38:
923:
as the study of classical modular forms and their generalisations developed. Of course the two topics were related, as can be seen for example in
Langlands' computation of the volume of certain fundamental domains using analytic methods. This classical theory culminated with the work of Siegel, who
4713:
is (roughly) a subgroup of an arithmetic group defined by taking all matrices satisfying certain equations modulo an integer, for example the group of 2 by 2 integer matrices with diagonal (respectively off-diagonal) coefficients congruent to 1 (respectively 0) modulo a positive integer. These are
3143:
in a Lie group is usually defined as a discrete subgroup with finite covolume. The terminology introduced above is coherent with this, as a theorem due to Borel and Harish-Chandra states that an arithmetic subgroup in a semisimple Lie group is of finite covolume (the discreteness is obvious).
966:
In the seventies
Margulis revolutionised the topic by proving that in "most" cases the arithmetic constructions account for all lattices in a given Lie group. Some limited results in this direction had been obtained earlier by Selberg, but Margulis' methods (the use of
4718:) is that this is true for (irreducible) arithmetic lattices in higher-rank groups and false in rank-one groups. It is still open in this generality but there are many results establishing it for specific lattices (in both its positive and negative cases).
971:
tools for actions on homogeneous spaces) were completely new in this context and were to be extremely influential on later developments, effectively renewing the old subject of geometry of numbers and allowing
Margulis himself to prove the
5574:. The possible lattices have been classified by Prasad and Yeung and the classification was completed by Cartwright and Steger who determined, by computer assisted computations, all the fake projective planes in each Prasad-Yeung class.
5399:
4983:
1218:
3677:
3605:
4673:
1744:
2059:
1923:
1294:
5140:
4901:
4834:
4325:
1647:
5534:
but is not biholomorphic to it; the first example was discovered by
Mumford. By work of Klingler (also proved independently by Yeung) all such are quotients of the 2-ball by arithmetic lattices in
3023:
4197:
4548:
4371:
2679:
2417:
2317:
2245:
2156:
854:
5073:
4459:
3533:
3351:
3296:
3129:
3086:
2830:
2360:
2199:
2110:
1433:
1094:
5532:
4726:
Instead of taking integral points in the definition of an arithmetic lattice one can take points which are only integral away from a finite number of primes. This leads to the notion of an
1393:
2922:
2574:
5475:
and numerous variations on her construction have appeared since. The isospectrality problem is in fact particularly amenable to study in the restricted setting of arithmetic manifolds.
2863:
2477:
1329:
1147:
4155:
2625:
4235:
3450:
5572:
4270:
3965:
3887:
3821:
3720:
1774:
2974:
1485:
4101:
4416:
5292:
5270:
3991:
3847:
3746:
3255:
3213:
3191:
2791:
2769:
1864:
1842:
1796:
1581:
1507:
1240:
1051:
483:
458:
421:
3781:
5443:(Lubotzky-Phillips-Sarnak). Such graphs are known to exist in abundance by probabilistic results but the explicit nature of these constructions makes them interesting.
987:
In another direction the classical topic of modular forms has blossomed into the modern theory of automorphic forms. The driving force behind this effort is mainly the
1349:
5323:
5312:
5228:
3925:
3377:
1453:
4714:
always finite-index subgroups and the congruence subgroup problem roughly asks whether all subgroups are obtained in this way. The conjecture (usually attributed to
2533:
2503:
5437:
3477:
1943:
1749:
On the other hand, the class of groups thus obtained is not larger than the class of arithmetic groups as defined above. Indeed, if we consider the algebraic group
2271:
888:
One of the origins of the mathematical theory of arithmetic groups is algebraic number theory. The classical reduction theory of quadratic and
Hermitian forms by
1983:
5248:
5208:
5180:
5160:
5030:
5010:
4906:
4767:
4745:
4693:
4595:
4575:
4502:
4482:
4391:
4121:
4077:
4057:
4034:
3404:
3233:
3169:
3043:
2942:
2883:
2747:
2727:
1820:
1667:
1601:
1559:
1535:
1114:
1152:
3610:
3538:
3310:
lattice is arithmetic. This result is true for all irreducible lattice in semisimple Lie groups of real rank larger than two. For example, all lattices in
4600:
2630:
Another classical example is given by the integral elements in the orthogonal group of a quadratic form defined over a number field, for example
6227:
785:
1672:
1245:
5078:
4839:
4772:
1988:
909:
3406:
has a factor of real rank one (otherwise the theorem always holds) and is not simple: it means that for any product decomposition
3306:
The spectacular result that
Margulis obtained is a partial converse to the Borel—Harish-Chandra theorem: for certain Lie groups
1869:
6159:
Prasad, Gopal; Rapinchuk, Andrei S. (2009). "Weakly commensurable arithmetic groups and isospectral locally symmetric spaces".
4275:
343:
5459:. It is in fact known that the Ramanujan property itself implies that the local girths of the graph are almost always large.
4675:
explained above. This amounts to classifying the algebraic groups whose real points are isomorphic up to a compact factor to
1606:
2979:
293:
4160:
939:
and others on algebraic groups. Shortly afterwards the finiteness of covolume was proven in full generality by Borel and
5782:"Harmonic analysis ans discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series"
3257:
will be co-compact in the associated orthogonal group if and only if the quadratic form does not vanish at any point in
963:
and others. The state of the art after this period was essentially fixed in
Raghunathan's treatise, published in 1972.
778:
288:
4507:
4330:
2633:
2376:
2276:
2204:
2115:
813:
5625:
Langlands, R. P. (1966), "The volume of the fundamental domain for some arithmetical subgroups of
Chevalley groups",
5035:
4421:
3482:
3313:
3260:
3091:
3048:
2796:
2322:
2161:
2072:
1398:
1056:
5497:
1358:
4008:
1011:
2888:
2538:
1352:
2835:
2428:
1301:
1119:
6262:
4126:
704:
2582:
771:
4504:
to be the
Hamilton quaternions at all real places. They exhaust all arithmetic commensurability classes in
1000:
4202:
3409:
3140:
913:
388:
202:
5537:
4244:
3930:
3852:
3786:
3685:
6071:
Abért, Miklós; Glasner, Yair; Virág, Bálint (2014). "Kesten's theorem for invariant random subgroups".
2947:
1458:
996:
120:
6247:
5416:
4014:
4002:
4082:
4396:
3682:
The
Margulis arithmeticity (and superrigidity) theorem holds for certain rank 1 Lie groups, namely
2690:
1752:
586:
320:
197:
85:
5472:
5275:
5253:
3970:
3826:
3725:
3238:
3196:
3174:
2774:
2752:
1847:
1825:
1779:
1564:
1490:
1223:
1034:
466:
441:
404:
3751:
1220:
In general it is not so obvious how to make precise sense of the notion of "integer points" of a
1007:
5887:"Harmonic maps into singular spaces and p-adic superrigidity for lattices in groups of rank one"
5394:{\displaystyle G=\mathrm {G} (\mathbb {R} )\times \prod _{p\in S}\mathrm {G} (\mathbb {Q} _{p})}
6257:
920:
736:
526:
6016:
5765:
Margulis, Grigori (1975). "Discrete groups of motions of manifolds of nonpositive curvature".
1334:
6252:
5315:
5297:
5213:
3892:
3356:
1438:
927:
For the modern theory to begin foundational work was needed, and was provided by the work of
869:
610:
2508:
2482:
908:
and the early development of the study of arithmetic invariant of number fields such as the
6190:
6124:
Vignéras, Marie-France (1980). "Variétés riemanniennes isospectrales et non isométriques".
6102:
6050:
5767:
Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974), Vol. 2
5695:
5634:
5468:
5456:
5422:
4238:
3455:
2698:
2577:
1928:
977:
973:
550:
538:
156:
90:
4978:{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )\times \mathrm {SL} _{2}(\mathbb {Q} _{p}).}
8:
6017:"Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups"
4710:
4704:
4577:
it is in theory possible to classify (up to commensurability) all arithmetic lattices in
2682:
2371:
2250:
905:
865:
125:
20:
3215:) is anisotropic. For example, the arithmetic lattice associated to a quadratic form in
1948:
6194:
6168:
6141:
6106:
6080:
6054:
6028:
5941:
5906:
5730:
5668:
5452:
5451:
Congruence covers of arithmetic surfaces are known to give rise to surfaces with large
5233:
5193:
5165:
5145:
5015:
4995:
4752:
4730:
4678:
4580:
4560:
4487:
4467:
4376:
4106:
4062:
4042:
4037:
4019:
3389:
3218:
3154:
3028:
2927:
2868:
2732:
2712:
2686:
1805:
1652:
1586:
1544:
1520:
1213:{\displaystyle \Gamma =\mathrm {GL} _{n}(\mathbb {Z} )\cap \mathrm {G} (\mathbb {Q} ).}
1099:
897:
110:
82:
5439:) of Lubotzky and Zimmer can be used to construct expander graphs (Margulis), or even
6221:
5910:
5886:
5672:
4715:
3672:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )\times \mathrm {SL} _{2}(\mathbb {Z} )}
3600:{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )\times \mathrm {SL} _{2}(\mathbb {R} )}
3148:
988:
960:
893:
515:
358:
252:
6110:
6058:
5836:
5609:
1509:
is associated a collection of "discrete" subgroups all commensurable to each other.
1242:-group, and the subgroup defined above can change when we take different embeddings
681:
6198:
6178:
6133:
6090:
6038:
5933:
5898:
5831:
5819:
5749:
5722:
5710:
5660:
5648:
5592:
5492:
5455:. Likewise the Ramanujan graphs constructed by Lubotzky-Phillips-Sarnak have large
5440:
1799:
1015:
992:
952:
877:
666:
658:
650:
642:
634:
622:
562:
502:
334:
276:
151:
5781:
5629:, Proc. Sympos. Pure Math., Providence, R.I.: Amer. Math. Soc., pp. 143–148,
6186:
6098:
6046:
5691:
5630:
5484:
4668:{\displaystyle G=\mathrm {SL} _{2}(\mathbb {R} ),\mathrm {SL} _{2}(\mathbb {C} )}
3452:
the lattice is not commensurable to a product of lattices in each of the factors
901:
889:
807:
750:
743:
729:
686:
574:
497:
327:
241:
181:
61:
2694:
968:
940:
857:
757:
693:
383:
363:
300:
265:
186:
176:
161:
146:
100:
77:
6182:
4836:. They are also naturally lattices in certain topological groups, for example
1006:
Finally arithmetic groups are often used to construct interesting examples of
6241:
6094:
6042:
5924:
Corlette, Kevin (1992). "Archimedean superrigidity and hyperbolic geometry".
3380:
2423:
2367:
2363:
981:
956:
861:
676:
598:
432:
305:
171:
3045:(but there are many more, corresponding to other embeddings); for instance,
1298:
Thus a better notion is to take for definition of an arithmetic subgroup of
932:
6008:
5488:
1538:
999:
originating in Selberg's work and developed in the most general setting by
948:
936:
928:
924:
showed the finiteness of the volume of a fundamental domain in many cases.
531:
230:
219:
166:
141:
136:
95:
66:
29:
3379:. The main new ingredient that Margulis used to prove his theorem was the
799:
6033:
5032:
a finite set of prime numbers is the same as for arithmetic groups with
6145:
6012:
5945:
5902:
5734:
5664:
4013:
An arithmetic Fuchsian group is constructed from the following data: a
1739:{\displaystyle \rho ^{-1}(\mathrm {GL} _{n}(O))\subset \mathrm {G} (F)}
698:
426:
5820:"Three-dimensional manifolds, Kleinian groups and hyperbolic geometry"
1517:
A natural generalisation of the construction above is as follows: let
912:. Arithmetic groups can be thought of as a vast generalisation of the
944:
943:. Meanwhile, there was progress on the general theory of lattices in
519:
6137:
5937:
5726:
4769:
stands for the set of primes inverted). The prototypical example is
3147:
The theorem is more precise: it says that the arithmetic lattice is
1010:
Riemannian manifolds. A particularly active research topic has been
3383:
of lattices in higher-rank groups that he proved for this purpose.
873:
56:
6173:
6085:
4464:
Arithmetic Kleinian groups are constructed similarly except that
2693:). Similar constructions can be performed with unitary groups of
398:
312:
1289:{\displaystyle \mathrm {G} \to \mathrm {GL} _{n}(\mathbb {Q} ).}
5135:{\displaystyle \mathrm {GL} _{n}\left(\mathbb {Z} \left\right)}
4896:{\displaystyle \mathrm {SL} _{2}\left(\mathbb {Z} \left\right)}
4829:{\displaystyle \mathrm {SL} _{2}\left(\mathbb {Z} \left\right)}
2704:
2054:{\displaystyle (\rho ')^{-1}(\mathrm {GL} _{nd}(\mathbb {Z} ))}
37:
6214:
COVOLUME DES GROUPES S-ARITHMÉTIQUES ET FAUX PLANS PROJECTIFS
5803:
Arthur, James (2005). "An introduction to the trace formula".
856:
They arise naturally in the study of arithmetic properties of
5960:
5867:
2832:
with compact kernel, the image of an arithmetic subgroup in
900:
for the action of certain arithmetic groups on the relevant
5805:
Harmonic analysis, the trace formula, and Shimura varieties
5185:
3927:. There are no known non-arithmetic lattices in the groups
3996:
1918:{\displaystyle \rho ':\mathrm {G} '\to \mathrm {GL} _{dn}}
4320:{\displaystyle (A^{\sigma }\otimes _{F}\mathbb {R} )^{1}}
2681:. A related construction is by taking the unit groups of
5980:
Discrete groups, expanding graphs and invariant measures
2729:
is a Lie group one can define an arithmetic lattice in
1642:{\displaystyle \rho :\mathrm {G} \to \mathrm {GL} _{n}}
5112:
4873:
4806:
864:. They also give rise to very interesting examples of
5540:
5500:
5425:
5326:
5300:
5278:
5256:
5236:
5216:
5196:
5168:
5148:
5081:
5038:
5018:
4998:
4909:
4842:
4775:
4755:
4733:
4681:
4603:
4583:
4563:
4510:
4490:
4470:
4424:
4399:
4379:
4333:
4278:
4247:
4205:
4163:
4129:
4109:
4085:
4065:
4045:
4022:
3973:
3933:
3895:
3855:
3829:
3789:
3754:
3728:
3688:
3613:
3541:
3485:
3458:
3412:
3392:
3359:
3316:
3263:
3241:
3221:
3199:
3177:
3157:
3094:
3051:
3031:
3018:{\displaystyle G\cap \mathrm {GL} _{n}(\mathbb {Z} )}
2982:
2950:
2930:
2891:
2871:
2838:
2799:
2777:
2755:
2735:
2715:
2636:
2585:
2541:
2511:
2485:
2431:
2379:
2325:
2279:
2253:
2207:
2164:
2118:
2075:
1991:
1951:
1931:
1872:
1850:
1828:
1808:
1782:
1755:
1675:
1655:
1609:
1589:
1567:
1547:
1523:
1493:
1461:
1455:
defined as above (with respect to any embedding into
1441:
1401:
1361:
1337:
1304:
1248:
1226:
1155:
1122:
1102:
1059:
1037:
816:
469:
444:
407:
5769:(in Russian). Canad. Math. Congress. pp. 21–34.
5708:
4461:are obtained in this way (up to commensurability).
4192:{\displaystyle A^{\sigma }\otimes _{F}\mathbb {R} }
3134:
5566:
5526:
5431:
5393:
5306:
5286:
5264:
5242:
5222:
5202:
5174:
5154:
5134:
5067:
5024:
5004:
4977:
4895:
4828:
4761:
4739:
4698:
4687:
4667:
4589:
4569:
4542:
4496:
4484:is required to have exactly one complex place and
4476:
4453:
4410:
4385:
4365:
4319:
4264:
4229:
4191:
4149:
4115:
4095:
4071:
4051:
4028:
3985:
3959:
3919:
3881:
3841:
3815:
3775:
3740:
3714:
3671:
3599:
3527:
3471:
3444:
3398:
3371:
3345:
3290:
3249:
3227:
3207:
3185:
3163:
3123:
3080:
3037:
3017:
2968:
2936:
2916:
2877:
2857:
2824:
2785:
2763:
2741:
2721:
2673:
2619:
2568:
2527:
2497:
2471:
2422:Other well-known and studied examples include the
2411:
2354:
2311:
2265:
2239:
2193:
2150:
2104:
2053:
1977:
1937:
1917:
1858:
1836:
1814:
1790:
1768:
1738:
1661:
1641:
1595:
1575:
1553:
1529:
1501:
1479:
1447:
1427:
1387:
1343:
1323:
1288:
1234:
1212:
1141:
1108:
1088:
1045:
876:. Finally, these two topics join in the theory of
848:
477:
452:
415:
6070:
3301:
6239:
6007:
5190:The Borel–Harish-Chandra theorem generalizes to
4543:{\displaystyle \mathrm {SL} _{2}(\mathbb {C} ).}
4366:{\displaystyle \mathrm {SL} _{2}(\mathbb {R} ),}
2674:{\displaystyle \mathrm {SO} (n,1)(\mathbb {Z} )}
2412:{\displaystyle \mathrm {Sp} _{2g}(\mathbb {Z} )}
2312:{\displaystyle \mathrm {PSL} _{2}(\mathbb {Z} )}
2240:{\displaystyle \mathrm {PGL} _{n}(\mathbb {Z} )}
2151:{\displaystyle \mathrm {PSL} _{n}(\mathbb {Z} )}
2069:The classical example of an arithmetic group is
1746:can legitimately be called an arithmetic group.
1487:). With this definition, to the algebraic group
849:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} ).}
806:is a group obtained as the integer points of an
6158:
5446:
5068:{\displaystyle \mathrm {GL} _{n}(\mathbb {Z} )}
4454:{\displaystyle \mathrm {SL} _{2}(\mathbb {R} )}
3528:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}
3346:{\displaystyle \mathrm {SL} _{n}(\mathbb {R} )}
3291:{\displaystyle \mathbb {Q} ^{n}\setminus \{0\}}
3124:{\displaystyle \mathrm {SL} _{n}(\mathbb {R} )}
3081:{\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )}
2825:{\displaystyle \mathrm {G} (\mathbb {R} )\to G}
2355:{\displaystyle \mathrm {SL} _{2}(\mathbb {Z} )}
2194:{\displaystyle \mathrm {GL} _{n}(\mathbb {Z} )}
2105:{\displaystyle \mathrm {SL} _{n}(\mathbb {Z} )}
1985:) then the group constructed above is equal to
1428:{\displaystyle \Lambda /(\Gamma \cap \Lambda )}
1089:{\displaystyle \mathrm {GL} _{n}(\mathbb {Q} )}
1018:wrote, "...often seem to have special beauty."
5527:{\displaystyle \mathbb {P} ^{2}(\mathbb {C} )}
4373:and it is co-compact in all cases except when
1388:{\displaystyle \Gamma /(\Gamma \cap \Lambda )}
1021:
916:of number fields to a noncommutative setting.
880:which is fundamental in modern number theory.
1116:then we can define an arithmetic subgroup of
779:
5958:
5884:
5865:
5627:Algebraic Groups and Discontinuous Subgroups
3285:
3279:
2917:{\displaystyle G=\mathrm {G} (\mathbb {R} )}
2705:Arithmetic lattices in semisimple Lie groups
5853:Discrete subgroups of semisimple Lie groups
5747:
5467:Arithmetic groups can be used to construct
5410:
2569:{\displaystyle \mathbb {Q} ({\sqrt {-m}}),}
6226:: CS1 maint: location missing publisher (
5711:"Arithmetic subgroups of algebraic groups"
2858:{\displaystyle \mathrm {G} (\mathbb {Q} )}
2472:{\displaystyle \mathrm {SL} _{2}(O_{-m}),}
1324:{\displaystyle \mathrm {G} (\mathbb {Q} )}
1142:{\displaystyle \mathrm {G} (\mathbb {Q} )}
786:
772:
6211:
6172:
6084:
6032:
5885:Gromov, Mikhail; Schoen, Richard (1992).
5835:
5646:
5624:
5517:
5503:
5478:
5378:
5342:
5258:
5103:
5058:
4959:
4929:
4864:
4797:
4658:
4629:
4530:
4444:
4401:
4353:
4303:
4220:
4185:
4150:{\displaystyle \sigma :F\to \mathbb {R} }
4143:
3662:
3633:
3590:
3561:
3505:
3336:
3266:
3243:
3179:
3114:
3071:
3008:
2907:
2848:
2809:
2779:
2664:
2543:
2402:
2345:
2302:
2230:
2184:
2141:
2095:
2041:
1968:
1852:
1830:
1784:
1314:
1276:
1228:
1200:
1181:
1132:
1079:
995:. One of the main tool used there is the
836:
471:
446:
409:
6123:
5977:
5923:
5850:
5817:
5764:
5462:
5186:Lattices in Lie groups over local fields
3783:. It is known not to hold in all groups
2620:{\displaystyle \mathrm {SL} _{2}(O_{m})}
5779:
3997:Arithmetic Fuchsian and Kleinian groups
904:. The topic was related to Minkowski's
6240:
5992:
5802:
5709:Borel, Armand; Harish-Chandra (1962).
5607:
5594:Introduction aux groupes arithmétiques
4721:
3386:Irreducibility only plays a role when
1512:
344:Classification of finite simple groups
5647:Borel, Armand; Tits, Jacques (1965).
5590:
4123:. It is asked that for one embedding
2535:is the ring of integers in the field
868:and hence are objects of interest in
5685:
5405:
4199:be isomorphic to the matrix algebra
2749:as follows: for any algebraic group
2689:over number fields (for example the
1026:
896:and others can be seen as computing
6161:Publ. Math. Inst. Hautes Études Sci
5891:Inst. Hautes Études Sci. Publ. Math
5653:Inst. Hautes Études Sci. Publ. Math
5611:Lectures on the geometry of numbers
4597:, in a manner similar to the cases
4230:{\displaystyle M_{2}(\mathbb {R} )}
3445:{\displaystyle G=G_{1}\times G_{2}}
13:
5995:Some applications of modular forms
5807:. Amer. Math. soc. pp. 1–263.
5567:{\displaystyle \mathrm {PU} (2,1)}
5545:
5542:
5369:
5334:
5301:
5280:
5217:
5210:-arithmetic groups as follows: if
5087:
5084:
5044:
5041:
4944:
4941:
4915:
4912:
4848:
4845:
4781:
4778:
4644:
4641:
4615:
4612:
4516:
4513:
4430:
4427:
4339:
4336:
4265:{\displaystyle {\mathcal {O}}^{1}}
4251:
4088:
3960:{\displaystyle \mathrm {SU} (n,1)}
3938:
3935:
3882:{\displaystyle \mathrm {SU} (n,1)}
3860:
3857:
3816:{\displaystyle \mathrm {SO} (n,1)}
3794:
3791:
3715:{\displaystyle \mathrm {Sp} (n,1)}
3693:
3690:
3648:
3645:
3619:
3616:
3576:
3573:
3547:
3544:
3491:
3488:
3322:
3319:
3201:
3100:
3097:
3057:
3054:
2994:
2991:
2956:
2953:
2899:
2840:
2801:
2757:
2641:
2638:
2591:
2588:
2437:
2434:
2385:
2382:
2331:
2328:
2288:
2285:
2282:
2216:
2213:
2210:
2170:
2167:
2127:
2124:
2121:
2081:
2078:
2024:
2021:
1902:
1899:
1886:
1758:
1723:
1697:
1694:
1629:
1626:
1617:
1569:
1495:
1467:
1464:
1442:
1419:
1413:
1402:
1379:
1373:
1362:
1338:
1306:
1262:
1259:
1250:
1192:
1167:
1164:
1156:
1124:
1065:
1062:
1039:
822:
819:
14:
6274:
5965:Introduction to arithmetic groups
5872:Introduction to arithmetic groups
4552:
3276:
2969:{\displaystyle \mathrm {GL} _{n}}
2578:Hilbert–Blumenthal modular groups
1480:{\displaystyle \mathrm {GL} _{n}}
1012:arithmetic hyperbolic 3-manifolds
919:The same groups also appeared in
6021:Journal of Differential Geometry
5751:Discrete subgroups of Lie groups
5162:is the product of the primes in
4009:Arithmetic hyperbolic 3-manifold
3135:The Borel–Harish-Chandra theorem
2112:, or the closely related groups
36:
6205:
6152:
6117:
6064:
6001:
5986:
5971:
5952:
5917:
5878:
5859:
5844:
5837:10.1090/s0273-0979-1982-15003-0
5811:
5796:
5690:. Birkhäuser. p. iii+126.
4699:The congruence subgroup problem
4557:For every semisimple Lie group
1603:. If we are given an embedding
1149:as the group of integer points
5773:
5758:
5741:
5702:
5679:
5640:
5618:
5601:
5584:
5561:
5549:
5521:
5513:
5388:
5373:
5346:
5338:
5062:
5054:
4969:
4954:
4933:
4925:
4662:
4654:
4633:
4625:
4534:
4526:
4448:
4440:
4357:
4349:
4308:
4279:
4224:
4216:
4139:
4096:{\displaystyle {\mathcal {O}}}
3954:
3942:
3876:
3864:
3810:
3798:
3709:
3697:
3666:
3658:
3637:
3629:
3594:
3586:
3565:
3557:
3522:
3519:
3509:
3501:
3340:
3332:
3302:Margulis arithmeticity theorem
3118:
3110:
3075:
3067:
3012:
3004:
2911:
2903:
2852:
2844:
2816:
2813:
2805:
2793:such that there is a morphism
2697:, a well-known example is the
2668:
2660:
2657:
2645:
2614:
2601:
2560:
2547:
2463:
2447:
2406:
2398:
2349:
2341:
2306:
2298:
2234:
2226:
2188:
2180:
2145:
2137:
2099:
2091:
2048:
2045:
2037:
2016:
2004:
1992:
1972:
1958:
1894:
1733:
1727:
1716:
1713:
1707:
1689:
1621:
1435:are finite sets) with a group
1422:
1410:
1382:
1370:
1318:
1310:
1280:
1272:
1254:
1204:
1196:
1185:
1177:
1136:
1128:
1083:
1075:
860:and other classical topics in
840:
832:
705:Infinite dimensional Lie group
1:
5997:. Cambridge University Press.
5824:Bull. Amer. Math. Soc. (N.S.)
5577:
5483:A fake projective plane is a
5471:. This was first realised by
4987:
4411:{\displaystyle \mathbb {Q} .}
3151:if and only if the "form" of
2505:is a square-free integer and
1769:{\displaystyle \mathrm {G} '}
6212:Rémy, Bertrand (2007–2008),
5978:Lubotzky, Alexander (1994).
5608:Siegel, Carl Ludwig (1989).
5447:Extremal surfaces and graphs
5287:{\displaystyle \mathrm {G} }
5265:{\displaystyle \mathbb {Q} }
4992:The formal definition of an
3986:{\displaystyle n\geqslant 4}
3842:{\displaystyle n\geqslant 2}
3741:{\displaystyle n\geqslant 1}
3250:{\displaystyle \mathbb {Q} }
3208:{\displaystyle \mathrm {G} }
3186:{\displaystyle \mathbb {Q} }
3171:used to define it (i.e. the
3088:is an arithmetic lattice in
3025:is an arithmetic lattice in
2865:is an arithmetic lattice in
2786:{\displaystyle \mathbb {Q} }
2764:{\displaystyle \mathrm {G} }
1859:{\displaystyle \mathbb {Q} }
1837:{\displaystyle \mathbb {Q} }
1791:{\displaystyle \mathbb {Q} }
1576:{\displaystyle \mathrm {G} }
1502:{\displaystyle \mathrm {G} }
1235:{\displaystyle \mathbb {Q} }
1053:is an algebraic subgroup of
1046:{\displaystyle \mathrm {G} }
478:{\displaystyle \mathbb {Z} }
453:{\displaystyle \mathbb {Z} }
416:{\displaystyle \mathbb {Z} }
7:
5959:Witte-Morris, Dave (2015).
5866:Witte-Morris, Dave (2015).
5688:Adèles and algebraic groups
4418:All arithmetic lattices in
4393:is the matrix algebra over
3776:{\displaystyle F_{4}^{-20}}
3479:. For example, the lattice
2370:. Similar examples are the
2064:
1022:Definition and construction
203:List of group theory topics
10:
6279:
5851:Margulis, Girgori (1991).
5818:Thurston, William (1982).
5748:Raghunathan, M.S. (1972).
4702:
4241:. Then the group of units
4237:and for all others to the
4006:
4000:
3748:and the exceptional group
883:
6183:10.1007/s10240-009-0019-6
4015:totally real number field
4003:Arithmetic Fuchsian group
980:) were later obtained by
6095:10.1215/00127094-2410064
5419:or the weaker property (
5411:Explicit expander graphs
2885:. Thus, for example, if
2691:Hurwitz quaternion order
2366:as it is related to the
1583:an algebraic group over
1344:{\displaystyle \Lambda }
321:Elementary abelian group
198:Glossary of group theory
5415:Arithmetic groups with
5307:{\displaystyle \Gamma }
5250:-arithmetic group in a
5223:{\displaystyle \Gamma }
4327:which is isomorphic to
3920:{\displaystyle n=1,2,3}
3372:{\displaystyle n\geq 3}
1448:{\displaystyle \Gamma }
6043:10.4310/jdg/1180135693
5993:Sarnak, Peter (1990).
5780:Selberg, Atle (1956).
5591:Borel, Armand (1969).
5568:
5528:
5479:Fake projective planes
5433:
5417:Kazhdan's property (T)
5395:
5308:
5288:
5266:
5244:
5224:
5204:
5176:
5156:
5136:
5069:
5026:
5012:-arithmetic group for
5006:
4979:
4897:
4830:
4763:
4741:
4689:
4669:
4591:
4571:
4544:
4498:
4478:
4455:
4412:
4387:
4367:
4321:
4266:
4231:
4193:
4151:
4117:
4097:
4073:
4053:
4030:
3987:
3961:
3921:
3883:
3843:
3817:
3777:
3742:
3716:
3673:
3607:is irreducible, while
3601:
3529:
3473:
3446:
3400:
3373:
3347:
3292:
3251:
3229:
3209:
3187:
3165:
3125:
3082:
3039:
3019:
2970:
2938:
2918:
2879:
2859:
2826:
2787:
2765:
2743:
2723:
2675:
2621:
2570:
2529:
2528:{\displaystyle O_{-m}}
2499:
2498:{\displaystyle m>0}
2473:
2413:
2356:
2313:
2267:
2241:
2195:
2152:
2106:
2055:
1979:
1939:
1919:
1860:
1838:
1816:
1792:
1770:
1740:
1663:
1643:
1597:
1577:
1555:
1541:with ring of integers
1531:
1503:
1481:
1449:
1429:
1389:
1355:(this means that both
1345:
1325:
1290:
1236:
1214:
1143:
1110:
1090:
1047:
921:analytic number theory
850:
737:Linear algebraic group
479:
454:
417:
6263:Differential geometry
5715:Annals of Mathematics
5569:
5529:
5473:Marie-France Vignéras
5469:isospectral manifolds
5463:Isospectral manifolds
5434:
5432:{\displaystyle \tau }
5396:
5316:locally compact group
5309:
5289:
5267:
5245:
5225:
5205:
5177:
5157:
5137:
5070:
5027:
5007:
4980:
4898:
4831:
4764:
4742:
4690:
4670:
4592:
4572:
4545:
4499:
4479:
4456:
4413:
4388:
4368:
4322:
4267:
4232:
4194:
4152:
4118:
4098:
4074:
4054:
4031:
3988:
3962:
3922:
3884:
3849:(ref to GPS) and for
3844:
3818:
3778:
3743:
3717:
3674:
3602:
3530:
3474:
3472:{\displaystyle G_{i}}
3447:
3401:
3374:
3348:
3293:
3252:
3230:
3210:
3188:
3166:
3126:
3083:
3040:
3020:
2971:
2939:
2919:
2880:
2860:
2827:
2788:
2766:
2744:
2724:
2676:
2622:
2571:
2530:
2500:
2474:
2414:
2372:Siegel modular groups
2357:
2314:
2268:
2242:
2196:
2153:
2107:
2056:
1980:
1940:
1938:{\displaystyle \rho }
1920:
1861:
1839:
1817:
1793:
1771:
1741:
1664:
1644:
1598:
1578:
1556:
1532:
1504:
1482:
1450:
1430:
1390:
1346:
1326:
1291:
1237:
1215:
1144:
1111:
1091:
1048:
870:differential geometry
851:
480:
455:
418:
6216:, séminaire Bourbaki
5686:Weil, André (1982).
5538:
5498:
5423:
5324:
5314:is a lattice in the
5298:
5276:
5254:
5234:
5214:
5194:
5166:
5146:
5079:
5036:
5016:
4996:
4907:
4840:
4773:
4753:
4731:
4679:
4601:
4581:
4561:
4508:
4488:
4468:
4422:
4397:
4377:
4331:
4276:
4245:
4239:Hamilton quaternions
4203:
4161:
4127:
4107:
4083:
4063:
4043:
4020:
3971:
3931:
3893:
3853:
3827:
3787:
3752:
3726:
3686:
3611:
3539:
3483:
3456:
3410:
3390:
3357:
3353:are arithmetic when
3314:
3261:
3239:
3219:
3197:
3175:
3155:
3092:
3049:
3029:
2980:
2948:
2928:
2889:
2869:
2836:
2797:
2775:
2753:
2733:
2713:
2699:Picard modular group
2634:
2583:
2539:
2509:
2483:
2429:
2377:
2323:
2277:
2251:
2205:
2162:
2116:
2073:
1989:
1949:
1929:
1870:
1848:
1826:
1806:
1780:
1753:
1673:
1653:
1607:
1587:
1565:
1545:
1521:
1491:
1459:
1439:
1399:
1359:
1335:
1302:
1246:
1224:
1153:
1120:
1100:
1057:
1035:
976:; stronger results (
974:Oppenheim conjecture
866:Riemannian manifolds
814:
467:
442:
405:
5786:J. Indian Math. Soc
5649:"Groupes réductifs"
5487:which has the same
4747:-arithmetic lattice
4722:S-arithmetic groups
4711:congruence subgroup
4705:Congruence subgroup
3772:
2687:quaternion algebras
2266:{\displaystyle n=2}
1800:restricting scalars
1513:Using number fields
969:ergodic-theoretical
906:geometry of numbers
898:fundamental domains
111:Group homomorphisms
21:Algebraic structure
5903:10.1007/bf02699433
5855:. Springer-Verlag.
5754:. Springer-Verlag.
5665:10.1007/bf02684375
5614:. Springer-Verlag.
5564:
5524:
5453:injectivity radius
5429:
5391:
5367:
5304:
5284:
5262:
5240:
5220:
5200:
5172:
5152:
5132:
5121:
5065:
5022:
5002:
4975:
4893:
4882:
4826:
4815:
4759:
4737:
4685:
4665:
4587:
4567:
4540:
4494:
4474:
4451:
4408:
4383:
4363:
4317:
4262:
4227:
4189:
4147:
4113:
4093:
4069:
4049:
4038:quaternion algebra
4026:
3983:
3957:
3917:
3879:
3839:
3813:
3773:
3755:
3738:
3712:
3669:
3597:
3525:
3469:
3442:
3396:
3369:
3343:
3288:
3247:
3225:
3205:
3183:
3161:
3121:
3078:
3035:
3015:
2966:
2934:
2914:
2875:
2855:
2822:
2783:
2761:
2739:
2719:
2671:
2617:
2566:
2525:
2495:
2469:
2409:
2352:
2309:
2263:
2237:
2191:
2148:
2102:
2051:
1978:{\displaystyle d=}
1975:
1935:
1915:
1856:
1834:
1812:
1788:
1766:
1736:
1669:then the subgroup
1659:
1639:
1593:
1573:
1551:
1527:
1499:
1477:
1445:
1425:
1385:
1341:
1321:
1286:
1232:
1210:
1139:
1106:
1086:
1043:
846:
587:Special orthogonal
475:
450:
413:
294:Lagrange's theorem
5406:Some applications
5352:
5272:-algebraic group
5243:{\displaystyle S}
5203:{\displaystyle S}
5175:{\displaystyle S}
5155:{\displaystyle N}
5120:
5025:{\displaystyle S}
5005:{\displaystyle S}
4881:
4814:
4762:{\displaystyle S}
4740:{\displaystyle S}
4716:Jean-Pierre Serre
4688:{\displaystyle G}
4590:{\displaystyle G}
4570:{\displaystyle G}
4497:{\displaystyle A}
4477:{\displaystyle F}
4386:{\displaystyle A}
4116:{\displaystyle A}
4072:{\displaystyle F}
4052:{\displaystyle A}
4029:{\displaystyle F}
3517:
3399:{\displaystyle G}
3228:{\displaystyle n}
3164:{\displaystyle G}
3038:{\displaystyle G}
2944:is a subgroup of
2937:{\displaystyle G}
2878:{\displaystyle G}
2742:{\displaystyle G}
2722:{\displaystyle G}
2558:
1815:{\displaystyle F}
1662:{\displaystyle F}
1596:{\displaystyle F}
1554:{\displaystyle O}
1530:{\displaystyle F}
1109:{\displaystyle n}
1027:Arithmetic groups
1008:locally symmetric
989:Langlands program
978:Ratner's theorems
961:M. S. Raghunathan
894:Hermann Minkowski
878:automorphic forms
796:
795:
371:
370:
253:Alternating group
210:
209:
6270:
6248:Algebraic groups
6232:
6231:
6225:
6217:
6209:
6203:
6202:
6176:
6156:
6150:
6149:
6121:
6115:
6114:
6088:
6068:
6062:
6061:
6036:
6011:; Schaps, Mary;
6009:Katz, Mikhail G.
6005:
5999:
5998:
5990:
5984:
5983:
5975:
5969:
5968:
5956:
5950:
5949:
5921:
5915:
5914:
5882:
5876:
5875:
5863:
5857:
5856:
5848:
5842:
5841:
5839:
5815:
5809:
5808:
5800:
5794:
5793:
5777:
5771:
5770:
5762:
5756:
5755:
5745:
5739:
5738:
5706:
5700:
5699:
5683:
5677:
5676:
5644:
5638:
5637:
5622:
5616:
5615:
5605:
5599:
5598:
5588:
5573:
5571:
5570:
5565:
5548:
5533:
5531:
5530:
5525:
5520:
5512:
5511:
5506:
5493:projective plane
5441:Ramanujan graphs
5438:
5436:
5435:
5430:
5400:
5398:
5397:
5392:
5387:
5386:
5381:
5372:
5366:
5345:
5337:
5313:
5311:
5310:
5305:
5293:
5291:
5290:
5285:
5283:
5271:
5269:
5268:
5263:
5261:
5249:
5247:
5246:
5241:
5229:
5227:
5226:
5221:
5209:
5207:
5206:
5201:
5181:
5179:
5178:
5173:
5161:
5159:
5158:
5153:
5141:
5139:
5138:
5133:
5131:
5127:
5126:
5122:
5113:
5106:
5096:
5095:
5090:
5074:
5072:
5071:
5066:
5061:
5053:
5052:
5047:
5031:
5029:
5028:
5023:
5011:
5009:
5008:
5003:
4984:
4982:
4981:
4976:
4968:
4967:
4962:
4953:
4952:
4947:
4932:
4924:
4923:
4918:
4903:is a lattice in
4902:
4900:
4899:
4894:
4892:
4888:
4887:
4883:
4874:
4867:
4857:
4856:
4851:
4835:
4833:
4832:
4827:
4825:
4821:
4820:
4816:
4807:
4800:
4790:
4789:
4784:
4768:
4766:
4765:
4760:
4746:
4744:
4743:
4738:
4694:
4692:
4691:
4686:
4674:
4672:
4671:
4666:
4661:
4653:
4652:
4647:
4632:
4624:
4623:
4618:
4596:
4594:
4593:
4588:
4576:
4574:
4573:
4568:
4549:
4547:
4546:
4541:
4533:
4525:
4524:
4519:
4503:
4501:
4500:
4495:
4483:
4481:
4480:
4475:
4460:
4458:
4457:
4452:
4447:
4439:
4438:
4433:
4417:
4415:
4414:
4409:
4404:
4392:
4390:
4389:
4384:
4372:
4370:
4369:
4364:
4356:
4348:
4347:
4342:
4326:
4324:
4323:
4318:
4316:
4315:
4306:
4301:
4300:
4291:
4290:
4272:is a lattice in
4271:
4269:
4268:
4263:
4261:
4260:
4255:
4254:
4236:
4234:
4233:
4228:
4223:
4215:
4214:
4198:
4196:
4195:
4190:
4188:
4183:
4182:
4173:
4172:
4156:
4154:
4153:
4148:
4146:
4122:
4120:
4119:
4114:
4102:
4100:
4099:
4094:
4092:
4091:
4078:
4076:
4075:
4070:
4058:
4056:
4055:
4050:
4035:
4033:
4032:
4027:
3992:
3990:
3989:
3984:
3966:
3964:
3963:
3958:
3941:
3926:
3924:
3923:
3918:
3888:
3886:
3885:
3880:
3863:
3848:
3846:
3845:
3840:
3822:
3820:
3819:
3814:
3797:
3782:
3780:
3779:
3774:
3771:
3763:
3747:
3745:
3744:
3739:
3721:
3719:
3718:
3713:
3696:
3678:
3676:
3675:
3670:
3665:
3657:
3656:
3651:
3636:
3628:
3627:
3622:
3606:
3604:
3603:
3598:
3593:
3585:
3584:
3579:
3564:
3556:
3555:
3550:
3534:
3532:
3531:
3526:
3518:
3513:
3508:
3500:
3499:
3494:
3478:
3476:
3475:
3470:
3468:
3467:
3451:
3449:
3448:
3443:
3441:
3440:
3428:
3427:
3405:
3403:
3402:
3397:
3378:
3376:
3375:
3370:
3352:
3350:
3349:
3344:
3339:
3331:
3330:
3325:
3297:
3295:
3294:
3289:
3275:
3274:
3269:
3256:
3254:
3253:
3248:
3246:
3234:
3232:
3231:
3226:
3214:
3212:
3211:
3206:
3204:
3192:
3190:
3189:
3184:
3182:
3170:
3168:
3167:
3162:
3130:
3128:
3127:
3122:
3117:
3109:
3108:
3103:
3087:
3085:
3084:
3079:
3074:
3066:
3065:
3060:
3044:
3042:
3041:
3036:
3024:
3022:
3021:
3016:
3011:
3003:
3002:
2997:
2975:
2973:
2972:
2967:
2965:
2964:
2959:
2943:
2941:
2940:
2935:
2923:
2921:
2920:
2915:
2910:
2902:
2884:
2882:
2881:
2876:
2864:
2862:
2861:
2856:
2851:
2843:
2831:
2829:
2828:
2823:
2812:
2804:
2792:
2790:
2789:
2784:
2782:
2770:
2768:
2767:
2762:
2760:
2748:
2746:
2745:
2740:
2728:
2726:
2725:
2720:
2680:
2678:
2677:
2672:
2667:
2644:
2626:
2624:
2623:
2618:
2613:
2612:
2600:
2599:
2594:
2575:
2573:
2572:
2567:
2559:
2551:
2546:
2534:
2532:
2531:
2526:
2524:
2523:
2504:
2502:
2501:
2496:
2478:
2476:
2475:
2470:
2462:
2461:
2446:
2445:
2440:
2418:
2416:
2415:
2410:
2405:
2397:
2396:
2388:
2362:, is called the
2361:
2359:
2358:
2353:
2348:
2340:
2339:
2334:
2318:
2316:
2315:
2310:
2305:
2297:
2296:
2291:
2272:
2270:
2269:
2264:
2246:
2244:
2243:
2238:
2233:
2225:
2224:
2219:
2200:
2198:
2197:
2192:
2187:
2179:
2178:
2173:
2157:
2155:
2154:
2149:
2144:
2136:
2135:
2130:
2111:
2109:
2108:
2103:
2098:
2090:
2089:
2084:
2060:
2058:
2057:
2052:
2044:
2036:
2035:
2027:
2015:
2014:
2002:
1984:
1982:
1981:
1976:
1971:
1944:
1942:
1941:
1936:
1924:
1922:
1921:
1916:
1914:
1913:
1905:
1893:
1889:
1880:
1865:
1863:
1862:
1857:
1855:
1843:
1841:
1840:
1835:
1833:
1821:
1819:
1818:
1813:
1797:
1795:
1794:
1789:
1787:
1775:
1773:
1772:
1767:
1765:
1761:
1745:
1743:
1742:
1737:
1726:
1706:
1705:
1700:
1688:
1687:
1668:
1666:
1665:
1660:
1648:
1646:
1645:
1640:
1638:
1637:
1632:
1620:
1602:
1600:
1599:
1594:
1582:
1580:
1579:
1574:
1572:
1560:
1558:
1557:
1552:
1536:
1534:
1533:
1528:
1508:
1506:
1505:
1500:
1498:
1486:
1484:
1483:
1478:
1476:
1475:
1470:
1454:
1452:
1451:
1446:
1434:
1432:
1431:
1426:
1409:
1394:
1392:
1391:
1386:
1369:
1350:
1348:
1347:
1342:
1330:
1328:
1327:
1322:
1317:
1309:
1295:
1293:
1292:
1287:
1279:
1271:
1270:
1265:
1253:
1241:
1239:
1238:
1233:
1231:
1219:
1217:
1216:
1211:
1203:
1195:
1184:
1176:
1175:
1170:
1148:
1146:
1145:
1140:
1135:
1127:
1115:
1113:
1112:
1107:
1095:
1093:
1092:
1087:
1082:
1074:
1073:
1068:
1052:
1050:
1049:
1044:
1042:
1016:William Thurston
993:Robert Langlands
953:Grigori Margulis
902:symmetric spaces
855:
853:
852:
847:
839:
831:
830:
825:
804:arithmetic group
788:
781:
774:
730:Algebraic groups
503:Hyperbolic group
493:Arithmetic group
484:
482:
481:
476:
474:
459:
457:
456:
451:
449:
422:
420:
419:
414:
412:
335:Schur multiplier
289:Cauchy's theorem
277:Quaternion group
225:
224:
51:
50:
40:
27:
16:
15:
6278:
6277:
6273:
6272:
6271:
6269:
6268:
6267:
6238:
6237:
6236:
6235:
6219:
6218:
6210:
6206:
6157:
6153:
6138:10.2307/1971319
6122:
6118:
6069:
6065:
6034:math.DG/0505007
6006:
6002:
5991:
5987:
5976:
5972:
5957:
5953:
5938:10.2307/2946567
5922:
5918:
5883:
5879:
5864:
5860:
5849:
5845:
5816:
5812:
5801:
5797:
5778:
5774:
5763:
5759:
5746:
5742:
5727:10.2307/1970210
5707:
5703:
5684:
5680:
5645:
5641:
5623:
5619:
5606:
5602:
5589:
5585:
5580:
5541:
5539:
5536:
5535:
5516:
5507:
5502:
5501:
5499:
5496:
5495:
5485:complex surface
5481:
5465:
5449:
5424:
5421:
5420:
5413:
5408:
5382:
5377:
5376:
5368:
5356:
5341:
5333:
5325:
5322:
5321:
5299:
5296:
5295:
5279:
5277:
5274:
5273:
5257:
5255:
5252:
5251:
5235:
5232:
5231:
5215:
5212:
5211:
5195:
5192:
5191:
5188:
5167:
5164:
5163:
5147:
5144:
5143:
5111:
5107:
5102:
5101:
5097:
5091:
5083:
5082:
5080:
5077:
5076:
5057:
5048:
5040:
5039:
5037:
5034:
5033:
5017:
5014:
5013:
4997:
4994:
4993:
4990:
4963:
4958:
4957:
4948:
4940:
4939:
4928:
4919:
4911:
4910:
4908:
4905:
4904:
4872:
4868:
4863:
4862:
4858:
4852:
4844:
4843:
4841:
4838:
4837:
4805:
4801:
4796:
4795:
4791:
4785:
4777:
4776:
4774:
4771:
4770:
4754:
4751:
4750:
4732:
4729:
4728:
4724:
4707:
4701:
4680:
4677:
4676:
4657:
4648:
4640:
4639:
4628:
4619:
4611:
4610:
4602:
4599:
4598:
4582:
4579:
4578:
4562:
4559:
4558:
4555:
4529:
4520:
4512:
4511:
4509:
4506:
4505:
4489:
4486:
4485:
4469:
4466:
4465:
4443:
4434:
4426:
4425:
4423:
4420:
4419:
4400:
4398:
4395:
4394:
4378:
4375:
4374:
4352:
4343:
4335:
4334:
4332:
4329:
4328:
4311:
4307:
4302:
4296:
4292:
4286:
4282:
4277:
4274:
4273:
4256:
4250:
4249:
4248:
4246:
4243:
4242:
4219:
4210:
4206:
4204:
4201:
4200:
4184:
4178:
4174:
4168:
4164:
4162:
4159:
4158:
4142:
4128:
4125:
4124:
4108:
4105:
4104:
4087:
4086:
4084:
4081:
4080:
4064:
4061:
4060:
4044:
4041:
4040:
4021:
4018:
4017:
4011:
4005:
3999:
3972:
3969:
3968:
3934:
3932:
3929:
3928:
3894:
3891:
3890:
3856:
3854:
3851:
3850:
3828:
3825:
3824:
3790:
3788:
3785:
3784:
3764:
3759:
3753:
3750:
3749:
3727:
3724:
3723:
3689:
3687:
3684:
3683:
3661:
3652:
3644:
3643:
3632:
3623:
3615:
3614:
3612:
3609:
3608:
3589:
3580:
3572:
3571:
3560:
3551:
3543:
3542:
3540:
3537:
3536:
3512:
3504:
3495:
3487:
3486:
3484:
3481:
3480:
3463:
3459:
3457:
3454:
3453:
3436:
3432:
3423:
3419:
3411:
3408:
3407:
3391:
3388:
3387:
3358:
3355:
3354:
3335:
3326:
3318:
3317:
3315:
3312:
3311:
3304:
3270:
3265:
3264:
3262:
3259:
3258:
3242:
3240:
3237:
3236:
3235:variables over
3220:
3217:
3216:
3200:
3198:
3195:
3194:
3178:
3176:
3173:
3172:
3156:
3153:
3152:
3137:
3113:
3104:
3096:
3095:
3093:
3090:
3089:
3070:
3061:
3053:
3052:
3050:
3047:
3046:
3030:
3027:
3026:
3007:
2998:
2990:
2989:
2981:
2978:
2977:
2960:
2952:
2951:
2949:
2946:
2945:
2929:
2926:
2925:
2906:
2898:
2890:
2887:
2886:
2870:
2867:
2866:
2847:
2839:
2837:
2834:
2833:
2808:
2800:
2798:
2795:
2794:
2778:
2776:
2773:
2772:
2756:
2754:
2751:
2750:
2734:
2731:
2730:
2714:
2711:
2710:
2707:
2695:hermitian forms
2663:
2637:
2635:
2632:
2631:
2608:
2604:
2595:
2587:
2586:
2584:
2581:
2580:
2550:
2542:
2540:
2537:
2536:
2516:
2512:
2510:
2507:
2506:
2484:
2481:
2480:
2454:
2450:
2441:
2433:
2432:
2430:
2427:
2426:
2401:
2389:
2381:
2380:
2378:
2375:
2374:
2344:
2335:
2327:
2326:
2324:
2321:
2320:
2319:, or sometimes
2301:
2292:
2281:
2280:
2278:
2275:
2274:
2252:
2249:
2248:
2229:
2220:
2209:
2208:
2206:
2203:
2202:
2183:
2174:
2166:
2165:
2163:
2160:
2159:
2140:
2131:
2120:
2119:
2117:
2114:
2113:
2094:
2085:
2077:
2076:
2074:
2071:
2070:
2067:
2040:
2028:
2020:
2019:
2007:
2003:
1995:
1990:
1987:
1986:
1967:
1950:
1947:
1946:
1930:
1927:
1926:
1906:
1898:
1897:
1885:
1884:
1873:
1871:
1868:
1867:
1851:
1849:
1846:
1845:
1829:
1827:
1824:
1823:
1807:
1804:
1803:
1783:
1781:
1778:
1777:
1757:
1756:
1754:
1751:
1750:
1722:
1701:
1693:
1692:
1680:
1676:
1674:
1671:
1670:
1654:
1651:
1650:
1633:
1625:
1624:
1616:
1608:
1605:
1604:
1588:
1585:
1584:
1568:
1566:
1563:
1562:
1546:
1543:
1542:
1522:
1519:
1518:
1515:
1494:
1492:
1489:
1488:
1471:
1463:
1462:
1460:
1457:
1456:
1440:
1437:
1436:
1405:
1400:
1397:
1396:
1365:
1360:
1357:
1356:
1336:
1333:
1332:
1313:
1305:
1303:
1300:
1299:
1275:
1266:
1258:
1257:
1249:
1247:
1244:
1243:
1227:
1225:
1222:
1221:
1199:
1191:
1180:
1171:
1163:
1162:
1154:
1151:
1150:
1131:
1123:
1121:
1118:
1117:
1101:
1098:
1097:
1078:
1069:
1061:
1060:
1058:
1055:
1054:
1038:
1036:
1033:
1032:
1029:
1024:
890:Charles Hermite
886:
858:quadratic forms
835:
826:
818:
817:
815:
812:
811:
808:algebraic group
792:
763:
762:
751:Abelian variety
744:Reductive group
732:
722:
721:
720:
719:
670:
662:
654:
646:
638:
611:Special unitary
522:
508:
507:
489:
488:
470:
468:
465:
464:
445:
443:
440:
439:
408:
406:
403:
402:
394:
393:
384:Discrete groups
373:
372:
328:Frobenius group
273:
260:
249:
242:Symmetric group
238:
222:
212:
211:
62:Normal subgroup
48:
28:
19:
12:
11:
5:
6276:
6266:
6265:
6260:
6255:
6250:
6234:
6233:
6204:
6151:
6116:
6063:
6027:(3): 399–422,
6000:
5985:
5970:
5951:
5932:(1): 165–182.
5916:
5877:
5858:
5843:
5830:(3): 357–381.
5810:
5795:
5788:. New Series.
5772:
5757:
5740:
5721:(3): 485–535.
5701:
5678:
5639:
5617:
5600:
5582:
5581:
5579:
5576:
5563:
5560:
5557:
5554:
5551:
5547:
5544:
5523:
5519:
5515:
5510:
5505:
5480:
5477:
5464:
5461:
5448:
5445:
5428:
5412:
5409:
5407:
5404:
5403:
5402:
5390:
5385:
5380:
5375:
5371:
5365:
5362:
5359:
5355:
5351:
5348:
5344:
5340:
5336:
5332:
5329:
5303:
5282:
5260:
5239:
5219:
5199:
5187:
5184:
5171:
5151:
5130:
5125:
5119:
5116:
5110:
5105:
5100:
5094:
5089:
5086:
5064:
5060:
5056:
5051:
5046:
5043:
5021:
5001:
4989:
4986:
4974:
4971:
4966:
4961:
4956:
4951:
4946:
4943:
4938:
4935:
4931:
4927:
4922:
4917:
4914:
4891:
4886:
4880:
4877:
4871:
4866:
4861:
4855:
4850:
4847:
4824:
4819:
4813:
4810:
4804:
4799:
4794:
4788:
4783:
4780:
4758:
4736:
4723:
4720:
4703:Main article:
4700:
4697:
4684:
4664:
4660:
4656:
4651:
4646:
4643:
4638:
4635:
4631:
4627:
4622:
4617:
4614:
4609:
4606:
4586:
4566:
4554:
4553:Classification
4551:
4539:
4536:
4532:
4528:
4523:
4518:
4515:
4493:
4473:
4450:
4446:
4442:
4437:
4432:
4429:
4407:
4403:
4382:
4362:
4359:
4355:
4351:
4346:
4341:
4338:
4314:
4310:
4305:
4299:
4295:
4289:
4285:
4281:
4259:
4253:
4226:
4222:
4218:
4213:
4209:
4187:
4181:
4177:
4171:
4167:
4145:
4141:
4138:
4135:
4132:
4112:
4090:
4068:
4048:
4025:
4007:Main article:
4001:Main article:
3998:
3995:
3982:
3979:
3976:
3956:
3953:
3950:
3947:
3944:
3940:
3937:
3916:
3913:
3910:
3907:
3904:
3901:
3898:
3878:
3875:
3872:
3869:
3866:
3862:
3859:
3838:
3835:
3832:
3812:
3809:
3806:
3803:
3800:
3796:
3793:
3770:
3767:
3762:
3758:
3737:
3734:
3731:
3711:
3708:
3705:
3702:
3699:
3695:
3692:
3668:
3664:
3660:
3655:
3650:
3647:
3642:
3639:
3635:
3631:
3626:
3621:
3618:
3596:
3592:
3588:
3583:
3578:
3575:
3570:
3567:
3563:
3559:
3554:
3549:
3546:
3524:
3521:
3516:
3511:
3507:
3503:
3498:
3493:
3490:
3466:
3462:
3439:
3435:
3431:
3426:
3422:
3418:
3415:
3395:
3368:
3365:
3362:
3342:
3338:
3334:
3329:
3324:
3321:
3303:
3300:
3287:
3284:
3281:
3278:
3273:
3268:
3245:
3224:
3203:
3181:
3160:
3136:
3133:
3120:
3116:
3112:
3107:
3102:
3099:
3077:
3073:
3069:
3064:
3059:
3056:
3034:
3014:
3010:
3006:
3001:
2996:
2993:
2988:
2985:
2963:
2958:
2955:
2933:
2913:
2909:
2905:
2901:
2897:
2894:
2874:
2854:
2850:
2846:
2842:
2821:
2818:
2815:
2811:
2807:
2803:
2781:
2759:
2738:
2718:
2706:
2703:
2670:
2666:
2662:
2659:
2656:
2653:
2650:
2647:
2643:
2640:
2616:
2611:
2607:
2603:
2598:
2593:
2590:
2565:
2562:
2557:
2554:
2549:
2545:
2522:
2519:
2515:
2494:
2491:
2488:
2468:
2465:
2460:
2457:
2453:
2449:
2444:
2439:
2436:
2424:Bianchi groups
2408:
2404:
2400:
2395:
2392:
2387:
2384:
2351:
2347:
2343:
2338:
2333:
2330:
2308:
2304:
2300:
2295:
2290:
2287:
2284:
2262:
2259:
2256:
2236:
2232:
2228:
2223:
2218:
2215:
2212:
2190:
2186:
2182:
2177:
2172:
2169:
2147:
2143:
2139:
2134:
2129:
2126:
2123:
2101:
2097:
2093:
2088:
2083:
2080:
2066:
2063:
2050:
2047:
2043:
2039:
2034:
2031:
2026:
2023:
2018:
2013:
2010:
2006:
2001:
1998:
1994:
1974:
1970:
1966:
1963:
1960:
1957:
1954:
1934:
1912:
1909:
1904:
1901:
1896:
1892:
1888:
1883:
1879:
1876:
1854:
1832:
1811:
1786:
1764:
1760:
1735:
1732:
1729:
1725:
1721:
1718:
1715:
1712:
1709:
1704:
1699:
1696:
1691:
1686:
1683:
1679:
1658:
1636:
1631:
1628:
1623:
1619:
1615:
1612:
1592:
1571:
1550:
1526:
1514:
1511:
1497:
1474:
1469:
1466:
1444:
1424:
1421:
1418:
1415:
1412:
1408:
1404:
1384:
1381:
1378:
1375:
1372:
1368:
1364:
1340:
1320:
1316:
1312:
1308:
1285:
1282:
1278:
1274:
1269:
1264:
1261:
1256:
1252:
1230:
1209:
1206:
1202:
1198:
1194:
1190:
1187:
1183:
1179:
1174:
1169:
1166:
1161:
1158:
1138:
1134:
1130:
1126:
1105:
1085:
1081:
1077:
1072:
1067:
1064:
1041:
1028:
1025:
1023:
1020:
941:Harish-Chandra
885:
882:
845:
842:
838:
834:
829:
824:
821:
810:, for example
794:
793:
791:
790:
783:
776:
768:
765:
764:
761:
760:
758:Elliptic curve
754:
753:
747:
746:
740:
739:
733:
728:
727:
724:
723:
718:
717:
714:
711:
707:
703:
702:
701:
696:
694:Diffeomorphism
690:
689:
684:
679:
673:
672:
668:
664:
660:
656:
652:
648:
644:
640:
636:
631:
630:
619:
618:
607:
606:
595:
594:
583:
582:
571:
570:
559:
558:
551:Special linear
547:
546:
539:General linear
535:
534:
529:
523:
514:
513:
510:
509:
506:
505:
500:
495:
487:
486:
473:
461:
448:
435:
433:Modular groups
431:
430:
429:
424:
411:
395:
392:
391:
386:
380:
379:
378:
375:
374:
369:
368:
367:
366:
361:
356:
353:
347:
346:
340:
339:
338:
337:
331:
330:
324:
323:
318:
309:
308:
306:Hall's theorem
303:
301:Sylow theorems
297:
296:
291:
283:
282:
281:
280:
274:
269:
266:Dihedral group
262:
261:
256:
250:
245:
239:
234:
223:
218:
217:
214:
213:
208:
207:
206:
205:
200:
192:
191:
190:
189:
184:
179:
174:
169:
164:
159:
157:multiplicative
154:
149:
144:
139:
131:
130:
129:
128:
123:
115:
114:
106:
105:
104:
103:
101:Wreath product
98:
93:
88:
86:direct product
80:
78:Quotient group
72:
71:
70:
69:
64:
59:
49:
46:
45:
42:
41:
33:
32:
9:
6:
4:
3:
2:
6275:
6264:
6261:
6259:
6258:Number theory
6256:
6254:
6251:
6249:
6246:
6245:
6243:
6229:
6223:
6215:
6208:
6200:
6196:
6192:
6188:
6184:
6180:
6175:
6170:
6166:
6162:
6155:
6147:
6143:
6139:
6135:
6131:
6128:(in French).
6127:
6126:Ann. of Math.
6120:
6112:
6108:
6104:
6100:
6096:
6092:
6087:
6082:
6078:
6074:
6067:
6060:
6056:
6052:
6048:
6044:
6040:
6035:
6030:
6026:
6022:
6018:
6014:
6010:
6004:
5996:
5989:
5982:. Birkhäuser.
5981:
5974:
5966:
5962:
5955:
5947:
5943:
5939:
5935:
5931:
5927:
5920:
5912:
5908:
5904:
5900:
5896:
5892:
5888:
5881:
5873:
5869:
5862:
5854:
5847:
5838:
5833:
5829:
5825:
5821:
5814:
5806:
5799:
5791:
5787:
5783:
5776:
5768:
5761:
5753:
5752:
5744:
5736:
5732:
5728:
5724:
5720:
5716:
5712:
5705:
5697:
5693:
5689:
5682:
5674:
5670:
5666:
5662:
5658:
5654:
5650:
5643:
5636:
5632:
5628:
5621:
5613:
5612:
5604:
5596:
5595:
5587:
5583:
5575:
5558:
5555:
5552:
5508:
5494:
5490:
5489:Betti numbers
5486:
5476:
5474:
5470:
5460:
5458:
5454:
5444:
5442:
5426:
5418:
5383:
5363:
5360:
5357:
5353:
5349:
5330:
5327:
5320:
5319:
5318:
5317:
5237:
5197:
5183:
5169:
5149:
5128:
5123:
5117:
5114:
5108:
5098:
5092:
5049:
5019:
4999:
4985:
4972:
4964:
4949:
4936:
4920:
4889:
4884:
4878:
4875:
4869:
4859:
4853:
4822:
4817:
4811:
4808:
4802:
4792:
4786:
4756:
4748:
4734:
4719:
4717:
4712:
4706:
4696:
4682:
4649:
4636:
4620:
4607:
4604:
4584:
4564:
4550:
4537:
4521:
4491:
4471:
4462:
4435:
4405:
4380:
4360:
4344:
4312:
4297:
4293:
4287:
4283:
4257:
4240:
4211:
4207:
4179:
4175:
4169:
4165:
4136:
4133:
4130:
4110:
4079:and an order
4066:
4046:
4039:
4023:
4016:
4010:
4004:
3994:
3980:
3977:
3974:
3951:
3948:
3945:
3914:
3911:
3908:
3905:
3902:
3899:
3896:
3873:
3870:
3867:
3836:
3833:
3830:
3807:
3804:
3801:
3768:
3765:
3760:
3756:
3735:
3732:
3729:
3706:
3703:
3700:
3680:
3653:
3640:
3624:
3581:
3568:
3552:
3514:
3496:
3464:
3460:
3437:
3433:
3429:
3424:
3420:
3416:
3413:
3393:
3384:
3382:
3381:superrigidity
3366:
3363:
3360:
3327:
3309:
3299:
3282:
3271:
3222:
3158:
3150:
3145:
3142:
3132:
3105:
3062:
3032:
2999:
2986:
2983:
2961:
2931:
2895:
2892:
2872:
2819:
2771:defined over
2736:
2716:
2702:
2700:
2696:
2692:
2688:
2684:
2654:
2651:
2648:
2628:
2609:
2605:
2596:
2579:
2563:
2555:
2552:
2520:
2517:
2513:
2492:
2489:
2486:
2466:
2458:
2455:
2451:
2442:
2425:
2420:
2393:
2390:
2373:
2369:
2368:modular curve
2365:
2364:modular group
2336:
2293:
2260:
2257:
2254:
2221:
2175:
2132:
2086:
2062:
2032:
2029:
2011:
2008:
1999:
1996:
1964:
1961:
1955:
1952:
1932:
1910:
1907:
1890:
1881:
1877:
1874:
1809:
1801:
1762:
1747:
1730:
1719:
1710:
1702:
1684:
1681:
1677:
1656:
1649:defined over
1634:
1613:
1610:
1590:
1548:
1540:
1524:
1510:
1472:
1416:
1406:
1376:
1366:
1354:
1353:commensurable
1296:
1283:
1267:
1207:
1188:
1172:
1159:
1103:
1070:
1019:
1017:
1013:
1009:
1004:
1002:
998:
997:trace formula
994:
991:initiated by
990:
985:
983:
982:Marina Ratner
979:
975:
970:
964:
962:
958:
957:David Kazhdan
954:
950:
946:
942:
938:
934:
930:
925:
922:
917:
915:
911:
907:
903:
899:
895:
891:
881:
879:
875:
871:
867:
863:
862:number theory
859:
843:
827:
809:
805:
801:
789:
784:
782:
777:
775:
770:
769:
767:
766:
759:
756:
755:
752:
749:
748:
745:
742:
741:
738:
735:
734:
731:
726:
725:
715:
712:
709:
708:
706:
700:
697:
695:
692:
691:
688:
685:
683:
680:
678:
675:
674:
671:
665:
663:
657:
655:
649:
647:
641:
639:
633:
632:
628:
624:
621:
620:
616:
612:
609:
608:
604:
600:
597:
596:
592:
588:
585:
584:
580:
576:
573:
572:
568:
564:
561:
560:
556:
552:
549:
548:
544:
540:
537:
536:
533:
530:
528:
525:
524:
521:
517:
512:
511:
504:
501:
499:
496:
494:
491:
490:
462:
437:
436:
434:
428:
425:
400:
397:
396:
390:
387:
385:
382:
381:
377:
376:
365:
362:
360:
357:
354:
351:
350:
349:
348:
345:
342:
341:
336:
333:
332:
329:
326:
325:
322:
319:
317:
315:
311:
310:
307:
304:
302:
299:
298:
295:
292:
290:
287:
286:
285:
284:
278:
275:
272:
267:
264:
263:
259:
254:
251:
248:
243:
240:
237:
232:
229:
228:
227:
226:
221:
220:Finite groups
216:
215:
204:
201:
199:
196:
195:
194:
193:
188:
185:
183:
180:
178:
175:
173:
170:
168:
165:
163:
160:
158:
155:
153:
150:
148:
145:
143:
140:
138:
135:
134:
133:
132:
127:
124:
122:
119:
118:
117:
116:
113:
112:
108:
107:
102:
99:
97:
94:
92:
89:
87:
84:
81:
79:
76:
75:
74:
73:
68:
65:
63:
60:
58:
55:
54:
53:
52:
47:Basic notions
44:
43:
39:
35:
34:
31:
26:
22:
18:
17:
6253:Group theory
6213:
6207:
6164:
6160:
6154:
6132:(1): 21–32.
6129:
6125:
6119:
6076:
6073:Duke Math. J
6072:
6066:
6024:
6020:
6003:
5994:
5988:
5979:
5973:
5964:
5954:
5929:
5926:Ann. of Math
5925:
5919:
5894:
5890:
5880:
5871:
5861:
5852:
5846:
5827:
5823:
5813:
5804:
5798:
5789:
5785:
5775:
5766:
5760:
5750:
5743:
5718:
5714:
5704:
5687:
5681:
5656:
5652:
5642:
5626:
5620:
5610:
5603:
5593:
5586:
5482:
5466:
5450:
5414:
5189:
5075:replaced by
4991:
4727:
4725:
4708:
4556:
4463:
4157:the algebra
4012:
3681:
3385:
3307:
3305:
3146:
3138:
2708:
2629:
2421:
2068:
1798:obtained by
1748:
1539:number field
1516:
1297:
1030:
1005:
1001:James Arthur
986:
965:
949:Atle Selberg
937:Jacques Tits
929:Armand Borel
926:
918:
910:discriminant
887:
803:
797:
626:
614:
602:
590:
578:
566:
554:
542:
492:
313:
270:
257:
246:
235:
231:Cyclic group
109:
96:Free product
67:Group action
30:Group theory
25:Group theory
24:
6167:: 113–184.
6013:Vishne, Uzi
5897:: 165–246.
1925:induced by
1866:-embedding
1014:, which as
914:unit groups
800:mathematics
516:Topological
355:alternating
6242:Categories
6079:(3): 465.
5659:: 55–150.
5597:. Hermann.
5578:References
4988:Definition
2273:the group
1331:any group
945:Lie groups
933:André Weil
623:Symplectic
563:Orthogonal
520:Lie groups
427:Free group
152:continuous
91:Direct sum
6174:0705.2891
6086:1201.3399
5911:118023776
5673:189767074
5427:τ
5361:∈
5354:∏
5350:×
5302:Γ
5218:Γ
4937:×
4294:⊗
4288:σ
4176:⊗
4170:σ
4140:→
4131:σ
3978:⩾
3834:⩾
3766:−
3733:⩾
3641:×
3569:×
3430:×
3364:≥
3277:∖
3149:cocompact
2987:∩
2817:→
2553:−
2518:−
2456:−
2009:−
1997:ρ
1933:ρ
1895:→
1875:ρ
1720:⊂
1682:−
1678:ρ
1622:→
1611:ρ
1443:Γ
1420:Λ
1417:∩
1414:Γ
1403:Λ
1380:Λ
1377:∩
1374:Γ
1363:Γ
1351:which is
1339:Λ
1255:→
1189:∩
1157:Γ
1096:for some
687:Conformal
575:Euclidean
182:nilpotent
6222:citation
6111:20839217
6059:18152345
6015:(2007),
5792:: 47–87.
3679:is not.
2576:and the
2065:Examples
2000:′
1891:′
1878:′
1844:and the
1763:′
874:topology
682:Poincaré
527:Solenoid
399:Integers
389:Lattices
364:sporadic
359:Lie type
187:solvable
177:dihedral
162:additive
147:infinite
57:Subgroup
6199:1153678
6191:2511587
6146:1971319
6103:3165420
6051:2331526
5946:2946567
5735:1970210
5696:0670072
5635:0213362
5491:as the
4749:(where
3193:-group
3141:lattice
1945:(where
884:History
677:Lorentz
599:Unitary
498:Lattice
438:PSL(2,
172:abelian
83:(Semi-)
6197:
6189:
6144:
6109:
6101:
6057:
6049:
5944:
5909:
5733:
5694:
5671:
5633:
5230:is an
5142:where
2683:orders
2479:where
2247:. For
532:Circle
463:SL(2,
352:cyclic
316:-group
167:cyclic
142:finite
137:simple
121:kernel
6195:S2CID
6169:arXiv
6142:JSTOR
6107:S2CID
6081:arXiv
6055:S2CID
6029:arXiv
5942:JSTOR
5907:S2CID
5731:JSTOR
5669:S2CID
5457:girth
5294:then
4059:over
3967:when
3889:when
2976:then
2709:When
1802:from
1776:over
1537:be a
802:, an
716:Sp(∞)
713:SU(∞)
126:image
6228:link
5961:"18"
5868:"16"
4036:, a
3823:for
3722:for
2924:and
2490:>
2201:and
1561:and
1395:and
872:and
710:O(∞)
699:Loop
518:and
6179:doi
6165:109
6134:doi
6130:112
6091:doi
6077:163
6039:doi
5934:doi
5930:135
5899:doi
5832:doi
5723:doi
5661:doi
4103:in
3535:in
3308:any
2685:in
1822:to
1031:If
955:,
947:by
798:In
625:Sp(
613:SU(
589:SO(
553:SL(
541:GL(
6244::
6224:}}
6220:{{
6193:.
6187:MR
6185:.
6177:.
6163:.
6140:.
6105:.
6099:MR
6097:.
6089:.
6075:.
6053:,
6047:MR
6045:,
6037:,
6025:76
6023:,
6019:,
5963:.
5940:.
5928:.
5905:.
5895:76
5893:.
5889:.
5870:.
5826:.
5822:.
5790:20
5784:.
5729:.
5719:75
5717:.
5713:.
5692:MR
5667:.
5657:27
5655:.
5651:.
5631:MR
5182:.
4709:A
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