25:
949:, for any nonzero prime of your Dedekind domain, there is a map from the Dedekind domain to the quotient of the Dedekind domain by the prime, which is a finite field for all finite primes. This induces a map from the fraction field to any such finite field. Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field.
103:
1789:
In the early 1940s, Weil used the first definition (over an arbitrary base field) but could not at first prove that it implied the second. Only in 1948 did he prove that complete algebraic groups can be embedded into projective space. Meanwhile, in order to make the proof of the
2904:
is a polarisation that is an isomorphism. Jacobians of curves are naturally equipped with a principal polarisation as soon as one picks an arbitrary rational base point on the curve, and the curve can be reconstructed from its polarised
Jacobian when the genus is
3823:
2895:
for abelian varieties and for which the pullback of the
Poincaré bundle along the associated graph morphism is ample (so it is analogous to a positive-definite quadratic form). Polarised abelian varieties have finite
3694:
2617:
2771:
2982:
3880:
3808:
3780:
3752:
3575:
2046:
1972:
1917:
899:. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory.
3909:-torsion points generate number fields with very little ramification and hence of small discriminant, while, on the other hand, there are lower bounds on discriminants of number fields.
3476:
3720:
3547:
2545:
3370:
2815:
2173:
3521:
2718:
3121:
2432:
2393:
4111:
3852:
1845:
1780:
548:
523:
486:
3412:
2679:
2648:
2498:
2467:
2345:
2300:
1381:
1438:
3159:
1686:
1258:
4019:
3907:
3308:
3240:. The fibers of an abelian scheme are abelian varieties, so one could think of an abelian scheme over S as being a family of abelian varieties parametrised by
3063:
3036:
2926:
1586:
1501:
1316:
1612:
independent periods; equivalently, it is a function in the function field of an abelian variety. For example, in the nineteenth century there was much interest in
3273:
2235:
2112:
1228:
2835:
1706:
1660:
3818:
1172:
1079:
laid the basis for the study of abelian functions in terms of complex tori. He also appears to be the first to use the name "abelian variety". It was
3135:; over the complex number this is equivalent to the definition of polarisation given above. A morphism of polarised abelian varieties is a morphism
1399:, one can make this condition more explicit. There are several equivalent formulations of this; all of them are known as the Riemann conditions.
1806:
and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings (see also the history section in the
1075:
By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions. Eventually, in the 1920s,
850:
3580:
4053:
2558:
2245:
to a product of abelian varieties of lower dimension. Any abelian variety is isogenous to a product of simple abelian varieties.
4173:
1550:
969:
408:
4297:
4207:
2723:
3949:
2942:
358:
3857:
3785:
3757:
3729:
3552:
2000:
1926:
989:
843:
353:
1874:
1204:
When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case
4254:
68:
46:
39:
3162:
2140:
3417:
3198:
1525:
with which the theory started, can be derived from the simpler, translation-invariant theory of differentials on
1339:
1136:
2080:-torsion, one considers only the geometric points, one obtains a new invariant for varieties in characteristic
1791:
769:
4374:
4333:
4289:
4231:
3939:
836:
3699:
3526:
2511:
1183:
that admits a positive line bundle. Since they are complex tori, abelian varieties carry the structure of a
1002:, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the
4379:
4328:
4226:
4030:
3954:
3944:
3934:
3341:
2780:
453:
267:
910:
that field. Historically the first abelian varieties to be studied were those defined over the field of
884:
4384:
4364:
1852:
1052:
After Abel and Jacobi, some of the most important contributors to the theory of abelian functions were
185:
2149:
1662:
is the quotient of the
Jacobian of some curve; that is, there is some surjection of abelian varieties
4369:
4323:
1168:
1061:
4221:
3481:
2691:
4115:
3280:
3080:
2405:
2366:
2132:
1856:
1818:
By the definitions, an abelian variety is a group variety. Its group of points can be proven to be
1026:
651:
385:
262:
150:
33:
4213:. A comprehensive treatment of the complex theory, with an overview of the history of the subject.
4094:
3835:
1828:
1763:
531:
506:
469:
937:
techniques lead naturally from abelian varieties defined over number fields to ones defined over
930:
876:
4288:, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.:
3375:
2068:
are not coprime, the same result can be recovered provided one interprets it as saying that the
1083:
in the 1940s who gave the subject its modern foundations in the language of algebraic geometry.
1069:
3323:
2657:
2626:
2476:
2445:
2324:
2278:
1613:
1360:
934:
801:
591:
50:
4033:, Jacobian varieties, in Arithmetic Geometry, eds Cornell and Silverman, Springer-Verlag, 1986
2685:
1417:
1030:
675:
3854:
with good reduction at all primes. Equivalently, there are no nonzero abelian schemes over
3138:
1665:
1276:
is an abelian variety, i.e., whether or not it can be embedded into a projective space. Let
1237:
1191:
of abelian varieties is a morphism of the underlying algebraic varieties that preserves the
1037:(i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an
4307:
4144:
4124:
4066:
3997:
3885:
3286:
3186:
3066:
3041:
3014:
2908:
2259:
2136:
1601:
1564:
1479:
1287:
615:
603:
221:
155:
4152:
4074:
8:
3250:
2774:
2214:
2091:
1848:
1782:
of complex numbers, these notions coincide with the previous definition. Over all bases,
1546:
1461:
1412:
1207:
1184:
1044:
903:
190:
85:
4128:
4088:
3829:
3319:
3190:
2897:
2846:
2820:
2144:
1807:
1746:
1691:
1645:
1091:
1015:
1011:
868:
175:
147:
4311:
4293:
4250:
4203:
4166:
3315:
2937:
2933:
1616:
that may be expressed in terms of elliptic integrals. This comes down to asking that
1164:
1076:
999:
995:
977:
580:
423:
317:
4341:
3975:
1065:
746:
4148:
4132:
4070:
2929:
2928:. Not all principally polarised abelian varieties are Jacobians of curves; see the
1803:
1729:
1628:
1521:
1264:
that the algebraic variety condition imposes extra constraints on a complex torus.
1261:
1192:
1132:
1099:
1087:
1057:
1053:
1007:
961:
953:
923:
896:
872:
731:
723:
715:
707:
699:
687:
627:
567:
557:
399:
341:
216:
964:
of other algebraic varieties. The group law of an abelian variety is necessarily
4303:
4264:
4246:
4199:
4140:
4062:
3922:
3233:
3229:
3218:
3214:
2681:
given by tensor product of line bundles, which makes it into an abelian variety.
1795:
1749:
1742:
1732:
1725:
1604:
on an abelian variety, which may be regarded therefore as a periodic function of
1554:
1148:
946:
933:
are a special case, which is important also from the viewpoint of number theory.
911:
888:
808:
794:
751:
639:
562:
392:
306:
246:
126:
3921:
is a commutative group variety which is an extension of an abelian variety by a
2303:
2118:
1783:
1350:
1342:
1231:
1095:
1072:. The subject was very popular at the time, already having a large literature.
973:
957:
822:
758:
448:
428:
365:
330:
251:
241:
226:
211:
165:
142:
3723:
2183:
of the abelian variety. Similar results hold for some other classes of fields
4358:
4281:
4238:
1860:
1819:
1273:
1116:
915:
741:
663:
497:
370:
236:
3782:, but the Néron model is not proper and hence is not an abelian scheme over
1080:
4217:
3221:
3005:
2870:
2866:
2862:
2125:
1979:
1799:
1388:
1144:
938:
596:
295:
284:
231:
206:
201:
160:
131:
94:
4315:
1708:
is a
Jacobian. This theorem remains true if the ground field is infinite.
1642:. It states that over an algebraically closed field every abelian variety
902:
An abelian variety can be defined by equations having coefficients in any
3193:. This allows for a uniform treatment of phenomena such as reduction mod
2853:
is coprime to the characteristic of the base. In general — for all
2315:
1802:
that he had announced in 1940 work, he had to introduce the notion of an
1003:
965:
942:
919:
864:
4136:
1989:
When the base field is an algebraically closed field of characteristic
1868:
1022:
763:
491:
3832:
independently proved that there are no nonzero abelian varieties over
2891:
from an abelian variety to its dual that is symmetric with respect to
4044:
1175:, one may equivalently define a complex abelian variety of dimension
1086:
Today, abelian varieties form an important tool in number theory, in
892:
584:
3689:{\displaystyle \operatorname {Proj} R/(y^{2}z-x^{3}-Axz^{2}-Bz^{3})}
1716:
Two equivalent definitions of abelian variety over a general field
1188:
121:
2612:{\displaystyle 1_{A}\times f\colon A\times T\to A\times A^{\vee }}
1272:
The following criterion by
Riemann decides whether or not a given
2887:
2242:
2057:
1354:
1197:
463:
377:
1014:. When those were replaced by polynomials of higher degree, say
102:
2302:(over the same field), which is the solution to the following
16:
A projective algebraic variety that is also an algebraic group
2175:
and a finite commutative group for some non-negative integer
1621:
1124:
1029:, the answer was formulated: this would involve functions of
976:
is an abelian variety of dimension 1. Abelian varieties have
2076:. If instead of looking at the full scheme structure on the
3326:, governed by the deformation properties of the associated
2211:, over the same field, is an abelian variety of dimension
1234:, and every complex torus gives rise to such a curve; for
2991:
2248:
1638:
One important structure theorem of abelian varieties is
3123:. A choice of an equivalence class of Riemann forms on
1230:, the notion of abelian variety is the same as that of
4349:, Oxford: Mathematical Institute, University of Oxford
4237:
2766:{\displaystyle f^{\vee }\colon B^{\vee }\to A^{\vee }}
2684:
This association is a duality in the sense that it is
2306:. A family of degree 0 line bundles parametrised by a
914:. Such abelian varieties turn out to be exactly those
4097:
4000:
3888:
3882:. The proof involves showing that the coordinates of
3860:
3838:
3788:
3760:
3732:
3702:
3583:
3555:
3529:
3484:
3420:
3414:
has no repeated complex roots. Then the discriminant
3378:
3344:
3289:
3253:
3141:
3083:
3044:
3017:
2977:{\displaystyle \mathrm {End} (A)\otimes \mathbb {Q} }
2945:
2911:
2823:
2783:
2726:
2694:
2660:
2629:
2561:
2514:
2479:
2448:
2408:
2369:
2327:
2281:
2217:
2152:
2094:
2072:-torsion defines a finite flat group scheme of rank 2
2003:
1929:
1877:
1831:
1766:
1694:
1668:
1648:
1567:
1482:
1420:
1402:
1363:
1290:
1240:
1210:
534:
509:
472:
4351:. Description of the Jacobian of the Covering Curves
2434:
is a trivial line bundle (here 0 is the identity of
1545:
over the complex numbers. From the point of view of
1338:
is an abelian variety if and only if there exists a
3201:), and parameter-families of abelian varieties. An
1813:
1167:over the field of complex numbers. By invoking the
4105:
4013:
3901:
3874:
3846:
3802:
3774:
3746:
3714:
3688:
3569:
3541:
3515:
3470:
3406:
3364:
3302:
3267:
3153:
3115:
3057:
3030:
2976:
2920:
2829:
2809:
2765:
2712:
2673:
2642:
2611:
2539:
2492:
2461:
2426:
2387:
2339:
2294:
2229:
2167:
2106:
2040:
1966:
1911:
1839:
1774:
1700:
1680:
1654:
1580:
1495:
1432:
1375:
1310:
1252:
1222:
945:. Since a number field is the fraction field of a
542:
517:
480:
3875:{\displaystyle \operatorname {Spec} \mathbb {Z} }
3803:{\displaystyle \operatorname {Spec} \mathbb {Z} }
3775:{\displaystyle \operatorname {Spec} \mathbb {Z} }
3747:{\displaystyle \operatorname {Spec} \mathbb {Z} }
3570:{\displaystyle \operatorname {Spec} \mathbb {Z} }
2041:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{2g}}
1967:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{2g}}
4356:
1912:{\displaystyle (\mathbb {Q} /\mathbb {Z} )^{2g}}
994:In the early nineteenth century, the theory of
4193:
4091:(1985). "Il n'y a pas de variété abélienne sur
998:succeeded in giving a basis for the theory of
4321:
844:
2415:
2409:
2382:
2376:
3471:{\displaystyle \Delta =-16(4A^{3}+27B^{2})}
2654:, so there is a natural group operation on
2650:correspond to line bundles of degree 0 on
851:
837:
4216:
4194:Birkenhake, Christina; Lange, H. (1992),
4099:
4042:
3868:
3840:
3796:
3768:
3740:
3563:
3492:
3358:
3322:of abelian schemes are, according to the
2970:
2155:
2021:
2008:
1947:
1934:
1892:
1882:
1833:
1768:
1155:. A complex abelian variety of dimension
1094:), and in algebraic geometry (especially
983:
906:; the variety is then said to be defined
536:
511:
474:
69:Learn how and when to remove this message
4087:
2837:(defined via the Poincaré bundle). The
1561:letters acting on the function field of
32:This article includes a list of general
4280:
2841:-torsion of an abelian variety and the
2688:, i.e., it associates to all morphisms
2473:, the Poincaré bundle, parametrised by
2253:
1711:
1322:is a complex vector space of dimension
4357:
4339:
3912:
3754:, which is a smooth group scheme over
3715:{\displaystyle \operatorname {Spec} R}
3542:{\displaystyle \operatorname {Spec} R}
3185:One can also define abelian varieties
2992:Polarisations over the complex numbers
2623:is a point, we see that the points of
2540:{\displaystyle f\colon T\to A^{\vee }}
2469:and a family of degree 0 line bundles
2139:. Hence, by the structure theorem for
1786:are abelian varieties of dimension 1.
1440:is associated with an abelian variety
952:Abelian varieties appear naturally as
409:Classification of finite simple groups
4322:Venkov, B.B.; Parshin, A.N. (2001) ,
3000:can be defined as an abelian variety
2249:Polarisation and dual abelian variety
2143:, it is isomorphic to a product of a
1633:
1515:. The study of differential forms on
1387:is usually called a (non-degenerate)
1267:
956:(the connected components of zero in
3950:Equations defining abelian varieties
2869:of each other. This generalises the
2773:in a compatible way, and there is a
1591:
18:
4167:"There is no Abelian scheme over Z"
3981:. Math Department Oxford University
3365:{\displaystyle A,B\in \mathbb {Z} }
3173:is equivalent to the given form on
3161:of abelian varieties such that the
2810:{\displaystyle (A^{\vee })^{\vee }}
1863:of an abelian variety of dimension
1179:to be a complex torus of dimension
1135:. It can always be obtained as the
1090:(more specifically in the study of
13:
3976:"N-Covers of Hyperelliptic Curves"
3507:
3421:
2953:
2950:
2947:
2195:The product of an abelian variety
1403:The Jacobian of an algebraic curve
1105:
990:History of manifolds and varieties
38:it lacks sufficient corresponding
14:
4396:
4262:
4243:Degeneration of Abelian Varieties
4179:from the original on 23 Aug 2020.
3973:
3180:
2619:. Applying this to the case when
2551:is isomorphic to the pullback of
2141:finitely generated abelian groups
1620:is a product of elliptic curves,
1448:, by means of an analytic map of
4343:N-COVERS OF HYPERELLIPTIC CURVES
3813:
2876:
2508:is associated a unique morphism
2168:{\displaystyle \mathbb {Z} ^{r}}
1997:-torsion is still isomorphic to
1814:Structure of the group of points
1159:is a complex torus of dimension
1131:that carries the structure of a
1041:): what would now be called the
101:
23:
3199:Arithmetic of abelian varieties
3069:if there are positive integers
1923:-torsion part is isomorphic to
929:Abelian varieties defined over
4159:
4081:
4049:over the ring of Witt vectors"
4036:
4024:
3967:
3683:
3619:
3611:
3593:
3516:{\displaystyle R=\mathbb {Z} }
3510:
3496:
3465:
3433:
3145:
2963:
2957:
2865:of dual abelian varieties are
2798:
2784:
2750:
2713:{\displaystyle f\colon A\to B}
2704:
2590:
2524:
2026:
2004:
1952:
1930:
1897:
1878:
1672:
770:Infinite dimensional Lie group
1:
4290:American Mathematical Society
3960:
3940:Timeline of abelian varieties
3116:{\displaystyle nH_{1}=mH_{2}}
2427:{\displaystyle \{0\}\times T}
2388:{\displaystyle A\times \{t\}}
1541:, for any non-singular curve
1472:as a group. More accurately,
1201:is a finite-to-one morphism.
1110:
4106:{\displaystyle \mathbb {Z} }
3847:{\displaystyle \mathbb {Q} }
3004:together with a choice of a
2996:Over the complex numbers, a
2885:of an abelian variety is an
1840:{\displaystyle \mathbb {C} }
1775:{\displaystyle \mathbb {C} }
1464:structure, and the image of
1284:-dimensional torus given as
1195:for the group structure. An
885:projective algebraic variety
543:{\displaystyle \mathbb {Z} }
518:{\displaystyle \mathbb {Z} }
481:{\displaystyle \mathbb {Z} }
7:
4329:Encyclopedia of Mathematics
4227:Encyclopedia of Mathematics
3945:Moduli of abelian varieties
3928:
3722:. It can be extended to a
2932:. A polarisation induces a
2190:
1760:When the base is the field
1608:complex variables, having 2
268:List of group theory topics
10:
4401:
4187:
3696:is an abelian scheme over
3407:{\displaystyle x^{3}+Ax+B}
3333:
3197:of abelian varieties (see
2395:is a degree 0 line bundle,
2257:
1853:algebraically closed field
1626:
1553:is the fixed field of the
1163:that is also a projective
1033:, having four independent
987:
4241:; Chai, Ching-Li (1990),
4196:Complex Abelian Varieties
4045:"Group schemes of period
4043:Abrashkin, V. A. (1985).
3310:-torsion points, for all
2998:polarised abelian variety
2845:-torsion of its dual are
2674:{\displaystyle A^{\vee }}
2643:{\displaystyle A^{\vee }}
2493:{\displaystyle A^{\vee }}
2462:{\displaystyle A^{\vee }}
2340:{\displaystyle A\times T}
2295:{\displaystyle A^{\vee }}
2203:, and an abelian variety
1519:, which give rise to the
1376:{\displaystyle L\times L}
1169:Kodaira embedding theorem
4116:Inventiones Mathematicae
3549:is an open subscheme of
3281:finite flat group scheme
3279:-torsion points forms a
2777:between the double dual
2686:contravariant functorial
2442:Then there is a variety
2237:. An abelian variety is
1974:, i.e., the product of 2
1391:. Choosing a basis for
1260:it has been known since
922:embedded into a complex
386:Elementary abelian group
263:Glossary of group theory
4340:Bruin, N; Flynn, E.V.,
3955:Horrocks–Mumford bundle
3165:of the Riemann form on
1614:hyperelliptic integrals
1433:{\displaystyle g\geq 1}
931:algebraic number fields
895:that can be defined by
877:algebraic number theory
53:more precise citations.
4277:. Online course notes.
4107:
4015:
3903:
3876:
3848:
3804:
3776:
3748:
3716:
3690:
3571:
3543:
3517:
3472:
3408:
3366:
3304:
3269:
3247:For an abelian scheme
3209:of relative dimension
3155:
3154:{\displaystyle A\to B}
3117:
3059:
3032:
2978:
2922:
2902:principal polarisation
2831:
2811:
2767:
2714:
2675:
2644:
2613:
2541:
2494:
2463:
2428:
2389:
2341:
2296:
2264:To an abelian variety
2231:
2169:
2108:
2042:
1968:
1913:
1841:
1776:
1702:
1682:
1681:{\displaystyle J\to A}
1656:
1582:
1529:. The abelian variety
1497:
1460:carries a commutative
1434:
1407:Every algebraic curve
1377:
1312:
1254:
1253:{\displaystyle g>1}
1224:
984:History and motivation
802:Linear algebraic group
544:
519:
482:
4108:
4054:Dokl. Akad. Nauk SSSR
4016:
4014:{\displaystyle C^{g}}
3904:
3902:{\displaystyle p^{n}}
3877:
3849:
3805:
3777:
3749:
3717:
3691:
3572:
3544:
3518:
3473:
3409:
3367:
3305:
3303:{\displaystyle p^{n}}
3270:
3156:
3118:
3060:
3058:{\displaystyle H_{2}}
3033:
3031:{\displaystyle H_{1}}
2979:
2923:
2921:{\displaystyle >1}
2873:for elliptic curves.
2832:
2812:
2768:
2715:
2676:
2645:
2614:
2542:
2495:
2464:
2429:
2390:
2359:, the restriction of
2342:
2297:
2232:
2170:
2109:
2043:
1969:
1914:
1842:
1777:
1720:are commonly in use:
1703:
1683:
1657:
1583:
1581:{\displaystyle C^{g}}
1498:
1496:{\displaystyle C^{g}}
1435:
1378:
1313:
1311:{\displaystyle X=V/L}
1255:
1225:
1143:-dimensional complex
1031:two complex variables
1018:, what would happen?
988:Further information:
545:
520:
483:
4375:Geometry of divisors
4198:, Berlin, New York:
4095:
3998:
3886:
3858:
3836:
3786:
3758:
3730:
3700:
3581:
3553:
3527:
3482:
3418:
3376:
3342:
3287:
3251:
3189:-theoretically and
3139:
3081:
3042:
3015:
3011:. Two Riemann forms
2943:
2909:
2821:
2781:
2724:
2692:
2658:
2627:
2559:
2512:
2477:
2446:
2406:
2367:
2325:
2279:
2274:dual abelian variety
2260:Dual abelian variety
2254:Dual abelian variety
2215:
2150:
2137:Mordell-Weil theorem
2092:
2001:
1927:
1875:
1829:
1764:
1712:Algebraic definition
1692:
1666:
1646:
1602:meromorphic function
1565:
1511:-tuple of points in
1480:
1418:
1361:
1288:
1238:
1208:
532:
507:
470:
4129:1985InMat..81..515F
4089:Fontaine, Jean-Marc
3919:semiabelian variety
3913:Semiabelian variety
3330:-divisible groups.
3283:. The union of the
3268:{\displaystyle A/S}
3205:over a base scheme
2898:automorphism groups
2849:to each other when
2775:natural isomorphism
2555:along the morphism
2500:such that a family
2398:the restriction of
2314:is defined to be a
2272:, one associates a
2230:{\displaystyle m+n}
2107:{\displaystyle n=p}
1849:Lefschetz principle
1847:, and hence by the
1640:Matsusaka's theorem
1547:birational geometry
1223:{\displaystyle g=1}
1127:of real dimension 2
1092:Hamiltonian systems
1045:hyperelliptic curve
1012:quartic polynomials
968:and the variety is
176:Group homomorphisms
86:Algebraic structure
4380:Algebraic surfaces
4137:10.1007/BF01388584
4103:
4011:
3899:
3872:
3844:
3830:Jean-Marc Fontaine
3800:
3772:
3744:
3712:
3686:
3567:
3539:
3513:
3468:
3404:
3362:
3324:Serre–Tate theorem
3300:
3265:
3191:relative to a base
3151:
3113:
3055:
3028:
2974:
2918:
2827:
2807:
2763:
2710:
2671:
2640:
2609:
2537:
2490:
2459:
2424:
2385:
2337:
2292:
2227:
2165:
2145:free abelian group
2133:finitely generated
2104:
2038:
1964:
1909:
1837:
1808:Algebraic Geometry
1792:Riemann hypothesis
1772:
1698:
1678:
1652:
1634:Important theorems
1578:
1493:
1430:
1373:
1308:
1268:Riemann conditions
1250:
1220:
1100:Albanese varieties
1000:elliptic integrals
996:elliptic functions
962:Albanese varieties
954:Jacobian varieties
869:algebraic geometry
867:, particularly in
652:Special orthogonal
540:
515:
478:
359:Lagrange's theorem
4385:Niels Henrik Abel
4365:Abelian varieties
4324:"Abelian_variety"
4299:978-81-85931-86-9
4286:Abelian varieties
4266:Abelian Varieties
4209:978-0-387-54747-3
3316:p-divisible group
3236:and of dimension
2938:endomorphism ring
2934:Rosati involution
2830:{\displaystyle A}
1701:{\displaystyle J}
1655:{\displaystyle A}
1592:Abelian functions
1522:abelian integrals
1383:. Such a form on
1340:positive definite
1165:algebraic variety
1088:dynamical systems
978:Kodaira dimension
897:regular functions
861:
860:
436:
435:
318:Alternating group
275:
274:
79:
78:
71:
4392:
4370:Algebraic curves
4350:
4348:
4336:
4318:
4276:
4275:
4273:
4259:
4234:
4222:"Abelian scheme"
4212:
4181:
4180:
4178:
4171:
4163:
4157:
4156:
4112:
4110:
4109:
4104:
4102:
4085:
4079:
4078:
4061:(6): 1289–1294.
4040:
4034:
4028:
4022:
4020:
4018:
4017:
4012:
4010:
4009:
3990:
3988:
3986:
3980:
3971:
3908:
3906:
3905:
3900:
3898:
3897:
3881:
3879:
3878:
3873:
3871:
3853:
3851:
3850:
3845:
3843:
3827:
3819:Viktor Abrashkin
3809:
3807:
3806:
3801:
3799:
3781:
3779:
3778:
3773:
3771:
3753:
3751:
3750:
3745:
3743:
3721:
3719:
3718:
3713:
3695:
3693:
3692:
3687:
3682:
3681:
3666:
3665:
3647:
3646:
3631:
3630:
3618:
3576:
3574:
3573:
3568:
3566:
3548:
3546:
3545:
3540:
3522:
3520:
3519:
3514:
3506:
3495:
3478:is nonzero. Let
3477:
3475:
3474:
3469:
3464:
3463:
3448:
3447:
3413:
3411:
3410:
3405:
3388:
3387:
3371:
3369:
3368:
3363:
3361:
3309:
3307:
3306:
3301:
3299:
3298:
3274:
3272:
3271:
3266:
3261:
3230:geometric fibers
3160:
3158:
3157:
3152:
3122:
3120:
3119:
3114:
3112:
3111:
3096:
3095:
3064:
3062:
3061:
3056:
3054:
3053:
3037:
3035:
3034:
3029:
3027:
3026:
2983:
2981:
2980:
2975:
2973:
2956:
2930:Schottky problem
2927:
2925:
2924:
2919:
2836:
2834:
2833:
2828:
2816:
2814:
2813:
2808:
2806:
2805:
2796:
2795:
2772:
2770:
2769:
2764:
2762:
2761:
2749:
2748:
2736:
2735:
2719:
2717:
2716:
2711:
2680:
2678:
2677:
2672:
2670:
2669:
2649:
2647:
2646:
2641:
2639:
2638:
2618:
2616:
2615:
2610:
2608:
2607:
2571:
2570:
2546:
2544:
2543:
2538:
2536:
2535:
2499:
2497:
2496:
2491:
2489:
2488:
2468:
2466:
2465:
2460:
2458:
2457:
2433:
2431:
2430:
2425:
2394:
2392:
2391:
2386:
2346:
2344:
2343:
2338:
2301:
2299:
2298:
2293:
2291:
2290:
2236:
2234:
2233:
2228:
2174:
2172:
2171:
2166:
2164:
2163:
2158:
2122:-rational points
2113:
2111:
2110:
2105:
2047:
2045:
2044:
2039:
2037:
2036:
2024:
2016:
2011:
1973:
1971:
1970:
1965:
1963:
1962:
1950:
1942:
1937:
1918:
1916:
1915:
1910:
1908:
1907:
1895:
1890:
1885:
1846:
1844:
1843:
1838:
1836:
1804:abstract variety
1781:
1779:
1778:
1773:
1771:
1707:
1705:
1704:
1699:
1687:
1685:
1684:
1679:
1661:
1659:
1658:
1653:
1629:abelian integral
1598:abelian function
1587:
1585:
1584:
1579:
1577:
1576:
1535:Jacobian variety
1502:
1500:
1499:
1494:
1492:
1491:
1439:
1437:
1436:
1431:
1382:
1380:
1379:
1374:
1330:is a lattice in
1317:
1315:
1314:
1309:
1304:
1259:
1257:
1256:
1251:
1229:
1227:
1226:
1221:
1193:identity element
1133:complex manifold
1096:Picard varieties
958:Picard varieties
924:projective space
887:that is also an
873:complex analysis
853:
846:
839:
795:Algebraic groups
568:Hyperbolic group
558:Arithmetic group
549:
547:
546:
541:
539:
524:
522:
521:
516:
514:
487:
485:
484:
479:
477:
400:Schur multiplier
354:Cauchy's theorem
342:Quaternion group
290:
289:
116:
115:
105:
92:
81:
80:
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
4400:
4399:
4395:
4394:
4393:
4391:
4390:
4389:
4355:
4354:
4346:
4300:
4271:
4269:
4257:
4247:Springer Verlag
4218:Dolgachev, I.V.
4210:
4200:Springer-Verlag
4190:
4185:
4184:
4176:
4169:
4165:
4164:
4160:
4098:
4096:
4093:
4092:
4086:
4082:
4041:
4037:
4029:
4025:
4005:
4001:
3999:
3996:
3995:
3984:
3982:
3978:
3972:
3968:
3963:
3931:
3915:
3893:
3889:
3887:
3884:
3883:
3867:
3859:
3856:
3855:
3839:
3837:
3834:
3833:
3821:
3816:
3795:
3787:
3784:
3783:
3767:
3759:
3756:
3755:
3739:
3731:
3728:
3727:
3701:
3698:
3697:
3677:
3673:
3661:
3657:
3642:
3638:
3626:
3622:
3614:
3582:
3579:
3578:
3562:
3554:
3551:
3550:
3528:
3525:
3524:
3502:
3491:
3483:
3480:
3479:
3459:
3455:
3443:
3439:
3419:
3416:
3415:
3383:
3379:
3377:
3374:
3373:
3357:
3343:
3340:
3339:
3336:
3294:
3290:
3288:
3285:
3284:
3275:, the group of
3257:
3252:
3249:
3248:
3183:
3140:
3137:
3136:
3107:
3103:
3091:
3087:
3082:
3079:
3078:
3049:
3045:
3043:
3040:
3039:
3022:
3018:
3016:
3013:
3012:
2994:
2969:
2946:
2944:
2941:
2940:
2910:
2907:
2906:
2879:
2822:
2819:
2818:
2801:
2797:
2791:
2787:
2782:
2779:
2778:
2757:
2753:
2744:
2740:
2731:
2727:
2725:
2722:
2721:
2720:dual morphisms
2693:
2690:
2689:
2665:
2661:
2659:
2656:
2655:
2634:
2630:
2628:
2625:
2624:
2603:
2599:
2566:
2562:
2560:
2557:
2556:
2531:
2527:
2513:
2510:
2509:
2484:
2480:
2478:
2475:
2474:
2453:
2449:
2447:
2444:
2443:
2407:
2404:
2403:
2368:
2365:
2364:
2326:
2323:
2322:
2286:
2282:
2280:
2277:
2276:
2262:
2256:
2251:
2216:
2213:
2212:
2193:
2159:
2154:
2153:
2151:
2148:
2147:
2093:
2090:
2089:
2084:(the so-called
2029:
2025:
2020:
2012:
2007:
2002:
1999:
1998:
1955:
1951:
1946:
1938:
1933:
1928:
1925:
1924:
1900:
1896:
1891:
1886:
1881:
1876:
1873:
1872:
1832:
1830:
1827:
1826:
1816:
1784:elliptic curves
1767:
1765:
1762:
1761:
1750:algebraic group
1733:algebraic group
1714:
1693:
1690:
1689:
1667:
1664:
1663:
1647:
1644:
1643:
1636:
1631:
1594:
1572:
1568:
1566:
1563:
1562:
1555:symmetric group
1503:: any point in
1487:
1483:
1481:
1478:
1477:
1419:
1416:
1415:
1405:
1362:
1359:
1358:
1300:
1289:
1286:
1285:
1270:
1239:
1236:
1235:
1209:
1206:
1205:
1113:
1108:
1106:Analytic theory
1039:abelian surface
1021:In the work of
992:
986:
947:Dedekind domain
920:holomorphically
912:complex numbers
889:algebraic group
881:abelian variety
857:
828:
827:
816:Abelian variety
809:Reductive group
797:
787:
786:
785:
784:
735:
727:
719:
711:
703:
676:Special unitary
587:
573:
572:
554:
553:
535:
533:
530:
529:
510:
508:
505:
504:
473:
471:
468:
467:
459:
458:
449:Discrete groups
438:
437:
393:Frobenius group
338:
325:
314:
307:Symmetric group
303:
287:
277:
276:
127:Normal subgroup
113:
93:
84:
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
4398:
4388:
4387:
4382:
4377:
4372:
4367:
4353:
4352:
4337:
4319:
4298:
4282:Mumford, David
4278:
4263:Milne, James,
4260:
4255:
4239:Faltings, Gerd
4235:
4214:
4208:
4189:
4186:
4183:
4182:
4158:
4123:(3): 515–538.
4101:
4080:
4035:
4023:
4008:
4004:
3994:is covered by
3965:
3964:
3962:
3959:
3958:
3957:
3952:
3947:
3942:
3937:
3930:
3927:
3914:
3911:
3896:
3892:
3870:
3866:
3863:
3842:
3815:
3812:
3798:
3794:
3791:
3770:
3766:
3763:
3742:
3738:
3735:
3711:
3708:
3705:
3685:
3680:
3676:
3672:
3669:
3664:
3660:
3656:
3653:
3650:
3645:
3641:
3637:
3634:
3629:
3625:
3621:
3617:
3613:
3610:
3607:
3604:
3601:
3598:
3595:
3592:
3589:
3586:
3565:
3561:
3558:
3538:
3535:
3532:
3512:
3509:
3505:
3501:
3498:
3494:
3490:
3487:
3467:
3462:
3458:
3454:
3451:
3446:
3442:
3438:
3435:
3432:
3429:
3426:
3423:
3403:
3400:
3397:
3394:
3391:
3386:
3382:
3360:
3356:
3353:
3350:
3347:
3335:
3332:
3297:
3293:
3264:
3260:
3256:
3203:abelian scheme
3182:
3181:Abelian scheme
3179:
3150:
3147:
3144:
3110:
3106:
3102:
3099:
3094:
3090:
3086:
3052:
3048:
3025:
3021:
2993:
2990:
2972:
2968:
2965:
2962:
2959:
2955:
2952:
2949:
2917:
2914:
2893:double-duality
2878:
2875:
2826:
2804:
2800:
2794:
2790:
2786:
2760:
2756:
2752:
2747:
2743:
2739:
2734:
2730:
2709:
2706:
2703:
2700:
2697:
2668:
2664:
2637:
2633:
2606:
2602:
2598:
2595:
2592:
2589:
2586:
2583:
2580:
2577:
2574:
2569:
2565:
2534:
2530:
2526:
2523:
2520:
2517:
2487:
2483:
2456:
2452:
2440:
2439:
2423:
2420:
2417:
2414:
2411:
2396:
2384:
2381:
2378:
2375:
2372:
2336:
2333:
2330:
2304:moduli problem
2289:
2285:
2258:Main article:
2255:
2252:
2250:
2247:
2226:
2223:
2220:
2192:
2189:
2162:
2157:
2103:
2100:
2097:
2035:
2032:
2028:
2023:
2019:
2015:
2010:
2006:
1978:copies of the
1961:
1958:
1954:
1949:
1945:
1941:
1936:
1932:
1906:
1903:
1899:
1894:
1889:
1884:
1880:
1857:characteristic
1835:
1825:For the field
1815:
1812:
1770:
1758:
1757:
1739:
1713:
1710:
1697:
1677:
1674:
1671:
1651:
1635:
1632:
1593:
1590:
1575:
1571:
1551:function field
1533:is called the
1490:
1486:
1476:is covered by
1456:. As a torus,
1429:
1426:
1423:
1404:
1401:
1372:
1369:
1366:
1351:imaginary part
1343:hermitian form
1307:
1303:
1299:
1296:
1293:
1269:
1266:
1249:
1246:
1243:
1232:elliptic curve
1219:
1216:
1213:
1173:Chow's theorem
1112:
1109:
1107:
1104:
1043:Jacobian of a
985:
982:
974:elliptic curve
891:, i.e., has a
859:
858:
856:
855:
848:
841:
833:
830:
829:
826:
825:
823:Elliptic curve
819:
818:
812:
811:
805:
804:
798:
793:
792:
789:
788:
783:
782:
779:
776:
772:
768:
767:
766:
761:
759:Diffeomorphism
755:
754:
749:
744:
738:
737:
733:
729:
725:
721:
717:
713:
709:
705:
701:
696:
695:
684:
683:
672:
671:
660:
659:
648:
647:
636:
635:
624:
623:
616:Special linear
612:
611:
604:General linear
600:
599:
594:
588:
579:
578:
575:
574:
571:
570:
565:
560:
552:
551:
538:
526:
513:
500:
498:Modular groups
496:
495:
494:
489:
476:
460:
457:
456:
451:
445:
444:
443:
440:
439:
434:
433:
432:
431:
426:
421:
418:
412:
411:
405:
404:
403:
402:
396:
395:
389:
388:
383:
374:
373:
371:Hall's theorem
368:
366:Sylow theorems
362:
361:
356:
348:
347:
346:
345:
339:
334:
331:Dihedral group
327:
326:
321:
315:
310:
304:
299:
288:
283:
282:
279:
278:
273:
272:
271:
270:
265:
257:
256:
255:
254:
249:
244:
239:
234:
229:
224:
222:multiplicative
219:
214:
209:
204:
196:
195:
194:
193:
188:
180:
179:
171:
170:
169:
168:
166:Wreath product
163:
158:
153:
151:direct product
145:
143:Quotient group
137:
136:
135:
134:
129:
124:
114:
111:
110:
107:
106:
98:
97:
77:
76:
31:
29:
22:
15:
9:
6:
4:
3:
2:
4397:
4386:
4383:
4381:
4378:
4376:
4373:
4371:
4368:
4366:
4363:
4362:
4360:
4345:
4344:
4338:
4335:
4331:
4330:
4325:
4320:
4317:
4313:
4309:
4305:
4301:
4295:
4291:
4287:
4283:
4279:
4268:
4267:
4261:
4258:
4256:3-540-52015-5
4252:
4248:
4244:
4240:
4236:
4233:
4229:
4228:
4223:
4219:
4215:
4211:
4205:
4201:
4197:
4192:
4191:
4175:
4168:
4162:
4154:
4150:
4146:
4142:
4138:
4134:
4130:
4126:
4122:
4118:
4117:
4090:
4084:
4076:
4072:
4068:
4064:
4060:
4056:
4055:
4050:
4048:
4039:
4032:
4027:
4006:
4002:
3993:
3977:
3970:
3966:
3956:
3953:
3951:
3948:
3946:
3943:
3941:
3938:
3936:
3933:
3932:
3926:
3924:
3920:
3910:
3894:
3890:
3864:
3861:
3831:
3825:
3820:
3814:Non-existence
3811:
3792:
3789:
3764:
3761:
3736:
3733:
3725:
3709:
3706:
3703:
3678:
3674:
3670:
3667:
3662:
3658:
3654:
3651:
3648:
3643:
3639:
3635:
3632:
3627:
3623:
3615:
3608:
3605:
3602:
3599:
3596:
3590:
3587:
3584:
3559:
3556:
3536:
3533:
3530:
3503:
3499:
3488:
3485:
3460:
3456:
3452:
3449:
3444:
3440:
3436:
3430:
3427:
3424:
3401:
3398:
3395:
3392:
3389:
3384:
3380:
3372:be such that
3354:
3351:
3348:
3345:
3331:
3329:
3325:
3321:
3317:
3313:
3295:
3291:
3282:
3278:
3262:
3258:
3254:
3245:
3243:
3239:
3235:
3231:
3227:
3223:
3220:
3216:
3212:
3208:
3204:
3200:
3196:
3192:
3188:
3178:
3176:
3172:
3168:
3164:
3148:
3142:
3134:
3130:
3126:
3108:
3104:
3100:
3097:
3092:
3088:
3084:
3076:
3072:
3068:
3050:
3046:
3023:
3019:
3010:
3007:
3003:
2999:
2989:
2987:
2966:
2960:
2939:
2935:
2931:
2915:
2912:
2903:
2899:
2894:
2890:
2889:
2884:
2877:Polarisations
2874:
2872:
2868:
2867:Cartier duals
2864:
2863:group schemes
2860:
2856:
2852:
2848:
2844:
2840:
2824:
2802:
2792:
2788:
2776:
2758:
2754:
2745:
2741:
2737:
2732:
2728:
2707:
2701:
2698:
2695:
2687:
2682:
2666:
2662:
2653:
2635:
2631:
2622:
2604:
2600:
2596:
2593:
2587:
2584:
2581:
2578:
2575:
2572:
2567:
2563:
2554:
2550:
2532:
2528:
2521:
2518:
2515:
2507:
2503:
2485:
2481:
2472:
2454:
2450:
2437:
2421:
2418:
2412:
2401:
2397:
2379:
2373:
2370:
2362:
2358:
2354:
2350:
2349:
2348:
2334:
2331:
2328:
2320:
2317:
2313:
2309:
2305:
2287:
2283:
2275:
2271:
2268:over a field
2267:
2261:
2246:
2244:
2241:if it is not
2240:
2224:
2221:
2218:
2210:
2207:of dimension
2206:
2202:
2199:of dimension
2198:
2188:
2186:
2182:
2178:
2160:
2146:
2142:
2138:
2134:
2130:
2127:
2123:
2121:
2117:The group of
2115:
2101:
2098:
2095:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2055:
2051:
2033:
2030:
2017:
2013:
1996:
1992:
1987:
1985:
1981:
1977:
1959:
1956:
1943:
1939:
1922:
1919:. Hence, its
1904:
1901:
1887:
1870:
1866:
1862:
1861:torsion group
1858:
1854:
1850:
1823:
1821:
1811:
1809:
1805:
1801:
1800:finite fields
1797:
1793:
1787:
1785:
1755:
1751:
1748:
1744:
1740:
1738:
1734:
1731:
1727:
1723:
1722:
1721:
1719:
1709:
1695:
1675:
1669:
1649:
1641:
1630:
1625:
1623:
1619:
1615:
1611:
1607:
1603:
1599:
1589:
1573:
1569:
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1532:
1528:
1524:
1523:
1518:
1514:
1510:
1507:comes from a
1506:
1488:
1484:
1475:
1471:
1467:
1463:
1459:
1455:
1451:
1447:
1444:of dimension
1443:
1427:
1424:
1421:
1414:
1410:
1400:
1398:
1394:
1390:
1386:
1370:
1367:
1364:
1356:
1352:
1348:
1344:
1341:
1337:
1333:
1329:
1325:
1321:
1305:
1301:
1297:
1294:
1291:
1283:
1279:
1275:
1274:complex torus
1265:
1263:
1247:
1244:
1241:
1233:
1217:
1214:
1211:
1202:
1200:
1199:
1194:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1162:
1158:
1154:
1150:
1146:
1142:
1138:
1134:
1130:
1126:
1122:
1119:of dimension
1118:
1117:complex torus
1103:
1101:
1097:
1093:
1089:
1084:
1082:
1078:
1073:
1071:
1067:
1063:
1059:
1055:
1050:
1048:
1046:
1040:
1036:
1032:
1028:
1024:
1019:
1017:
1013:
1009:
1005:
1001:
997:
991:
981:
979:
975:
971:
967:
963:
959:
955:
950:
948:
944:
940:
939:finite fields
936:
932:
927:
925:
921:
917:
913:
909:
905:
900:
898:
894:
890:
886:
882:
878:
874:
870:
866:
854:
849:
847:
842:
840:
835:
834:
832:
831:
824:
821:
820:
817:
814:
813:
810:
807:
806:
803:
800:
799:
796:
791:
790:
780:
777:
774:
773:
771:
765:
762:
760:
757:
756:
753:
750:
748:
745:
743:
740:
739:
736:
730:
728:
722:
720:
714:
712:
706:
704:
698:
697:
693:
689:
686:
685:
681:
677:
674:
673:
669:
665:
662:
661:
657:
653:
650:
649:
645:
641:
638:
637:
633:
629:
626:
625:
621:
617:
614:
613:
609:
605:
602:
601:
598:
595:
593:
590:
589:
586:
582:
577:
576:
569:
566:
564:
561:
559:
556:
555:
527:
502:
501:
499:
493:
490:
465:
462:
461:
455:
452:
450:
447:
446:
442:
441:
430:
427:
425:
422:
419:
416:
415:
414:
413:
410:
407:
406:
401:
398:
397:
394:
391:
390:
387:
384:
382:
380:
376:
375:
372:
369:
367:
364:
363:
360:
357:
355:
352:
351:
350:
349:
343:
340:
337:
332:
329:
328:
324:
319:
316:
313:
308:
305:
302:
297:
294:
293:
292:
291:
286:
285:Finite groups
281:
280:
269:
266:
264:
261:
260:
259:
258:
253:
250:
248:
245:
243:
240:
238:
235:
233:
230:
228:
225:
223:
220:
218:
215:
213:
210:
208:
205:
203:
200:
199:
198:
197:
192:
189:
187:
184:
183:
182:
181:
178:
177:
173:
172:
167:
164:
162:
159:
157:
154:
152:
149:
146:
144:
141:
140:
139:
138:
133:
130:
128:
125:
123:
120:
119:
118:
117:
112:Basic notions
109:
108:
104:
100:
99:
96:
91:
87:
83:
82:
73:
70:
62:
59:February 2013
52:
48:
42:
41:
35:
30:
21:
20:
4342:
4327:
4285:
4270:, retrieved
4265:
4242:
4225:
4195:
4161:
4120:
4114:
4083:
4058:
4052:
4046:
4038:
4026:
3991:
3983:. Retrieved
3969:
3918:
3916:
3817:
3337:
3327:
3320:Deformations
3311:
3276:
3246:
3241:
3237:
3225:
3222:group scheme
3210:
3206:
3202:
3194:
3184:
3174:
3170:
3166:
3132:
3129:polarisation
3128:
3127:is called a
3124:
3074:
3070:
3008:
3006:Riemann form
3001:
2997:
2995:
2985:
2901:
2892:
2886:
2883:polarisation
2882:
2880:
2871:Weil pairing
2858:
2857:— the
2854:
2850:
2842:
2838:
2683:
2651:
2620:
2552:
2548:
2505:
2501:
2470:
2441:
2435:
2399:
2360:
2356:
2352:
2318:
2311:
2307:
2273:
2269:
2265:
2263:
2238:
2208:
2204:
2200:
2196:
2194:
2184:
2180:
2176:
2128:
2126:global field
2119:
2116:
2085:
2081:
2077:
2073:
2069:
2065:
2061:
2053:
2049:
1994:
1990:
1988:
1983:
1980:cyclic group
1975:
1920:
1864:
1824:
1817:
1788:
1759:
1753:
1736:
1717:
1715:
1639:
1637:
1624:an isogeny.
1617:
1609:
1605:
1597:
1595:
1558:
1542:
1538:
1534:
1530:
1526:
1520:
1516:
1512:
1508:
1504:
1473:
1469:
1465:
1457:
1453:
1449:
1445:
1441:
1408:
1406:
1396:
1392:
1389:Riemann form
1384:
1346:
1335:
1331:
1327:
1323:
1319:
1281:
1277:
1271:
1203:
1196:
1180:
1176:
1160:
1156:
1152:
1145:vector space
1140:
1128:
1120:
1114:
1085:
1074:
1051:
1042:
1038:
1034:
1020:
1004:square roots
993:
970:non-singular
951:
943:local fields
941:and various
935:Localization
928:
918:that can be
916:complex tori
907:
901:
880:
862:
815:
691:
679:
667:
655:
643:
631:
619:
607:
378:
335:
322:
311:
300:
296:Cyclic group
174:
161:Free product
132:Group action
95:Group theory
90:Group theory
89:
65:
56:
37:
4031:Milne, J.S.
3822: [
3724:Néron model
3065:are called
2316:line bundle
2179:called the
2088:-rank when
1820:commutative
1058:Weierstrass
1027:Carl Jacobi
966:commutative
865:mathematics
581:Topological
420:alternating
51:introducing
4359:Categories
4153:0612.14043
4075:0593.14029
3985:14 January
3974:Bruin, N.
3961:References
3314:, forms a
3077:such that
3067:equivalent
2347:such that
1869:isomorphic
1859:zero, the
1851:for every
1810:article).
1747:projective
1627:See also:
1468:generates
1357:values on
1111:Definition
1081:André Weil
1047:of genus 2
1023:Niels Abel
688:Symplectic
628:Orthogonal
585:Lie groups
492:Free group
217:continuous
156:Direct sum
34:references
4334:EMS Press
4284:(2008) ,
4272:6 October
4232:EMS Press
4220:(2001) ,
3865:
3793:
3765:
3737:
3707:
3668:−
3649:−
3636:−
3588:
3560:
3534:
3508:Δ
3428:−
3422:Δ
3355:∈
3234:connected
3146:→
2967:⊗
2861:-torsion
2803:∨
2793:∨
2759:∨
2751:→
2746:∨
2738::
2733:∨
2705:→
2699::
2667:∨
2636:∨
2605:∨
2597:×
2591:→
2585:×
2579::
2573:×
2533:∨
2525:→
2519::
2486:∨
2455:∨
2419:×
2374:×
2332:×
2310:-variety
2288:∨
2243:isogenous
1982:of order
1743:connected
1726:connected
1673:→
1425:≥
1368:×
1151:of rank 2
1077:Lefschetz
1062:Frobenius
893:group law
752:Conformal
640:Euclidean
247:nilpotent
4174:Archived
3929:See also
3577:. Then
3163:pullback
2547:so that
2351:for all
2191:Products
1730:complete
1355:integral
1189:morphism
1137:quotient
1066:Poincaré
1016:quintics
747:Poincaré
592:Solenoid
464:Integers
454:Lattices
429:sporadic
424:Lie type
252:solvable
242:dihedral
227:additive
212:infinite
122:Subgroup
4308:0282985
4188:Sources
4145:0807070
4125:Bibcode
4067:0802862
3935:Motives
3334:Example
2936:on the
2888:isogeny
2135:by the
2060:. When
2058:coprime
1334:. Then
1262:Riemann
1198:isogeny
1149:lattice
1054:Riemann
1035:periods
742:Lorentz
664:Unitary
563:Lattice
503:PSL(2,
237:abelian
148:(Semi-)
47:improve
4316:138290
4314:
4306:
4296:
4253:
4206:
4151:
4143:
4073:
4065:
3228:whose
3219:smooth
3215:proper
3187:scheme
2239:simple
2124:for a
1993:, the
1796:curves
1688:where
1549:, its
1353:takes
1349:whose
1318:where
1070:Picard
1068:, and
960:) and
597:Circle
528:SL(2,
417:cyclic
381:-group
232:cyclic
207:finite
202:simple
186:kernel
36:, but
4347:(PDF)
4177:(PDF)
4170:(PDF)
3979:(PDF)
3923:torus
3826:]
3726:over
3523:, so
3224:over
3213:is a
2048:when
1798:over
1752:over
1735:over
1622:up to
1600:is a
1462:group
1452:into
1413:genus
1280:be a
1185:group
1147:by a
1139:of a
1125:torus
1123:is a
1008:cubic
972:. An
904:field
883:is a
879:, an
781:Sp(∞)
778:SU(∞)
191:image
4312:OCLC
4294:ISBN
4274:2016
4251:ISBN
4204:ISBN
3987:2015
3862:Spec
3828:and
3790:Spec
3762:Spec
3734:Spec
3704:Spec
3585:Proj
3557:Spec
3531:Spec
3338:Let
3232:are
3073:and
3038:and
2913:>
2900:. A
2847:dual
2817:and
2181:rank
2064:and
2056:are
2052:and
1794:for
1745:and
1728:and
1395:and
1326:and
1245:>
1187:. A
1171:and
1098:and
1025:and
1010:and
908:over
875:and
775:O(∞)
764:Loop
583:and
4149:Zbl
4133:doi
4113:".
4071:Zbl
4059:283
3169:to
3131:of
2984:of
2504:on
2402:to
2363:to
2355:in
2321:on
2131:is
2114:).
1871:to
1867:is
1855:of
1596:An
1557:on
1537:of
1411:of
1345:on
1102:).
1006:of
980:0.
926:.
863:In
690:Sp(
678:SU(
654:SO(
618:SL(
606:GL(
4361::
4332:,
4326:,
4310:,
4304:MR
4302:,
4292:,
4249:,
4245:,
4230:,
4224:,
4202:,
4172:.
4147:.
4141:MR
4139:.
4131:.
4121:81
4119:.
4069:.
4063:MR
4057:.
4051:.
3925:.
3917:A
3824:ru
3810:.
3453:27
3431:16
3318:.
3244:.
3217:,
3177:.
2988:.
2881:A
2438:).
2187:.
1986:.
1822:.
1741:a
1724:a
1588:.
1115:A
1064:,
1060:,
1056:,
1049:.
871:,
666:U(
642:E(
630:O(
88:→
4155:.
4135::
4127::
4100:Z
4077:.
4047:p
4021::
4007:g
4003:C
3992:J
3989:.
3895:n
3891:p
3869:Z
3841:Q
3797:Z
3769:Z
3741:Z
3710:R
3684:)
3679:3
3675:z
3671:B
3663:2
3659:z
3655:x
3652:A
3644:3
3640:x
3633:z
3628:2
3624:y
3620:(
3616:/
3612:]
3609:z
3606:,
3603:y
3600:,
3597:x
3594:[
3591:R
3564:Z
3537:R
3511:]
3504:/
3500:1
3497:[
3493:Z
3489:=
3486:R
3466:)
3461:2
3457:B
3450:+
3445:3
3441:A
3437:4
3434:(
3425:=
3402:B
3399:+
3396:x
3393:A
3390:+
3385:3
3381:x
3359:Z
3352:B
3349:,
3346:A
3328:p
3312:n
3296:n
3292:p
3277:n
3263:S
3259:/
3255:A
3242:S
3238:g
3226:S
3211:g
3207:S
3195:p
3175:A
3171:A
3167:B
3149:B
3143:A
3133:A
3125:A
3109:2
3105:H
3101:m
3098:=
3093:1
3089:H
3085:n
3075:m
3071:n
3051:2
3047:H
3024:1
3020:H
3009:H
3002:A
2986:A
2971:Q
2964:)
2961:A
2958:(
2954:d
2951:n
2948:E
2916:1
2859:n
2855:n
2851:n
2843:n
2839:n
2825:A
2799:)
2789:A
2785:(
2755:A
2742:B
2729:f
2708:B
2702:A
2696:f
2663:A
2652:A
2632:A
2621:T
2601:A
2594:A
2588:T
2582:A
2576:f
2568:A
2564:1
2553:P
2549:L
2529:A
2522:T
2516:f
2506:T
2502:L
2482:A
2471:P
2451:A
2436:A
2422:T
2416:}
2413:0
2410:{
2400:L
2383:}
2380:t
2377:{
2371:A
2361:L
2357:T
2353:t
2335:T
2329:A
2319:L
2312:T
2308:k
2284:A
2270:k
2266:A
2225:n
2222:+
2219:m
2209:n
2205:B
2201:m
2197:A
2185:k
2177:r
2161:r
2156:Z
2129:k
2120:k
2102:p
2099:=
2096:n
2086:p
2082:p
2078:n
2074:g
2070:n
2066:p
2062:n
2054:p
2050:n
2034:g
2031:2
2027:)
2022:Z
2018:n
2014:/
2009:Z
2005:(
1995:n
1991:p
1984:n
1976:g
1960:g
1957:2
1953:)
1948:Z
1944:n
1940:/
1935:Z
1931:(
1921:n
1905:g
1902:2
1898:)
1893:Z
1888:/
1883:Q
1879:(
1865:g
1834:C
1769:C
1756:.
1754:k
1737:k
1718:k
1696:J
1676:A
1670:J
1650:A
1618:J
1610:n
1606:n
1574:g
1570:C
1559:g
1543:C
1539:C
1531:J
1527:J
1517:C
1513:C
1509:g
1505:J
1489:g
1485:C
1474:J
1470:J
1466:C
1458:J
1454:J
1450:C
1446:g
1442:J
1428:1
1422:g
1409:C
1397:L
1393:V
1385:X
1371:L
1365:L
1347:V
1336:X
1332:V
1328:L
1324:g
1320:V
1306:L
1302:/
1298:V
1295:=
1292:X
1282:g
1278:X
1248:1
1242:g
1218:1
1215:=
1212:g
1181:g
1177:g
1161:g
1157:g
1153:g
1141:g
1129:g
1121:g
852:e
845:t
838:v
734:8
732:E
726:7
724:E
718:6
716:E
710:4
708:F
702:2
700:G
694:)
692:n
682:)
680:n
670:)
668:n
658:)
656:n
646:)
644:n
634:)
632:n
622:)
620:n
610:)
608:n
550:)
537:Z
525:)
512:Z
488:)
475:Z
466:(
379:p
344:Q
336:n
333:D
323:n
320:A
312:n
309:S
301:n
298:Z
72:)
66:(
61:)
57:(
43:.
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