
1370:
1346:
1320:
1249:
1229:
1204:
1177:
1155:
1134:
1107:
1079:
1029:
929:
876:
850:
830:
801:
774:
754:
726:
695:
671:
634:
614:
594:
574:
496:
476:
433:
389:
360:
329:
292:
221:is isomorphic to the
216:
185:
147:
104:
82:
53:
6332:
6043:
5958:
5887:
5816:
5755:
5720:
5665:
5492:
5463:
5443:
5421:
5401:
5379:
5350:
5328:
5292:
5267:
5247:
5225:
5203:
5179:
5143:
5121:
5099:
5079:
5054:
5032:
5010:
4988:
4968:
4940:
4920:
4900:
4880:
4854:
4834:
4810:
4772:
4732:
4706:
4663:
4615:
4595:
4575:
4553:
4531:
4507:
4474:
4450:
4430:
4406:
4373:
4336:
4314:
4283:
4248:
4225:
4205:
4170:
4119:
4050:
4021:
3999:
3979:
3959:
3920:
3859:
3855:in the vector space
3839:
3817:
3795:
3742:
3718:
3714:which is split over
3696:
3672:
3648:
3644:which is split over
3628:
3606:
3583:
3559:
3537:
3505:
3483:
3320:
3159:
3087:
3021:
2974:
2927:
2856:
2827:
2795:
2775:
2749:
2710:
2690:
2670:
2650:
2630:
2610:
2590:
2566:
2538:
2369:
2343:
2317:
2288:
2233:
2204:
2171:
2134:
2100:
2066:
2040:
1897:
1877:
1857:
1824:
1785:
1744:
1703:
1666:
1639:
1617:
1595:
1573:
1547:
1521:
1501:
1444:
1416:
1387:
1359:
1329:
1258:
1238:
1216:
1186:
1166:
1144:
1116:
1096:
1038:
938:
885:
859:
839:
828:{\displaystyle xy=1}
810:
783:
763:
735:
704:
684:
678:multiplicative group
676:is a field then the
660:
652:Multiplicative group
623:
603:
583:
505:
485:
465:
398:
387:{\displaystyle U(1)}
369:
338:
301:
228:
194:
190:the algebraic torus
172:
125:
91:
62:
31:
6616:On Tamagawa Numbers
5534:is defined to be a
4721:{\displaystyle n-1}
3407:
2764:{\displaystyle d-1}
2555:{\displaystyle E/F}
2534:More generally, if
2358:{\displaystyle q+1}
2332:{\displaystyle q-1}
2277:Over a finite field
2011:
1971:
1916:
1433:{\displaystyle E/F}
1212:In other words, if
874:{\displaystyle x,y}
752:{\displaystyle E/F}
438:as real manifolds.
6481:Mathematics portal
6426:
6291:torus is equal to
6149:
6014:
5936:
5822:
5802:
5741:
5703:
5500:
5478:
5449:
5429:
5407:
5387:
5365:
5336:
5300:
5275:
5253:
5233:
5211:
5197:simplicial complex
5185:
5161:
5129:
5107:
5085:
5062:
5040:
5018:
4996:
4974:
4960:Q-rank of lattices
4946:
4926:
4906:
4886:
4866:
4840:
4816:
4793:
4758:
4718:
4692:
4649:
4601:
4581:
4561:
4539:
4513:
4480:
4456:
4436:
4412:
4379:
4351:
4322:
4300:
4277:anisotropic kernel
4265:
4231:
4211:
4191:
4152:
4105:
4036:
4007:
3985:
3965:
3945:
3906:
3845:
3825:
3803:
3778:Cartan subalgebras
3748:
3724:
3704:
3678:
3654:
3634:
3612:
3601:(sometimes called
3589:
3565:
3545:
3511:
3491:
3430:
3371:
3246:
3129:
3073:
2996:
2949:
2895:
2842:
2813:
2781:
2761:
2731:
2696:
2676:
2656:
2636:
2616:
2596:
2572:
2552:
2522:
2410:
2355:
2329:
2303:
2263:
2219:
2186:
2154:
2127:or as the special
2117:
2081:
2048:
2018:
1994:
1954:
1900:
1883:
1863:
1843:
1797:
1771:
1730:
1672:
1645:
1625:
1603:
1579:
1555:
1529:
1507:
1487:
1430:
1395:
1365:
1341:
1315:
1244:
1224:
1199:
1172:
1150:
1129:
1102:
1074:
1024:
924:
871:
845:
825:
796:
769:
749:
721:
690:
666:
629:
609:
589:
569:
491:
471:
428:
384:
355:
324:
287:
211:
180:
142:
113:commonly found in
99:
77:
48:
6650:978-0-9865716-0-2
6611:Exp. VIII–X
6605:A. Grothendieck,
6590:Witte-Morris 2015
6578:Witte-Morris 2015
5825:{\displaystyle X}
5770:
5715:projective scheme
5671:
5452:{\displaystyle X}
5410:{\displaystyle X}
5256:{\displaystyle M}
5243:. In particular,
5188:{\displaystyle M}
4977:{\displaystyle G}
4964:If the Lie group
4949:{\displaystyle X}
4929:{\displaystyle X}
4909:{\displaystyle X}
4889:{\displaystyle T}
4843:{\displaystyle G}
4819:{\displaystyle X}
4604:{\displaystyle r}
4584:{\displaystyle G}
4516:{\displaystyle G}
4494:Tori and geometry
4483:{\displaystyle F}
4459:{\displaystyle F}
4439:{\displaystyle F}
4424:quasi-split group
4415:{\displaystyle F}
4400:Galois cohomology
4382:{\displaystyle F}
4181:
3988:{\displaystyle F}
3968:{\displaystyle F}
3751:{\displaystyle F}
3727:{\displaystyle F}
3681:{\displaystyle F}
3657:{\displaystyle F}
3637:{\displaystyle G}
3615:{\displaystyle F}
3592:{\displaystyle F}
3568:{\displaystyle F}
3514:{\displaystyle F}
2784:{\displaystyle F}
2699:{\displaystyle F}
2679:{\displaystyle d}
2659:{\displaystyle F}
2639:{\displaystyle E}
2619:{\displaystyle F}
2599:{\displaystyle E}
2575:{\displaystyle d}
1924:
1886:{\displaystyle k}
1866:{\displaystyle n}
1841:
1675:{\displaystyle F}
1648:{\displaystyle F}
1582:{\displaystyle F}
1510:{\displaystyle F}
1368:{\displaystyle r}
1297:
1277:
1247:{\displaystyle F}
1197:
1175:{\displaystyle F}
1153:{\displaystyle F}
1127:
1105:{\displaystyle F}
855:with coordinates
848:{\displaystyle F}
772:{\displaystyle E}
693:{\displaystyle F}
669:{\displaystyle F}
632:{\displaystyle r}
612:{\displaystyle G}
592:{\displaystyle r}
494:{\displaystyle p}
474:{\displaystyle G}
249:
6680:
6654:
6633:
6593:
6587:
6581:
6575:
6569:
6568:, Theorem 2.7.1.
6563:
6557:
6556:
6548:
6542:
6541:
6539:
6532:
6523:
6483:
6478:
6477:
6435:
6433:
6432:
6427:
6422:
6421:
6416:
6401:
6400:
6370:
6369:
6357:
6356:
6344:
6343:
6309:is a torus over
6287:evaluated on an
6169:Tamagawa numbers
6158:
6156:
6155:
6150:
6139:
6138:
6126:
6125:
6110:
6109:
6097:
6096:
6084:
6083:
6082:
6076:
6055:
6054:
6023:
6021:
6020:
6015:
6007:
5999:
5998:
5983:
5982:
5970:
5969:
5945:
5943:
5942:
5937:
5929:
5921:
5920:
5899:
5898:
5831:
5829:
5828:
5823:
5811:
5809:
5808:
5803:
5771:
5768:
5750:
5748:
5747:
5742:
5740:
5739:
5734:
5712:
5710:
5709:
5704:
5702:
5701:
5690:
5672:
5669:
5509:
5507:
5506:
5501:
5499:
5487:
5485:
5484:
5479:
5477:
5476:
5471:
5459:is equal to the
5458:
5456:
5455:
5450:
5438:
5436:
5435:
5430:
5428:
5416:
5414:
5413:
5408:
5396:
5394:
5393:
5388:
5386:
5374:
5372:
5371:
5366:
5364:
5363:
5358:
5345:
5343:
5342:
5337:
5335:
5309:
5307:
5306:
5301:
5299:
5285:is anisotropic.
5284:
5282:
5281:
5276:
5274:
5262:
5260:
5259:
5254:
5242:
5240:
5239:
5234:
5232:
5220:
5218:
5217:
5212:
5210:
5194:
5192:
5191:
5186:
5170:
5168:
5167:
5162:
5138:
5136:
5135:
5130:
5128:
5116:
5114:
5113:
5108:
5106:
5094:
5092:
5091:
5086:
5074:arithmetic group
5071:
5069:
5068:
5063:
5061:
5049:
5047:
5046:
5041:
5039:
5027:
5025:
5024:
5019:
5017:
5005:
5003:
5002:
4997:
4995:
4983:
4981:
4980:
4975:
4955:
4953:
4952:
4947:
4935:
4933:
4932:
4927:
4915:
4913:
4912:
4907:
4895:
4893:
4892:
4887:
4875:
4873:
4872:
4867:
4849:
4847:
4846:
4841:
4825:
4823:
4822:
4817:
4802:
4800:
4799:
4794:
4767:
4765:
4764:
4759:
4742:
4727:
4725:
4724:
4719:
4701:
4699:
4698:
4693:
4688:
4680:
4679:
4674:
4658:
4656:
4655:
4650:
4642:
4641:
4632:
4631:
4626:
4610:
4608:
4607:
4602:
4590:
4588:
4587:
4582:
4570:
4568:
4567:
4562:
4560:
4548:
4546:
4545:
4540:
4538:
4522:
4520:
4519:
4514:
4489:
4487:
4486:
4481:
4465:
4463:
4462:
4457:
4445:
4443:
4442:
4437:
4421:
4419:
4418:
4413:
4388:
4386:
4385:
4380:
4360:
4358:
4357:
4352:
4347:
4346:
4341:
4331:
4329:
4328:
4323:
4321:
4309:
4307:
4306:
4301:
4299:
4294:
4293:
4288:
4274:
4272:
4271:
4266:
4264:
4259:
4258:
4253:
4240:
4238:
4237:
4232:
4220:
4218:
4217:
4212:
4200:
4198:
4197:
4192:
4187:
4182:
4174:
4161:
4159:
4158:
4153:
4136:
4135:
4130:
4114:
4112:
4111:
4106:
4101:
4096:
4095:
4086:
4085:
4070:
4062:
4061:
4045:
4043:
4042:
4037:
4032:
4031:
4026:
4016:
4014:
4013:
4008:
4006:
3995:-Lie algebra of
3994:
3992:
3991:
3986:
3974:
3972:
3971:
3966:
3954:
3952:
3951:
3946:
3944:
3936:
3931:
3930:
3925:
3915:
3913:
3912:
3907:
3905:
3900:
3899:
3898:
3885:
3877:
3876:
3854:
3852:
3851:
3846:
3834:
3832:
3831:
3826:
3824:
3812:
3810:
3809:
3804:
3802:
3758:-rank is zero).
3757:
3755:
3754:
3749:
3733:
3731:
3730:
3725:
3713:
3711:
3710:
3705:
3703:
3687:
3685:
3684:
3679:
3663:
3661:
3660:
3655:
3643:
3641:
3640:
3635:
3621:
3619:
3618:
3613:
3598:
3596:
3595:
3590:
3574:
3572:
3571:
3566:
3554:
3552:
3551:
3546:
3544:
3520:
3518:
3517:
3512:
3500:
3498:
3497:
3492:
3490:
3439:
3437:
3436:
3431:
3417:
3416:
3406:
3405:
3404:
3394:
3393:
3392:
3382:
3361:
3360:
3356:
3347:
3332:
3331:
3255:
3253:
3252:
3247:
3230:
3229:
3218:
3203:
3180:
3179:
3138:
3136:
3135:
3130:
3082:
3080:
3079:
3074:
3072:
3055:
3054:
3033:
3032:
3005:
3003:
3002:
2997:
2986:
2985:
2958:
2956:
2955:
2950:
2939:
2938:
2904:
2902:
2901:
2896:
2894:
2893:
2888:
2882:
2877:
2876:
2875:
2874:
2864:
2851:
2849:
2848:
2843:
2841:
2840:
2835:
2822:
2820:
2819:
2814:
2812:
2807:
2802:
2790:
2788:
2787:
2782:
2770:
2768:
2767:
2762:
2740:
2738:
2737:
2732:
2730:
2729:
2725:
2705:
2703:
2702:
2697:
2685:
2683:
2682:
2677:
2665:
2663:
2662:
2657:
2645:
2643:
2642:
2637:
2625:
2623:
2622:
2617:
2605:
2603:
2602:
2597:
2584:Weil restriction
2581:
2579:
2578:
2573:
2561:
2559:
2558:
2553:
2548:
2531:
2529:
2528:
2523:
2515:
2514:
2509:
2500:
2499:
2494:
2482:
2478:
2471:
2470:
2455:
2454:
2442:
2441:
2436:
2415:
2414:
2364:
2362:
2361:
2356:
2338:
2336:
2335:
2330:
2312:
2310:
2309:
2304:
2302:
2301:
2296:
2272:
2270:
2269:
2264:
2262:
2261:
2256:
2247:
2246:
2241:
2228:
2226:
2225:
2220:
2218:
2217:
2212:
2195:
2193:
2192:
2187:
2185:
2184:
2179:
2163:
2161:
2160:
2155:
2144:
2129:orthogonal group
2126:
2124:
2123:
2118:
2107:
2090:
2088:
2087:
2082:
2080:
2079:
2074:
2062:the split torus
2057:
2055:
2054:
2049:
2047:
2027:
2025:
2024:
2019:
2010:
2002:
1990:
1989:
1970:
1962:
1950:
1949:
1931:
1930:
1925:
1922:
1915:
1910:
1905:
1892:
1890:
1889:
1884:
1872:
1870:
1869:
1864:
1852:
1850:
1849:
1844:
1842:
1834:
1806:
1804:
1803:
1798:
1780:
1778:
1777:
1772:
1770:
1762:
1758:
1739:
1737:
1736:
1731:
1729:
1725:
1716:
1681:
1679:
1678:
1673:
1654:
1652:
1651:
1646:
1634:
1632:
1631:
1626:
1624:
1612:
1610:
1609:
1604:
1602:
1588:
1586:
1585:
1580:
1564:
1562:
1561:
1556:
1554:
1538:
1536:
1535:
1530:
1528:
1516:
1514:
1513:
1508:
1496:
1494:
1493:
1488:
1486:
1485:
1476:
1475:
1451:
1439:
1437:
1436:
1431:
1426:
1404:
1402:
1401:
1396:
1394:
1374:
1372:
1371:
1366:
1350:
1348:
1347:
1342:
1324:
1322:
1321:
1316:
1314:
1313:
1304:
1303:
1298:
1290:
1278:
1270:
1265:
1253:
1251:
1250:
1245:
1233:
1231:
1230:
1225:
1223:
1208:
1206:
1205:
1200:
1198:
1190:
1181:
1179:
1178:
1173:
1159:
1157:
1156:
1151:
1138:
1136:
1135:
1130:
1128:
1120:
1111:
1109:
1108:
1103:
1083:
1081:
1080:
1075:
1033:
1031:
1030:
1025:
1020:
1006:
983:
972:
933:
931:
930:
925:
923:
922:
910:
909:
897:
896:
880:
878:
877:
872:
854:
852:
851:
846:
834:
832:
831:
826:
805:
803:
802:
797:
795:
794:
778:
776:
775:
770:
758:
756:
755:
750:
745:
730:
728:
727:
722:
720:
719:
718:
712:
699:
697:
696:
691:
675:
673:
672:
667:
638:
636:
635:
630:
618:
616:
615:
610:
598:
596:
595:
590:
578:
576:
575:
570:
559:
558:
540:
539:
530:
529:
528:
527:
500:
498:
497:
492:
480:
478:
477:
472:
443:symmetric spaces
437:
435:
434:
429:
427:
426:
421:
393:
391:
390:
385:
364:
362:
361:
356:
354:
353:
352:
346:
333:
331:
330:
325:
323:
296:
294:
293:
288:
280:
279:
258:
250:
247:
242:
241:
236:
220:
218:
217:
212:
210:
209:
208:
202:
189:
187:
186:
181:
179:
151:
149:
148:
143:
141:
140:
139:
133:
108:
106:
105:
100:
98:
86:
84:
83:
78:
76:
75:
70:
57:
55:
54:
49:
47:
46:
45:
39:
6688:
6687:
6683:
6682:
6681:
6679:
6678:
6677:
6658:
6657:
6651:
6602:
6597:
6596:
6588:
6584:
6576:
6572:
6564:
6560:
6549:
6545:
6537:
6530:
6524:
6517:
6512:
6479:
6472:
6469:
6450:
6417:
6412:
6411:
6396:
6392:
6365:
6361:
6352:
6348:
6339:
6335:
6333:
6330:
6329:
6319:
6297:
6286:
6268:
6261:
6254:
6247:
6240:
6233:
6226:
6215:
6208:
6185:
6167:In his work on
6165:
6134:
6130:
6121:
6117:
6105:
6101:
6092:
6088:
6078:
6077:
6066:
6065:
6050:
6046:
6044:
6041:
6040:
6030:
6003:
5994:
5990:
5978:
5974:
5965:
5961:
5959:
5956:
5955:
5925:
5916:
5912:
5894:
5890:
5888:
5885:
5884:
5838:
5817:
5814:
5813:
5767:
5756:
5753:
5752:
5735:
5730:
5729:
5721:
5718:
5717:
5691:
5686:
5685:
5668:
5666:
5663:
5662:
5656:
5634:
5605:
5596:
5554:
5521:
5516:
5495:
5493:
5490:
5489:
5472:
5467:
5466:
5464:
5461:
5460:
5444:
5441:
5440:
5424:
5422:
5419:
5418:
5402:
5399:
5398:
5382:
5380:
5377:
5376:
5359:
5354:
5353:
5351:
5348:
5347:
5331:
5329:
5326:
5325:
5322:
5316:
5295:
5293:
5290:
5289:
5270:
5268:
5265:
5264:
5248:
5245:
5244:
5228:
5226:
5223:
5222:
5206:
5204:
5201:
5200:
5180:
5177:
5176:
5173:asymptotic cone
5144:
5141:
5140:
5124:
5122:
5119:
5118:
5102:
5100:
5097:
5096:
5080:
5077:
5076:
5057:
5055:
5052:
5051:
5035:
5033:
5030:
5029:
5013:
5011:
5008:
5007:
4991:
4989:
4986:
4985:
4969:
4966:
4965:
4962:
4941:
4938:
4937:
4921:
4918:
4917:
4901:
4898:
4897:
4881:
4878:
4877:
4855:
4852:
4851:
4835:
4832:
4831:
4828:symmetric space
4811:
4808:
4807:
4773:
4770:
4769:
4735:
4733:
4730:
4729:
4707:
4704:
4703:
4684:
4675:
4667:
4666:
4664:
4661:
4660:
4637:
4633:
4627:
4622:
4621:
4616:
4613:
4612:
4596:
4593:
4592:
4576:
4573:
4572:
4556:
4554:
4551:
4550:
4534:
4532:
4529:
4528:
4508:
4505:
4504:
4501:
4496:
4475:
4472:
4471:
4451:
4448:
4447:
4431:
4428:
4427:
4407:
4404:
4403:
4374:
4371:
4370:
4369:Two semisimple
4342:
4340:
4339:
4337:
4334:
4333:
4317:
4315:
4312:
4311:
4295:
4289:
4287:
4286:
4284:
4281:
4280:
4260:
4254:
4252:
4251:
4249:
4246:
4245:
4226:
4223:
4222:
4206:
4203:
4202:
4183:
4173:
4171:
4168:
4167:
4131:
4129:
4128:
4120:
4117:
4116:
4097:
4091:
4087:
4081:
4077:
4066:
4057:
4053:
4051:
4048:
4047:
4027:
4025:
4024:
4022:
4019:
4018:
4002:
4000:
3997:
3996:
3980:
3977:
3976:
3960:
3957:
3956:
3940:
3932:
3926:
3924:
3923:
3921:
3918:
3917:
3901:
3894:
3893:
3889:
3881:
3872:
3868:
3860:
3857:
3856:
3840:
3837:
3836:
3820:
3818:
3815:
3814:
3798:
3796:
3793:
3792:
3786:Dynkin diagrams
3770:
3764:
3743:
3740:
3739:
3719:
3716:
3715:
3699:
3697:
3694:
3693:
3673:
3670:
3669:
3649:
3646:
3645:
3629:
3626:
3625:
3607:
3604:
3603:
3584:
3581:
3580:
3560:
3557:
3556:
3540:
3538:
3535:
3534:
3506:
3503:
3502:
3486:
3484:
3481:
3480:
3477:
3459:
3454:
3412:
3408:
3400:
3396:
3395:
3388:
3384:
3383:
3372:
3352:
3348:
3337:
3336:
3327:
3323:
3321:
3318:
3317:
3219:
3214:
3213:
3193:
3175:
3171:
3160:
3157:
3156:
3088:
3085:
3084:
3068:
3050:
3046:
3028:
3024:
3022:
3019:
3018:
3012:
2981:
2977:
2975:
2972:
2971:
2969:
2934:
2930:
2928:
2925:
2924:
2911:
2889:
2884:
2883:
2878:
2870:
2866:
2865:
2860:
2859:
2857:
2854:
2853:
2836:
2831:
2830:
2828:
2825:
2824:
2808:
2803:
2798:
2796:
2793:
2792:
2776:
2773:
2772:
2750:
2747:
2746:
2721:
2717:
2713:
2711:
2708:
2707:
2691:
2688:
2687:
2671:
2668:
2667:
2666:-torus of rank
2651:
2648:
2647:
2631:
2628:
2627:
2611:
2608:
2607:
2591:
2588:
2587:
2567:
2564:
2563:
2544:
2539:
2536:
2535:
2510:
2505:
2504:
2495:
2487:
2486:
2466:
2462:
2450:
2446:
2437:
2432:
2431:
2409:
2408:
2403:
2397:
2396:
2388:
2378:
2377:
2376:
2372:
2370:
2367:
2366:
2344:
2341:
2340:
2318:
2315:
2314:
2297:
2292:
2291:
2289:
2286:
2285:
2279:
2257:
2252:
2251:
2242:
2237:
2236:
2234:
2231:
2230:
2213:
2208:
2207:
2205:
2202:
2201:
2180:
2175:
2174:
2172:
2169:
2168:
2137:
2135:
2132:
2131:
2103:
2101:
2098:
2097:
2075:
2070:
2069:
2067:
2064:
2063:
2043:
2041:
2038:
2037:
2034:
2003:
1998:
1985:
1981:
1963:
1958:
1945:
1941:
1926:
1921:
1920:
1911:
1906:
1901:
1898:
1895:
1894:
1878:
1875:
1874:
1858:
1855:
1854:
1833:
1825:
1822:
1821:
1818:
1813:
1786:
1783:
1782:
1766:
1754:
1753:
1745:
1742:
1741:
1721:
1720:
1712:
1704:
1701:
1700:
1689:
1667:
1664:
1663:
1640:
1637:
1636:
1620:
1618:
1615:
1614:
1598:
1596:
1593:
1592:
1574:
1571:
1570:
1550:
1548:
1545:
1544:
1541:splitting field
1524:
1522:
1519:
1518:
1502:
1499:
1498:
1481:
1477:
1471:
1467:
1447:
1445:
1442:
1441:
1422:
1417:
1414:
1413:
1390:
1388:
1385:
1384:
1360:
1357:
1356:
1330:
1327:
1326:
1309:
1305:
1299:
1289:
1288:
1269:
1261:
1259:
1256:
1255:
1239:
1236:
1235:
1219:
1217:
1214:
1213:
1189:
1187:
1184:
1183:
1167:
1164:
1163:
1145:
1142:
1141:
1119:
1117:
1114:
1113:
1097:
1094:
1093:
1090:
1039:
1036:
1035:
1013:
999:
976:
965:
939:
936:
935:
918:
914:
905:
901:
892:
888:
886:
883:
882:
860:
857:
856:
840:
837:
836:
811:
808:
807:
790:
786:
784:
781:
780:
764:
761:
760:
741:
736:
733:
732:
714:
713:
708:
707:
705:
702:
701:
685:
682:
681:
661:
658:
657:
654:
648:
624:
621:
620:
604:
601:
600:
584:
581:
580:
554:
553:
535:
534:
523:
519:
518:
514:
506:
503:
502:
486:
483:
482:
466:
463:
462:
455:
422:
417:
416:
399:
396:
395:
370:
367:
366:
348:
347:
342:
341:
339:
336:
335:
334:. In fact, any
319:
302:
299:
298:
272:
268:
254:
246:
237:
232:
231:
229:
226:
225:
204:
203:
198:
197:
195:
192:
191:
175:
173:
170:
169:
166:Cartan subgroup
135:
134:
129:
128:
126:
123:
122:
111:algebraic group
94:
92:
89:
88:
71:
66:
65:
63:
60:
59:
41:
40:
35:
34:
32:
29:
28:
25:algebraic torus
17:
12:
11:
5:
6686:
6676:
6675:
6670:
6656:
6655:
6649:
6634:
6625:
6618:
6612:
6601:
6598:
6595:
6594:
6582:
6570:
6558:
6543:
6514:
6513:
6511:
6508:
6507:
6506:
6501:
6496:
6491:
6489:Toric geometry
6485:
6484:
6468:
6465:
6448:
6425:
6420:
6415:
6410:
6407:
6404:
6399:
6395:
6391:
6388:
6385:
6382:
6379:
6376:
6373:
6368:
6364:
6360:
6355:
6351:
6347:
6342:
6338:
6326:
6325:
6317:
6303:
6295:
6284:
6270:
6266:
6259:
6252:
6245:
6238:
6231:
6224:
6213:
6206:
6183:
6164:
6161:
6148:
6145:
6142:
6137:
6133:
6129:
6124:
6120:
6116:
6113:
6108:
6104:
6100:
6095:
6091:
6087:
6081:
6075:
6072:
6069:
6064:
6061:
6058:
6053:
6049:
6028:
6013:
6010:
6006:
6002:
5997:
5993:
5989:
5986:
5981:
5977:
5973:
5968:
5964:
5935:
5932:
5928:
5924:
5919:
5915:
5911:
5908:
5905:
5902:
5897:
5893:
5837:
5834:
5821:
5801:
5798:
5795:
5792:
5789:
5786:
5783:
5780:
5777:
5774:
5766:
5763:
5760:
5738:
5733:
5728:
5725:
5700:
5697:
5694:
5689:
5684:
5681:
5678:
5675:
5655:
5652:
5630:
5601:
5591:
5550:
5520:
5517:
5515:
5512:
5498:
5475:
5470:
5448:
5427:
5417:associated to
5406:
5385:
5362:
5357:
5334:
5318:Main article:
5315:
5312:
5298:
5273:
5252:
5231:
5209:
5184:
5160:
5157:
5154:
5151:
5148:
5127:
5105:
5095:associated to
5084:
5060:
5038:
5016:
4994:
4973:
4961:
4958:
4945:
4925:
4905:
4885:
4865:
4862:
4859:
4839:
4830:associated to
4815:
4792:
4789:
4786:
4783:
4780:
4777:
4757:
4754:
4751:
4748:
4745:
4741:
4738:
4717:
4714:
4711:
4691:
4687:
4683:
4678:
4673:
4670:
4648:
4645:
4640:
4636:
4630:
4625:
4620:
4600:
4580:
4559:
4537:
4512:
4500:
4497:
4495:
4492:
4479:
4455:
4435:
4411:
4392:
4391:
4378:
4350:
4345:
4320:
4298:
4292:
4263:
4257:
4230:
4210:
4190:
4186:
4180:
4177:
4151:
4148:
4145:
4142:
4139:
4134:
4127:
4124:
4115:induces a map
4104:
4100:
4094:
4090:
4084:
4080:
4076:
4073:
4069:
4065:
4060:
4056:
4035:
4030:
4005:
3984:
3964:
3943:
3939:
3935:
3929:
3904:
3897:
3892:
3888:
3884:
3880:
3875:
3871:
3867:
3864:
3844:
3823:
3801:
3766:Main article:
3763:
3760:
3747:
3723:
3702:
3677:
3666:
3665:
3653:
3633:
3611:
3588:
3576:
3564:
3543:
3510:
3489:
3476:
3473:
3469:
3468:
3458:
3455:
3453:
3450:
3441:
3440:
3429:
3426:
3423:
3420:
3415:
3411:
3403:
3399:
3391:
3387:
3381:
3378:
3375:
3370:
3367:
3364:
3359:
3355:
3351:
3346:
3343:
3340:
3335:
3330:
3326:
3257:
3256:
3245:
3242:
3239:
3236:
3233:
3228:
3225:
3222:
3217:
3212:
3209:
3206:
3202:
3199:
3196:
3192:
3189:
3186:
3183:
3178:
3174:
3170:
3167:
3164:
3128:
3125:
3122:
3119:
3116:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3071:
3067:
3064:
3061:
3058:
3053:
3049:
3045:
3042:
3039:
3036:
3031:
3027:
3010:
2995:
2992:
2989:
2984:
2980:
2967:
2948:
2945:
2942:
2937:
2933:
2910:
2907:
2892:
2887:
2881:
2873:
2869:
2863:
2839:
2834:
2811:
2806:
2801:
2780:
2760:
2757:
2754:
2728:
2724:
2720:
2716:
2695:
2675:
2655:
2635:
2615:
2595:
2571:
2551:
2547:
2543:
2521:
2518:
2513:
2508:
2503:
2498:
2493:
2490:
2485:
2481:
2477:
2474:
2469:
2465:
2461:
2458:
2453:
2449:
2445:
2440:
2435:
2430:
2427:
2424:
2421:
2418:
2413:
2407:
2404:
2402:
2399:
2398:
2395:
2392:
2389:
2387:
2384:
2383:
2381:
2375:
2354:
2351:
2348:
2328:
2325:
2322:
2300:
2295:
2278:
2275:
2260:
2255:
2250:
2245:
2240:
2216:
2211:
2198:
2197:
2183:
2178:
2153:
2150:
2147:
2143:
2140:
2116:
2113:
2110:
2106:
2091:
2078:
2073:
2046:
2033:
2030:
2017:
2014:
2009:
2006:
2001:
1997:
1993:
1988:
1984:
1980:
1977:
1974:
1969:
1966:
1961:
1957:
1953:
1948:
1944:
1940:
1937:
1934:
1929:
1919:
1914:
1909:
1904:
1882:
1862:
1840:
1837:
1832:
1829:
1817:
1814:
1812:
1809:
1796:
1793:
1790:
1769:
1765:
1761:
1757:
1752:
1749:
1728:
1724:
1719:
1715:
1711:
1708:
1688:
1685:
1684:
1683:
1682:-rank is zero.
1671:
1656:
1644:
1623:
1601:
1578:
1566:
1553:
1527:
1506:
1484:
1480:
1474:
1470:
1466:
1463:
1460:
1457:
1454:
1450:
1429:
1425:
1421:
1406:
1393:
1375:is called the
1364:
1340:
1337:
1334:
1312:
1308:
1302:
1296:
1293:
1287:
1284:
1281:
1276:
1273:
1268:
1264:
1243:
1222:
1196:
1193:
1171:
1149:
1126:
1123:
1101:
1089:
1086:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1052:
1049:
1046:
1043:
1023:
1019:
1016:
1012:
1009:
1005:
1002:
998:
995:
992:
989:
986:
982:
979:
975:
971:
968:
964:
961:
958:
955:
952:
949:
946:
943:
921:
917:
913:
908:
904:
900:
895:
891:
870:
867:
864:
844:
824:
821:
818:
815:
793:
789:
768:
748:
744:
740:
717:
711:
689:
665:
650:Main article:
647:
644:
628:
608:
588:
568:
565:
562:
557:
552:
549:
546:
543:
538:
533:
526:
522:
517:
513:
510:
490:
470:
454:
451:
425:
420:
415:
412:
409:
406:
403:
383:
380:
377:
374:
351:
345:
322:
318:
315:
312:
309:
306:
286:
283:
278:
275:
271:
267:
264:
261:
257:
253:
245:
240:
235:
207:
201:
178:
138:
132:
119:toric geometry
97:
74:
69:
44:
38:
15:
9:
6:
4:
3:
2:
6685:
6674:
6671:
6669:
6666:
6665:
6663:
6652:
6646:
6642:
6641:
6635:
6631:
6626:
6623:
6619:
6617:
6613:
6610:
6609:
6604:
6603:
6592:, p. 25.
6591:
6586:
6580:, p. 22.
6579:
6574:
6567:
6562:
6554:
6547:
6536:
6529:
6522:
6520:
6515:
6505:
6502:
6500:
6497:
6495:
6492:
6490:
6487:
6486:
6482:
6476:
6471:
6464:
6460:
6458:
6454:
6451:torsors over
6447:
6443:
6439:
6418:
6408:
6405:
6397:
6393:
6389:
6386:
6383:
6374:
6366:
6362:
6358:
6353:
6349:
6340:
6336:
6323:
6316:
6312:
6308:
6304:
6301:
6294:
6290:
6283:
6279:
6275:
6271:
6265:
6258:
6251:
6244:
6237:
6230:
6223:
6219:
6212:
6205:
6201:
6200:
6199:
6197:
6193:
6189:
6182:
6178:
6174:
6170:
6160:
6135:
6131:
6122:
6118:
6114:
6106:
6102:
6093:
6089:
6062:
6059:
6051:
6047:
6036:
6034:
6027:
5995:
5991:
5987:
5984:
5979:
5975:
5966:
5962:
5953:
5950:over a field
5949:
5917:
5913:
5909:
5906:
5903:
5895:
5891:
5882:
5878:
5874:
5870:
5866:
5863:Given a rank
5861:
5859:
5855:
5851:
5847:
5843:
5833:
5819:
5799:
5787:
5781:
5775:
5761:
5758:
5736:
5726:
5723:
5716:
5698:
5695:
5692:
5682:
5676:
5661:
5651:
5648:
5646:
5642:
5638:
5633:
5629:
5625:
5621:
5617:
5613:
5609:
5604:
5600:
5595:
5590:
5586:
5582:
5578:
5574:
5570:
5567: →
5566:
5562:
5558:
5553:
5549:
5545:
5541:
5537:
5533:
5529:
5526:
5523:Given a base
5511:
5473:
5446:
5404:
5360:
5321:
5311:
5286:
5250:
5198:
5182:
5174:
5158:
5149:
5146:
5075:
4971:
4957:
4943:
4923:
4903:
4883:
4863:
4860:
4857:
4837:
4829:
4813:
4804:
4787:
4784:
4781:
4752:
4749:
4746:
4715:
4712:
4709:
4676:
4646:
4638:
4628:
4598:
4578:
4526:
4510:
4491:
4477:
4469:
4453:
4446:; then every
4433:
4425:
4409:
4401:
4397:
4390:
4376:
4367:
4366:
4365:
4362:
4343:
4290:
4278:
4255:
4242:
4188:
4184:
4175:
4165:
4146:
4140:
4132:
4092:
4082:
4078:
4058:
4054:
4028:
3982:
3962:
3955:is a maximal
3937:
3927:
3890:
3873:
3869:
3865:
3862:
3789:
3787:
3783:
3779:
3775:
3769:
3759:
3745:
3737:
3721:
3691:
3675:
3651:
3631:
3623:
3609:
3600:
3586:
3577:
3562:
3532:
3531:absolute rank
3528:
3524:
3523:
3522:
3508:
3472:
3467:
3464:
3463:
3462:
3449:
3446:
3427:
3421:
3413:
3409:
3401:
3397:
3389:
3385:
3368:
3362:
3357:
3353:
3349:
3328:
3324:
3316:
3315:
3314:
3312:
3311:Galois module
3308:
3304:
3300:
3296:
3291:
3289:
3285:
3281:
3277:
3273:
3269:
3266:When a field
3264:
3262:
3261:formal groups
3243:
3234:
3226:
3223:
3220:
3210:
3207:
3190:
3184:
3176:
3168:
3162:
3155:
3154:
3153:
3151:
3146:
3142:
3123:
3120:
3117:
3111:
3108:
3099:
3096:
3093:
3059:
3051:
3047:
3043:
3037:
3029:
3025:
3016:
3013: →
3009:
2990:
2982:
2978:
2966:
2963: →
2962:
2943:
2935:
2931:
2923:
2920:
2916:
2906:
2890:
2879:
2871:
2867:
2837:
2804:
2778:
2758:
2755:
2752:
2744:
2726:
2722:
2718:
2714:
2693:
2673:
2653:
2633:
2613:
2593:
2585:
2569:
2549:
2545:
2541:
2532:
2519:
2511:
2496:
2483:
2479:
2475:
2472:
2467:
2463:
2459:
2456:
2451:
2447:
2443:
2438:
2428:
2425:
2422:
2419:
2416:
2411:
2405:
2400:
2393:
2390:
2385:
2379:
2373:
2352:
2349:
2346:
2326:
2323:
2320:
2298:
2284:
2274:
2258:
2248:
2243:
2214:
2181:
2167:
2148:
2130:
2111:
2096:
2095:unitary group
2092:
2076:
2061:
2060:
2059:
2029:
2007:
2004:
1999:
1995:
1991:
1986:
1982:
1978:
1975:
1972:
1967:
1964:
1959:
1955:
1951:
1946:
1942:
1935:
1927:
1917:
1912:
1907:
1880:
1860:
1835:
1830:
1827:
1808:
1794:
1791:
1788:
1759:
1750:
1747:
1726:
1709:
1706:
1698:
1694:
1669:
1661:
1657:
1642:
1590:
1576:
1567:
1542:
1504:
1482:
1472:
1468:
1461:
1455:
1427:
1423:
1419:
1411:
1407:
1383:of the torus
1382:
1381:absolute rank
1378:
1362:
1354:
1353:
1352:
1338:
1335:
1332:
1310:
1300:
1291:
1282:
1271:
1241:
1210:
1191:
1169:
1161:
1147:
1121:
1099:
1085:
1068:
1065:
1062:
1050:
1047:
1044:
1017:
1014:
1010:
1007:
1003:
1000:
996:
980:
977:
973:
969:
966:
959:
953:
950:
947:
919:
915:
906:
902:
898:
893:
889:
868:
865:
862:
842:
822:
819:
816:
813:
791:
787:
766:
746:
742:
738:
687:
679:
663:
653:
643:
640:
626:
606:
586:
563:
544:
531:
524:
520:
511:
488:
468:
460:
450:
448:
444:
439:
423:
413:
407:
401:
378:
372:
316:
310:
304:
276:
273:
269:
265:
262:
243:
238:
224:
167:
163:
159:
155:
120:
116:
112:
72:
26:
22:
6639:
6629:
6621:
6615:
6606:
6585:
6573:
6561:
6552:
6546:
6504:Hopf algebra
6461:
6452:
6445:
6441:
6438:Picard group
6327:
6321:
6314:
6310:
6306:
6299:
6292:
6288:
6281:
6277:
6273:
6263:
6256:
6249:
6242:
6235:
6228:
6221:
6217:
6210:
6203:
6195:
6191:
6187:
6180:
6176:
6166:
6037:
6032:
6025:
5951:
5947:
5880:
5876:
5872:
5868:
5864:
5862:
5858:Grothendieck
5849:
5845:
5841:
5839:
5657:
5649:
5644:
5636:
5631:
5627:
5623:
5619:
5615:
5611:
5607:
5602:
5598:
5593:
5588:
5584:
5580:
5576:
5572:
5568:
5564:
5560:
5556:
5551:
5547:
5544:fpqc locally
5539:
5536:group scheme
5531:
5527:
5522:
5323:
5287:
4963:
4805:
4768:is equal to
4702:is equal to
4524:
4502:
4393:
4368:
4363:
4276:
4243:
3790:
3782:root systems
3771:
3735:
3689:
3667:
3602:
3579:
3530:
3526:
3478:
3470:
3465:
3460:
3444:
3442:
3313:isomorphism
3309:, we have a
3306:
3302:
3301:and a torus
3298:
3294:
3292:
3287:
3283:
3279:
3275:
3271:
3267:
3265:
3258:
3149:
3144:
3140:
3014:
3007:
2964:
2960:
2914:
2912:
2533:
2283:finite field
2280:
2199:
2035:
1819:
1696:
1690:
1659:
1569:
1540:
1409:
1380:
1376:
1355:The integer
1211:
1140:
1091:
677:
655:
641:
456:
440:
223:group scheme
164:theory (see
157:
24:
18:
5660:affine cone
4396:Tits (1966)
3736:anisotropic
3622:-split rank
1660:anisotropic
1517:over which
934:defined by
21:mathematics
6673:Lie groups
6662:Categories
6600:References
5519:Definition
4468:inner form
4402:groups of
4164:Tits index
3768:Tits index
2743:field norm
1781:such that
1088:Definition
501:, the maps
6566:Tits 1966
6384:≅
6367:∙
6123:∙
6094:∙
5797:→
5782:−
5759:π
5727:⊂
5683:⊂
5488:-rank of
5314:Buildings
5221:-rank of
5156:∖
5153:Γ
5083:Γ
5050:-rank of
5028:then the
4861:⊂
4713:−
4644:→
4629:×
4525:real rank
4349:Φ
4229:Φ
4209:Φ
4179:¯
4141:∪
4138:Φ
4126:→
4123:Φ
4083:∗
4075:→
4059:∗
4034:Φ
3938:⊂
3891:⊗
3874:∗
3843:Φ
3414:∙
3369:≅
3329:∙
3121:∘
3112:
3106:↦
3083:given by
3066:→
3052:∙
3044:×
3030:∙
2983:∙
2936:∙
2756:−
2582:then the
2484:⊂
2457:−
2429:∈
2324:−
2281:Over the
2249:×
2244:×
2215:×
2077:×
2005:−
1976:…
1965:−
1839:¯
1795:ϕ
1792:∘
1789:ψ
1764:→
1748:ψ
1718:→
1707:ϕ
1697:isogenous
1687:Isogenies
1473:×
1462:≅
1336:≥
1325:for some
1301:×
1295:¯
1283:≅
1275:¯
1195:¯
1139:. Then a
1125:¯
1057:↦
991:↦
912:→
899:×
792:×
551:→
512:⋅
447:buildings
424:∗
414:⊂
317:⊂
274:−
239:∗
162:Lie group
6620:T. Ono,
6614:T. Ono,
6535:Archived
6467:See also
5654:Examples
5542:that is
4162:and the
1811:Examples
1760:′
1727:′
1018:′
1004:′
981:′
970:′
152:. These
6526:Milne.
6234:×
5836:Weights
4826:is the
4527:is the
4275:is the
2922:lattice
2741:of its
1693:isogeny
1662:if its
459:perfect
6647:
6324:) = 1.
6173:T. Ono
5867:torus
5525:scheme
3521:then:
2919:weight
2771:. Any
2646:is an
1234:is an
1160:-torus
154:groups
6608:SGA 3
6538:(PDF)
6531:(PDF)
6510:Notes
6494:Torus
6216:over
6190:, as
5871:over
5713:of a
5559:over
5538:over
4426:over
3690:split
3599:-rank
3305:over
2586:from
2166:torus
1589:-rank
1410:split
680:over
87:, or
23:, an
6645:ISBN
6241:) =
6209:and
5641:rank
4850:and
3784:and
3578:its
3529:(or
3527:rank
3525:its
2686:and
1923:Spec
1568:The
1377:rank
1092:Let
759:the
445:and
248:Spec
158:tori
117:and
6440:of
5769:Aff
5670:Aff
5324:If
5175:of
4896:in
4806:If
4776:min
4503:If
4310:in
4201:on
3791:If
3479:If
3443:If
3109:deg
2606:to
1691:An
1591:of
1543:of
1440:if
1379:or
656:If
639:.
160:in
19:In
6664::
6533:.
6518:^
6459:.
6280:,
6255:)
6220:,
6171:,
6035:.
5832:.
5647:.
5597:=
5592:1,
5589:GL
5510:.
4956:.
4803:.
4361:.
3290:.
3191::=
2905:.
2028:.
1084:.
449:.
58:,
6653:.
6453:T
6449:m
6446:G
6442:T
6424:)
6419:m
6414:G
6409:,
6406:T
6403:(
6398:1
6394:t
6390:x
6387:E
6381:)
6378:)
6375:T
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6350:G
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6300:K
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6289:L
6285:L
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6278:K
6276:/
6274:L
6269:)
6267:2
6264:T
6262:(
6260:K
6257:f
6253:1
6250:T
6248:(
6246:K
6243:f
6239:2
6236:T
6232:1
6229:T
6227:(
6225:K
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6218:K
6214:2
6211:T
6207:1
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6196:k
6192:K
6188:K
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6181:f
6177:k
6147:)
6144:)
6141:)
6136:2
6132:T
6128:(
6119:X
6115:,
6112:)
6107:1
6103:T
6099:(
6090:X
6086:(
6080:Z
6074:m
6071:o
6068:H
6063:,
6060:S
6057:(
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6001:(
5996:n
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5988:G
5985:,
5980:K
5976:G
5972:(
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5927:Z
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5904:S
5901:(
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5877:S
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5869:T
5865:n
5850:S
5846:S
5842:S
5820:X
5800:X
5794:)
5791:}
5788:0
5785:{
5779:)
5776:X
5773:(
5765:(
5762::
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5699:1
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5688:A
5680:)
5677:X
5674:(
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5637:L
5635:/
5632:m
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5616:K
5612:S
5608:U
5606:/
5603:m
5599:G
5594:U
5585:U
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5577:U
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5569:S
5565:X
5561:S
5557:S
5555:/
5552:m
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5540:S
5532:S
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5497:G
5474:p
5469:Q
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5426:G
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5361:p
5356:Q
5333:G
5297:Q
5272:G
5251:M
5230:G
5208:Q
5183:M
5159:X
5150:=
5147:M
5126:G
5104:G
5059:G
5037:Q
5015:Q
4993:G
4972:G
4944:X
4924:X
4904:X
4884:T
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4782:p
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4756:)
4753:q
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4682:(
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4377:F
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4319:G
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4291:F
4262:T
4256:F
4189:F
4185:/
4176:F
4150:}
4147:0
4144:{
4133:F
4103:)
4099:T
4093:F
4089:(
4079:X
4072:)
4068:T
4064:(
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4004:G
3983:F
3963:F
3942:T
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3928:F
3903:R
3896:Z
3887:)
3883:T
3879:(
3870:X
3866:=
3863:V
3822:G
3800:T
3746:F
3722:F
3701:G
3676:F
3664:.
3652:F
3632:G
3610:F
3587:F
3563:F
3542:G
3509:F
3488:G
3445:T
3428:.
3425:)
3422:T
3419:(
3410:X
3402:K
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3380:d
3377:n
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3363:T
3358:K
3354:/
3350:L
3345:s
3342:e
3339:R
3334:(
3325:X
3307:L
3303:T
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3297:/
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3288:K
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3280:K
3276:K
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3241:)
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3235:X
3232:(
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3224:,
3221:m
3216:G
3211:,
3208:M
3205:(
3201:m
3198:o
3195:H
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3185:X
3182:(
3177:S
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3169:M
3166:(
3163:D
3150:Z
3145:n
3141:n
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3124:g
3118:f
3115:(
3103:)
3100:g
3097:,
3094:f
3091:(
3070:Z
3063:)
3060:T
3057:(
3048:X
3041:)
3038:T
3035:(
3026:X
3015:T
3011:m
3008:G
2994:)
2991:T
2988:(
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2968:m
2965:G
2961:T
2947:)
2944:T
2941:(
2932:X
2915:T
2891:q
2886:F
2880:/
2872:2
2868:q
2862:F
2838:q
2833:F
2810:R
2805:/
2800:C
2779:F
2759:1
2753:d
2727:F
2723:/
2719:E
2715:N
2694:F
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2614:F
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2550:F
2546:/
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2520:.
2517:)
2512:q
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2502:(
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2480:}
2476:1
2473:=
2468:2
2464:u
2460:d
2452:2
2448:t
2444:,
2439:q
2434:F
2426:u
2423:,
2420:t
2417::
2412:)
2406:t
2401:u
2394:u
2391:d
2386:t
2380:(
2374:{
2353:1
2350:+
2347:q
2327:1
2321:q
2299:q
2294:F
2259:1
2254:T
2239:R
2210:C
2182:1
2177:T
2152:)
2149:2
2146:(
2142:O
2139:S
2115:)
2112:1
2109:(
2105:U
2072:R
2045:R
2016:)
2013:]
2008:1
2000:n
1996:t
1992:,
1987:n
1983:t
1979:,
1973:,
1968:1
1960:1
1956:t
1952:,
1947:1
1943:t
1939:[
1936:k
1933:(
1928:k
1918:=
1913:n
1908:m
1903:G
1881:k
1861:n
1836:k
1831:=
1828:k
1768:T
1756:T
1751::
1723:T
1714:T
1710::
1670:F
1643:F
1622:T
1600:T
1577:F
1565:.
1552:T
1526:T
1505:F
1483:r
1479:)
1469:E
1465:(
1459:)
1456:E
1453:(
1449:T
1428:F
1424:/
1420:E
1405:.
1392:T
1363:r
1339:1
1333:r
1311:r
1307:)
1292:F
1286:(
1280:)
1272:F
1267:(
1263:T
1242:F
1221:T
1192:F
1170:F
1148:F
1122:F
1100:F
1072:)
1069:x
1066:,
1063:y
1060:(
1054:)
1051:y
1048:,
1045:x
1042:(
1022:)
1015:y
1011:y
1008:,
1001:x
997:x
994:(
988:)
985:)
978:y
974:,
967:x
963:(
960:,
957:)
954:y
951:,
948:x
945:(
942:(
920:2
916:F
907:2
903:F
894:2
890:F
869:y
866:,
863:x
843:F
823:1
820:=
817:y
814:x
788:E
767:E
747:F
743:/
739:E
716:m
710:G
688:F
664:F
627:r
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564:G
561:(
556:O
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542:(
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509:(
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408:1
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277:1
270:t
266:,
263:t
260:[
256:C
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