36:
960:, 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.
114:
1800:
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
2915:
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
3834:
2906:
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
3705:
2628:
2782:
2993:
3891:
3819:
3791:
3763:
3586:
2057:
1983:
1928:
910:. 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.
3920:-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.
3487:
3731:
3558:
2556:
3381:
2826:
2184:
3532:
2729:
3132:
2443:
2404:
4122:
3863:
1856:
1791:
559:
534:
497:
3423:
2690:
2659:
2509:
2478:
2356:
2311:
1392:
1449:
3170:
1697:
1269:
4030:
3918:
3319:
3251:. 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
3074:
3047:
2937:
1597:
1512:
1327:
1623:
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
3284:
2246:
2123:
1239:
861:
2846:
1717:
1671:
4064:
1090:
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
3146:; over the complex number this is equivalent to the definition of polarisation given above. A morphism of polarised abelian varieties is a morphism
1183:
1410:, one can make this condition more explicit. There are several equivalent formulations of this; all of them are known as the Riemann conditions.
1817:
and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings (see also the history section in the
1561:
1086:
By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions. Eventually, in the 1920s,
980:
419:
4308:
4218:
3960:
3591:
369:
57:
2569:
1000:
854:
364:
4265:
79:
3173:
2256:
to a product of abelian varieties of lower dimension. Any abelian variety is isogenous to a product of simple abelian varieties.
2151:
4184:
3209:
1350:
1147:
1802:
780:
2734:
2953:
4385:
4344:
4300:
4242:
3950:
847:
3965:
3868:
3796:
3768:
3740:
3563:
2011:
1937:
4390:
4339:
4237:
4041:
3955:
3945:
1885:
1215:
When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case
464:
278:
50:
44:
4395:
4375:
1863:
4380:
3428:
1536:
with which the theory started, can be derived from the simpler, translation-invariant theory of differentials on
1179:
1072:
61:
4126:
3291:
2143:
2091:-torsion, one considers only the geometric points, one obtains a new invariant for varieties in characteristic
1867:
1037:
662:
396:
273:
161:
3710:
3537:
2522:
1194:
that admits a positive line bundle. Since they are complex tori, abelian varieties carry the structure of a
1080:
941:
887:
3334:
1013:, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the
1624:
945:
812:
602:
3352:
2791:
2696:
1041:
921:
that field. Historically the first abelian varieties to be studied were those defined over the field of
686:
1063:
After Abel and Jacobi, some of the most important contributors to the theory of abelian functions were
3829:
2160:
1673:
is the quotient of the
Jacobian of some curve; that is, there is some surjection of abelian varieties
4334:
4135:
3197:
3077:
2270:
2147:
1612:
626:
614:
232:
166:
4232:
3492:
2702:
3091:
2785:
2416:
2377:
1859:
1829:
By the definitions, an abelian variety is a group variety. Its group of points can be proven to be
1557:
1472:
1423:
1195:
1055:
914:
201:
96:
4224:. A comprehensive treatment of the complex theory, with an overview of the history of the subject.
4105:
3846:
1839:
1774:
542:
517:
480:
4099:
3840:
3330:
3201:
2908:
2857:
2155:
1818:
1757:
1102:
1026:
1022:
948:
techniques lead naturally from abelian varieties defined over number fields to ones defined over
879:
186:
158:
4299:, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.:
3386:
2079:
are not coprime, the same result can be recovered provided one interprets it as saying that the
1094:
in the 1940s who gave the subject its modern foundations in the language of algebraic geometry.
2668:
2637:
2487:
2456:
2335:
2289:
1371:
4322:
4304:
4261:
4214:
4044:, Jacobian varieties, in Arithmetic Geometry, eds Cornell and Silverman, Springer-Verlag, 1986
3326:
2948:
2944:
1175:
1087:
1076:
1010:
1006:
988:
895:
757:
591:
434:
328:
1428:
4159:
4143:
4081:
3865:
with good reduction at all primes. Equivalently, there are no nonzero abelian schemes over
3149:
2940:
1814:
1740:
1676:
1639:
1532:
1287:
is an abelian variety, i.e., whether or not it can be embedded into a projective space. Let
1272:
1248:
1203:
1202:
of abelian varieties is a morphism of the underlying algebraic varieties that preserves the
1143:
1110:
1098:
1068:
1064:
1048:(i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an
1018:
972:
964:
934:
907:
883:
742:
734:
726:
718:
710:
698:
638:
578:
568:
410:
352:
227:
196:
4318:
4155:
4077:
4008:
3896:
3297:
3052:
3025:
2919:
1575:
1490:
1298:
4314:
4257:
4210:
4163:
4151:
4085:
4073:
3933:
3244:
3240:
3229:
3225:
1806:
1760:
1753:
1743:
1736:
1565:
1159:
957:
922:
899:
819:
805:
762:
650:
573:
403:
317:
257:
137:
3261:
2225:
2102:
1793:
of complex numbers, these notions coincide with the previous definition. Over all bases,
1218:
4139:
3734:
2831:
2314:
2129:
1794:
1702:
1656:
1361:
1353:
1242:
1106:
984:
968:
833:
769:
459:
439:
376:
341:
262:
252:
237:
222:
176:
153:
4369:
4292:
4249:
4177:
1871:
1830:
1627:
that may be expressed in terms of elliptic integrals. This comes down to asking that
1284:
1127:
1091:
926:
752:
674:
508:
381:
247:
4352:
3986:
4228:
3232:
3016:
2939:. Not all principally polarised abelian varieties are Jacobians of curves; see the
2881:
2877:
2873:
2136:
1990:
1810:
1399:
1275:
that the algebraic variety condition imposes extra constraints on a complex torus.
1155:
949:
607:
306:
295:
242:
217:
212:
171:
142:
105:
975:
of other algebraic varieties. The group law of an abelian variety is necessarily
4275:
2692:
given by tensor product of line bundles, which makes it into an abelian variety.
1615:
on an abelian variety, which may be regarded therefore as a periodic function of
944:
are a special case, which is important also from the viewpoint of number theory.
17:
3932:
is a commutative group variety which is an extension of an abelian variety by a
2326:
1014:
976:
953:
930:
875:
1083:. The subject was very popular at the time, already having a large literature.
2194:
of the abelian variety. Similar results hold for some other classes of fields
1879:
1033:
774:
502:
3793:, but the Néron model is not proper and hence is not an abelian scheme over
903:
595:
4326:
1719:
is a
Jacobian. This theorem remains true if the ground field is infinite.
1653:. It states that over an algebraically closed field every abelian variety
913:
An abelian variety can be defined by equations having coefficients in any
113:
3204:. This allows for a uniform treatment of phenomena such as reduction mod
2864:
is coprime to the characteristic of the base. In general — for all
1813:
that he had announced in 1940 work, he had to introduce the notion of an
1199:
132:
4147:
2898:
2253:
2068:
2000:
When the base field is an algebraically closed field of characteristic
1365:
1208:
474:
388:
3843:
independently proved that there are no nonzero abelian varieties over
2902:
from an abelian variety to its dual that is symmetric with respect to
4055:
1186:, one may equivalently define a complex abelian variety of dimension
1097:
Today, abelian varieties form an important tool in number theory, in
3700:{\displaystyle \operatorname {Proj} R/(y^{2}z-x^{3}-Axz^{2}-Bz^{3})}
1727:
Two equivalent definitions of abelian variety over a general field
2623:{\displaystyle 1_{A}\times f\colon A\times T\to A\times A^{\vee }}
1632:
1283:
The following criterion by
Riemann decides whether or not a given
1135:
1025:. When those were replaced by polynomials of higher degree, say
2313:(over the same field), which is the solution to the following
29:
27:
A projective algebraic variety that is also an algebraic group
2186:
and a finite commutative group for some non-negative integer
1040:, the answer was formulated: this would involve functions of
987:
is an abelian variety of dimension 1. Abelian varieties have
2087:. If instead of looking at the full scheme structure on the
3337:, governed by the deformation properties of the associated
2222:, over the same field, is an abelian variety of dimension
1245:, and every complex torus gives rise to such a curve; for
1649:
One important structure theorem of abelian varieties is
3134:. A choice of an equivalence class of Riemann forms on
1241:, the notion of abelian variety is the same as that of
4360:, Oxford: Mathematical Institute, University of Oxford
2777:{\displaystyle f^{\vee }\colon B^{\vee }\to A^{\vee }}
2695:
This association is a duality in the sense that it is
2317:. A family of degree 0 line bundles parametrised by a
925:. Such abelian varieties turn out to be exactly those
4108:
4011:
3899:
3893:. The proof involves showing that the coordinates of
3871:
3849:
3799:
3771:
3743:
3713:
3594:
3566:
3540:
3495:
3431:
3425:
has no repeated complex roots. Then the discriminant
3389:
3355:
3300:
3264:
3152:
3094:
3055:
3028:
2988:{\displaystyle \mathrm {End} (A)\otimes \mathbb {Q} }
2956:
2922:
2834:
2794:
2737:
2705:
2671:
2640:
2572:
2525:
2490:
2459:
2419:
2380:
2338:
2292:
2228:
2163:
2105:
2083:-torsion defines a finite flat group scheme of rank 2
2014:
1940:
1888:
1842:
1777:
1705:
1679:
1659:
1578:
1493:
1431:
1374:
1301:
1251:
1221:
545:
520:
483:
4362:. Description of the Jacobian of the Covering Curves
2445:
is a trivial line bundle (here 0 is the identity of
1556:
over the complex numbers. From the point of view of
1349:
is an abelian variety if and only if there exists a
3212:), and parameter-families of abelian varieties. An
1178:over the field of complex numbers. By invoking the
4116:
4024:
3912:
3885:
3857:
3813:
3785:
3757:
3725:
3699:
3580:
3552:
3526:
3481:
3417:
3375:
3313:
3278:
3164:
3126:
3068:
3041:
2987:
2931:
2840:
2820:
2776:
2723:
2684:
2653:
2622:
2550:
2503:
2472:
2437:
2398:
2350:
2305:
2240:
2178:
2117:
2051:
1977:
1922:
1850:
1785:
1711:
1691:
1665:
1591:
1506:
1443:
1386:
1321:
1263:
1233:
956:. Since a number field is the fraction field of a
553:
528:
491:
3886:{\displaystyle \operatorname {Spec} \mathbb {Z} }
3814:{\displaystyle \operatorname {Spec} \mathbb {Z} }
3786:{\displaystyle \operatorname {Spec} \mathbb {Z} }
3758:{\displaystyle \operatorname {Spec} \mathbb {Z} }
3581:{\displaystyle \operatorname {Spec} \mathbb {Z} }
2052:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{2g}}
1978:{\displaystyle (\mathbb {Z} /n\mathbb {Z} )^{2g}}
1923:{\displaystyle (\mathbb {Q} /\mathbb {Z} )^{2g}}
1005:In the early nineteenth century, the theory of
4102:(1985). "Il n'y a pas de variété abélienne sur
1009:succeeded in giving a basis for the theory of
855:
8:
2426:
2420:
2393:
2387:
3482:{\displaystyle \Delta =-16(4A^{3}+27B^{2})}
2665:, so there is a natural group operation on
2661:correspond to line bundles of degree 0 on
862:
848:
300:
126:
91:
4205:Birkenhake, Christina; Lange, H. (1992),
4110:
4109:
4107:
4016:
4010:
3904:
3898:
3879:
3878:
3870:
3851:
3850:
3848:
3807:
3806:
3798:
3779:
3778:
3770:
3751:
3750:
3742:
3712:
3688:
3672:
3653:
3637:
3625:
3593:
3574:
3573:
3565:
3539:
3513:
3503:
3502:
3494:
3470:
3454:
3430:
3394:
3388:
3369:
3368:
3354:
3333:of abelian schemes are, according to the
3305:
3299:
3268:
3263:
3151:
3118:
3102:
3093:
3060:
3054:
3033:
3027:
2981:
2980:
2957:
2955:
2921:
2833:
2812:
2802:
2793:
2768:
2755:
2742:
2736:
2704:
2676:
2670:
2645:
2639:
2614:
2577:
2571:
2542:
2524:
2495:
2489:
2464:
2458:
2418:
2379:
2337:
2297:
2291:
2227:
2170:
2166:
2165:
2162:
2104:
2040:
2032:
2031:
2023:
2019:
2018:
2013:
1966:
1958:
1957:
1949:
1945:
1944:
1939:
1911:
1903:
1902:
1897:
1893:
1892:
1887:
1844:
1843:
1841:
1779:
1778:
1776:
1704:
1678:
1658:
1583:
1577:
1498:
1492:
1430:
1373:
1311:
1300:
1250:
1220:
1166:. A complex abelian variety of dimension
1105:), and in algebraic geometry (especially
917:; the variety is then said to be defined
547:
546:
544:
522:
521:
519:
485:
484:
482:
80:Learn how and when to remove this message
2848:(defined via the Poincaré bundle). The
1572:letters acting on the function field of
43:This article includes a list of general
3977:
2852:-torsion of an abelian variety and the
2699:, i.e., it associates to all morphisms
2484:, the Poincaré bundle, parametrised by
1333:is a complex vector space of dimension
418:
184:
94:
3765:, which is a smooth group scheme over
3726:{\displaystyle \operatorname {Spec} R}
3553:{\displaystyle \operatorname {Spec} R}
3196:One can also define abelian varieties
3003:Polarisations over the complex numbers
2634:is a point, we see that the points of
2551:{\displaystyle f\colon T\to A^{\vee }}
2480:and a family of degree 0 line bundles
2150:. Hence, by the structure theorem for
1797:are abelian varieties of dimension 1.
1451:is associated with an abelian variety
963:Abelian varieties appear naturally as
420:Classification of finite simple groups
4333:Venkov, B.B.; Parshin, A.N. (2001) ,
3011:can be defined as an abelian variety
2260:Polarisation and dual abelian variety
2154:, it is isomorphic to a product of a
1526:. The study of differential forms on
1398:is usually called a (non-degenerate)
967:(the connected components of zero in
7:
3961:Equations defining abelian varieties
2880:of each other. This generalises the
2784:in a compatible way, and there is a
4178:"There is no Abelian scheme over Z"
3992:. Math Department Oxford University
3376:{\displaystyle A,B\in \mathbb {Z} }
3184:is equivalent to the given form on
3172:of abelian varieties such that the
2821:{\displaystyle (A^{\vee })^{\vee }}
1874:of an abelian variety of dimension
1190:to be a complex torus of dimension
1146:. It can always be obtained as the
1101:(more specifically in the study of
3987:"N-Covers of Hyperelliptic Curves"
3518:
3432:
2964:
2961:
2958:
2206:The product of an abelian variety
1414:The Jacobian of an algebraic curve
1001:History of manifolds and varieties
49:it lacks sufficient corresponding
25:
4254:Degeneration of Abelian Varieties
4190:from the original on 23 Aug 2020.
2630:. Applying this to the case when
2562:is isomorphic to the pullback of
2152:finitely generated abelian groups
1631:is a product of elliptic curves,
1459:, by means of an analytic map of
4354:N-COVERS OF HYPERELLIPTIC CURVES
2519:is associated a unique morphism
2179:{\displaystyle \mathbb {Z} ^{r}}
2008:-torsion is still isomorphic to
1825:Structure of the group of points
1170:is a complex torus of dimension
1142:that carries the structure of a
1052:): what would now be called the
112:
34:
3210:Arithmetic of abelian varieties
3080:if there are positive integers
1934:-torsion part is isomorphic to
940:Abelian varieties defined over
4060:over the ring of Witt vectors"
3694:
3630:
3622:
3604:
3527:{\displaystyle R=\mathbb {Z} }
3521:
3507:
3476:
3444:
3156:
2974:
2968:
2876:of dual abelian varieties are
2809:
2795:
2761:
2724:{\displaystyle f\colon A\to B}
2715:
2601:
2535:
2037:
2015:
1963:
1941:
1908:
1889:
1683:
781:Infinite dimensional Lie group
1:
4301:American Mathematical Society
3951:Timeline of abelian varieties
3127:{\displaystyle nH_{1}=mH_{2}}
2438:{\displaystyle \{0\}\times T}
2399:{\displaystyle A\times \{t\}}
1552:, for any non-singular curve
1483:as a group. More accurately,
1212:is a finite-to-one morphism.
4117:{\displaystyle \mathbb {Z} }
3858:{\displaystyle \mathbb {Q} }
3015:together with a choice of a
3007:Over the complex numbers, a
2896:of an abelian variety is an
1851:{\displaystyle \mathbb {C} }
1786:{\displaystyle \mathbb {C} }
1475:structure, and the image of
1295:-dimensional torus given as
1206:for the group structure. An
896:projective algebraic variety
554:{\displaystyle \mathbb {Z} }
529:{\displaystyle \mathbb {Z} }
492:{\displaystyle \mathbb {Z} }
4340:Encyclopedia of Mathematics
4238:Encyclopedia of Mathematics
3956:Moduli of abelian varieties
3733:. It can be extended to a
2943:. A polarisation induces a
1771:When the base is the field
1619:complex variables, having 2
279:List of group theory topics
4412:
3707:is an abelian scheme over
3418:{\displaystyle x^{3}+Ax+B}
3208:of abelian varieties (see
2406:is a degree 0 line bundle,
2268:
1864:algebraically closed field
1637:
1564:is the fixed field of the
1174:that is also a projective
1044:, having four independent
998:
4252:; Chai, Ching-Li (1990),
4207:Complex Abelian Varieties
4056:"Group schemes of period
4054:Abrashkin, V. A. (1985).
3321:-torsion points, for all
3009:polarised abelian variety
2856:-torsion of its dual are
2685:{\displaystyle A^{\vee }}
2654:{\displaystyle A^{\vee }}
2504:{\displaystyle A^{\vee }}
2473:{\displaystyle A^{\vee }}
2351:{\displaystyle A\times T}
2306:{\displaystyle A^{\vee }}
2214:, and an abelian variety
1530:, which give rise to the
1387:{\displaystyle L\times L}
1180:Kodaira embedding theorem
4127:Inventiones Mathematicae
3560:is an open subscheme of
3292:finite flat group scheme
3290:-torsion points forms a
2788:between the double dual
2697:contravariant functorial
2453:Then there is a variety
2248:. An abelian variety is
1985:, i.e., the product of 2
1402:. Choosing a basis for
1271:it has been known since
933:embedded into a complex
397:Elementary abelian group
274:Glossary of group theory
4351:Bruin, N; Flynn, E.V.,
3966:Horrocks–Mumford bundle
3176:of the Riemann form on
1625:hyperelliptic integrals
1444:{\displaystyle g\geq 1}
942:algebraic number fields
906:that can be defined by
888:algebraic number theory
64:more precise citations.
4288:. Online course notes.
4118:
4026:
3914:
3887:
3859:
3815:
3787:
3759:
3727:
3701:
3582:
3554:
3528:
3483:
3419:
3377:
3315:
3280:
3258:For an abelian scheme
3220:of relative dimension
3166:
3165:{\displaystyle A\to B}
3128:
3070:
3043:
2989:
2933:
2913:principal polarisation
2842:
2822:
2778:
2725:
2686:
2655:
2624:
2552:
2505:
2474:
2439:
2400:
2352:
2307:
2275:To an abelian variety
2242:
2180:
2119:
2053:
1979:
1924:
1852:
1787:
1713:
1693:
1692:{\displaystyle J\to A}
1667:
1593:
1540:. The abelian variety
1508:
1471:carries a commutative
1445:
1418:Every algebraic curve
1388:
1323:
1265:
1264:{\displaystyle g>1}
1235:
995:History and motivation
813:Linear algebraic group
555:
530:
493:
4119:
4065:Dokl. Akad. Nauk SSSR
4027:
4025:{\displaystyle C^{g}}
3915:
3913:{\displaystyle p^{n}}
3888:
3860:
3816:
3788:
3760:
3728:
3702:
3583:
3555:
3529:
3484:
3420:
3378:
3316:
3314:{\displaystyle p^{n}}
3281:
3167:
3129:
3071:
3069:{\displaystyle H_{2}}
3044:
3042:{\displaystyle H_{1}}
2990:
2934:
2932:{\displaystyle >1}
2884:for elliptic curves.
2843:
2823:
2779:
2726:
2687:
2656:
2625:
2553:
2506:
2475:
2440:
2401:
2370:, the restriction of
2353:
2308:
2243:
2181:
2120:
2054:
1980:
1925:
1853:
1788:
1731:are commonly in use:
1714:
1694:
1668:
1594:
1592:{\displaystyle C^{g}}
1509:
1507:{\displaystyle C^{g}}
1446:
1389:
1324:
1322:{\displaystyle X=V/L}
1266:
1236:
1154:-dimensional complex
1042:two complex variables
1029:, what would happen?
999:Further information:
556:
531:
494:
4386:Geometry of divisors
4209:, Berlin, New York:
4106:
4009:
3897:
3869:
3847:
3797:
3769:
3741:
3711:
3592:
3564:
3538:
3493:
3429:
3387:
3353:
3298:
3262:
3200:-theoretically and
3150:
3092:
3053:
3026:
3022:. Two Riemann forms
2954:
2920:
2832:
2792:
2735:
2703:
2669:
2638:
2570:
2523:
2488:
2457:
2417:
2378:
2336:
2290:
2285:dual abelian variety
2271:Dual abelian variety
2265:Dual abelian variety
2226:
2161:
2148:Mordell-Weil theorem
2103:
2012:
1938:
1886:
1840:
1775:
1723:Algebraic definition
1703:
1677:
1657:
1613:meromorphic function
1576:
1522:-tuple of points in
1491:
1429:
1372:
1299:
1249:
1219:
543:
518:
481:
4140:1985InMat..81..515F
4100:Fontaine, Jean-Marc
3930:semiabelian variety
3924:Semiabelian variety
3341:-divisible groups.
3294:. The union of the
3279:{\displaystyle A/S}
3216:over a base scheme
2909:automorphism groups
2860:to each other when
2786:natural isomorphism
2566:along the morphism
2511:such that a family
2409:the restriction of
2325:is defined to be a
2283:, one associates a
2241:{\displaystyle m+n}
2118:{\displaystyle n=p}
1860:Lefschetz principle
1858:, and hence by the
1651:Matsusaka's theorem
1558:birational geometry
1234:{\displaystyle g=1}
1138:of real dimension 2
1103:Hamiltonian systems
1056:hyperelliptic curve
1023:quartic polynomials
979:and the variety is
187:Group homomorphisms
97:Algebraic structure
4391:Algebraic surfaces
4148:10.1007/BF01388584
4114:
4022:
3910:
3883:
3855:
3841:Jean-Marc Fontaine
3811:
3783:
3755:
3723:
3697:
3578:
3550:
3524:
3479:
3415:
3373:
3335:Serre–Tate theorem
3311:
3276:
3202:relative to a base
3162:
3124:
3066:
3039:
2985:
2929:
2838:
2818:
2774:
2721:
2682:
2651:
2620:
2548:
2501:
2470:
2435:
2396:
2348:
2303:
2238:
2176:
2156:free abelian group
2144:finitely generated
2115:
2049:
1975:
1920:
1848:
1819:Algebraic Geometry
1803:Riemann hypothesis
1783:
1709:
1689:
1663:
1645:Important theorems
1589:
1504:
1441:
1384:
1319:
1279:Riemann conditions
1261:
1231:
1111:Albanese varieties
1011:elliptic integrals
1007:elliptic functions
973:Albanese varieties
965:Jacobian varieties
880:algebraic geometry
878:, particularly in
663:Special orthogonal
551:
526:
489:
370:Lagrange's theorem
4396:Niels Henrik Abel
4376:Abelian varieties
4335:"Abelian_variety"
4310:978-81-85931-86-9
4297:Abelian varieties
4277:Abelian Varieties
4220:978-0-387-54747-3
3327:p-divisible group
3247:and of dimension
2949:endomorphism ring
2945:Rosati involution
2841:{\displaystyle A}
1712:{\displaystyle J}
1666:{\displaystyle A}
1603:Abelian functions
1533:abelian integrals
1394:. Such a form on
1351:positive definite
1176:algebraic variety
1099:dynamical systems
989:Kodaira dimension
908:regular functions
872:
871:
447:
446:
329:Alternating group
286:
285:
90:
89:
82:
18:Abelian varieties
16:(Redirected from
4403:
4381:Algebraic curves
4361:
4359:
4347:
4329:
4287:
4286:
4284:
4270:
4245:
4233:"Abelian scheme"
4223:
4192:
4191:
4189:
4182:
4174:
4168:
4167:
4123:
4121:
4120:
4115:
4113:
4096:
4090:
4089:
4072:(6): 1289–1294.
4051:
4045:
4039:
4033:
4031:
4029:
4028:
4023:
4021:
4020:
4001:
3999:
3997:
3991:
3982:
3919:
3917:
3916:
3911:
3909:
3908:
3892:
3890:
3889:
3884:
3882:
3864:
3862:
3861:
3856:
3854:
3838:
3830:Viktor Abrashkin
3820:
3818:
3817:
3812:
3810:
3792:
3790:
3789:
3784:
3782:
3764:
3762:
3761:
3756:
3754:
3732:
3730:
3729:
3724:
3706:
3704:
3703:
3698:
3693:
3692:
3677:
3676:
3658:
3657:
3642:
3641:
3629:
3587:
3585:
3584:
3579:
3577:
3559:
3557:
3556:
3551:
3533:
3531:
3530:
3525:
3517:
3506:
3489:is nonzero. Let
3488:
3486:
3485:
3480:
3475:
3474:
3459:
3458:
3424:
3422:
3421:
3416:
3399:
3398:
3382:
3380:
3379:
3374:
3372:
3320:
3318:
3317:
3312:
3310:
3309:
3285:
3283:
3282:
3277:
3272:
3241:geometric fibers
3171:
3169:
3168:
3163:
3133:
3131:
3130:
3125:
3123:
3122:
3107:
3106:
3075:
3073:
3072:
3067:
3065:
3064:
3048:
3046:
3045:
3040:
3038:
3037:
2994:
2992:
2991:
2986:
2984:
2967:
2941:Schottky problem
2938:
2936:
2935:
2930:
2847:
2845:
2844:
2839:
2827:
2825:
2824:
2819:
2817:
2816:
2807:
2806:
2783:
2781:
2780:
2775:
2773:
2772:
2760:
2759:
2747:
2746:
2730:
2728:
2727:
2722:
2691:
2689:
2688:
2683:
2681:
2680:
2660:
2658:
2657:
2652:
2650:
2649:
2629:
2627:
2626:
2621:
2619:
2618:
2582:
2581:
2557:
2555:
2554:
2549:
2547:
2546:
2510:
2508:
2507:
2502:
2500:
2499:
2479:
2477:
2476:
2471:
2469:
2468:
2444:
2442:
2441:
2436:
2405:
2403:
2402:
2397:
2357:
2355:
2354:
2349:
2312:
2310:
2309:
2304:
2302:
2301:
2247:
2245:
2244:
2239:
2185:
2183:
2182:
2177:
2175:
2174:
2169:
2133:-rational points
2124:
2122:
2121:
2116:
2058:
2056:
2055:
2050:
2048:
2047:
2035:
2027:
2022:
1984:
1982:
1981:
1976:
1974:
1973:
1961:
1953:
1948:
1929:
1927:
1926:
1921:
1919:
1918:
1906:
1901:
1896:
1857:
1855:
1854:
1849:
1847:
1815:abstract variety
1792:
1790:
1789:
1784:
1782:
1718:
1716:
1715:
1710:
1698:
1696:
1695:
1690:
1672:
1670:
1669:
1664:
1640:abelian integral
1609:abelian function
1598:
1596:
1595:
1590:
1588:
1587:
1546:Jacobian variety
1513:
1511:
1510:
1505:
1503:
1502:
1450:
1448:
1447:
1442:
1393:
1391:
1390:
1385:
1341:is a lattice in
1328:
1326:
1325:
1320:
1315:
1270:
1268:
1267:
1262:
1240:
1238:
1237:
1232:
1204:identity element
1144:complex manifold
1107:Picard varieties
969:Picard varieties
935:projective space
898:that is also an
884:complex analysis
864:
857:
850:
806:Algebraic groups
579:Hyperbolic group
569:Arithmetic group
560:
558:
557:
552:
550:
535:
533:
532:
527:
525:
498:
496:
495:
490:
488:
411:Schur multiplier
365:Cauchy's theorem
353:Quaternion group
301:
127:
116:
103:
92:
85:
78:
74:
71:
65:
60:this article by
51:inline citations
38:
37:
30:
21:
4411:
4410:
4406:
4405:
4404:
4402:
4401:
4400:
4366:
4365:
4357:
4350:
4332:
4311:
4291:
4282:
4280:
4273:
4268:
4258:Springer Verlag
4248:
4229:Dolgachev, I.V.
4227:
4221:
4211:Springer-Verlag
4204:
4201:
4196:
4195:
4187:
4180:
4176:
4175:
4171:
4104:
4103:
4098:
4097:
4093:
4053:
4052:
4048:
4040:
4036:
4012:
4007:
4006:
3995:
3993:
3989:
3984:
3983:
3979:
3974:
3942:
3926:
3900:
3895:
3894:
3867:
3866:
3845:
3844:
3832:
3827:
3795:
3794:
3767:
3766:
3739:
3738:
3709:
3708:
3684:
3668:
3649:
3633:
3590:
3589:
3562:
3561:
3536:
3535:
3491:
3490:
3466:
3450:
3427:
3426:
3390:
3385:
3384:
3351:
3350:
3347:
3301:
3296:
3295:
3286:, the group of
3260:
3259:
3194:
3148:
3147:
3114:
3098:
3090:
3089:
3056:
3051:
3050:
3029:
3024:
3023:
3005:
2952:
2951:
2918:
2917:
2890:
2830:
2829:
2808:
2798:
2790:
2789:
2764:
2751:
2738:
2733:
2732:
2731:dual morphisms
2701:
2700:
2672:
2667:
2666:
2641:
2636:
2635:
2610:
2573:
2568:
2567:
2538:
2521:
2520:
2491:
2486:
2485:
2460:
2455:
2454:
2415:
2414:
2376:
2375:
2334:
2333:
2293:
2288:
2287:
2273:
2267:
2262:
2224:
2223:
2204:
2164:
2159:
2158:
2101:
2100:
2095:(the so-called
2036:
2010:
2009:
1962:
1936:
1935:
1907:
1884:
1883:
1838:
1837:
1827:
1795:elliptic curves
1773:
1772:
1761:algebraic group
1744:algebraic group
1725:
1701:
1700:
1675:
1674:
1655:
1654:
1647:
1642:
1605:
1579:
1574:
1573:
1566:symmetric group
1514:: any point in
1494:
1489:
1488:
1427:
1426:
1416:
1370:
1369:
1297:
1296:
1281:
1247:
1246:
1217:
1216:
1124:
1119:
1117:Analytic theory
1050:abelian surface
1032:In the work of
1003:
997:
958:Dedekind domain
931:holomorphically
923:complex numbers
900:algebraic group
892:abelian variety
868:
839:
838:
827:Abelian variety
820:Reductive group
808:
798:
797:
796:
795:
746:
738:
730:
722:
714:
687:Special unitary
598:
584:
583:
565:
564:
541:
540:
516:
515:
479:
478:
470:
469:
460:Discrete groups
449:
448:
404:Frobenius group
349:
336:
325:
318:Symmetric group
314:
298:
288:
287:
138:Normal subgroup
124:
104:
95:
86:
75:
69:
66:
56:Please help to
55:
39:
35:
28:
23:
22:
15:
12:
11:
5:
4409:
4407:
4399:
4398:
4393:
4388:
4383:
4378:
4368:
4367:
4364:
4363:
4348:
4330:
4309:
4293:Mumford, David
4289:
4274:Milne, James,
4271:
4266:
4250:Faltings, Gerd
4246:
4225:
4219:
4200:
4197:
4194:
4193:
4169:
4134:(3): 515–538.
4112:
4091:
4046:
4034:
4019:
4015:
4005:is covered by
3976:
3975:
3973:
3970:
3969:
3968:
3963:
3958:
3953:
3948:
3941:
3938:
3925:
3922:
3907:
3903:
3881:
3877:
3874:
3853:
3826:
3823:
3809:
3805:
3802:
3781:
3777:
3774:
3753:
3749:
3746:
3722:
3719:
3716:
3696:
3691:
3687:
3683:
3680:
3675:
3671:
3667:
3664:
3661:
3656:
3652:
3648:
3645:
3640:
3636:
3632:
3628:
3624:
3621:
3618:
3615:
3612:
3609:
3606:
3603:
3600:
3597:
3576:
3572:
3569:
3549:
3546:
3543:
3523:
3520:
3516:
3512:
3509:
3505:
3501:
3498:
3478:
3473:
3469:
3465:
3462:
3457:
3453:
3449:
3446:
3443:
3440:
3437:
3434:
3414:
3411:
3408:
3405:
3402:
3397:
3393:
3371:
3367:
3364:
3361:
3358:
3346:
3343:
3308:
3304:
3275:
3271:
3267:
3214:abelian scheme
3193:
3192:Abelian scheme
3190:
3161:
3158:
3155:
3121:
3117:
3113:
3110:
3105:
3101:
3097:
3063:
3059:
3036:
3032:
3004:
3001:
2983:
2979:
2976:
2973:
2970:
2966:
2963:
2960:
2928:
2925:
2904:double-duality
2889:
2886:
2837:
2815:
2811:
2805:
2801:
2797:
2771:
2767:
2763:
2758:
2754:
2750:
2745:
2741:
2720:
2717:
2714:
2711:
2708:
2679:
2675:
2648:
2644:
2617:
2613:
2609:
2606:
2603:
2600:
2597:
2594:
2591:
2588:
2585:
2580:
2576:
2545:
2541:
2537:
2534:
2531:
2528:
2498:
2494:
2467:
2463:
2451:
2450:
2434:
2431:
2428:
2425:
2422:
2407:
2395:
2392:
2389:
2386:
2383:
2347:
2344:
2341:
2315:moduli problem
2300:
2296:
2269:Main article:
2266:
2263:
2261:
2258:
2237:
2234:
2231:
2203:
2200:
2173:
2168:
2114:
2111:
2108:
2046:
2043:
2039:
2034:
2030:
2026:
2021:
2017:
1989:copies of the
1972:
1969:
1965:
1960:
1956:
1952:
1947:
1943:
1917:
1914:
1910:
1905:
1900:
1895:
1891:
1868:characteristic
1846:
1836:For the field
1826:
1823:
1781:
1769:
1768:
1750:
1724:
1721:
1708:
1688:
1685:
1682:
1662:
1646:
1643:
1604:
1601:
1586:
1582:
1562:function field
1544:is called the
1501:
1497:
1487:is covered by
1467:. As a torus,
1440:
1437:
1434:
1415:
1412:
1383:
1380:
1377:
1362:imaginary part
1354:hermitian form
1318:
1314:
1310:
1307:
1304:
1280:
1277:
1260:
1257:
1254:
1243:elliptic curve
1230:
1227:
1224:
1184:Chow's theorem
1123:
1120:
1118:
1115:
1054:Jacobian of a
996:
993:
985:elliptic curve
902:, i.e., has a
870:
869:
867:
866:
859:
852:
844:
841:
840:
837:
836:
834:Elliptic curve
830:
829:
823:
822:
816:
815:
809:
804:
803:
800:
799:
794:
793:
790:
787:
783:
779:
778:
777:
772:
770:Diffeomorphism
766:
765:
760:
755:
749:
748:
744:
740:
736:
732:
728:
724:
720:
716:
712:
707:
706:
695:
694:
683:
682:
671:
670:
659:
658:
647:
646:
635:
634:
627:Special linear
623:
622:
615:General linear
611:
610:
605:
599:
590:
589:
586:
585:
582:
581:
576:
571:
563:
562:
549:
537:
524:
511:
509:Modular groups
507:
506:
505:
500:
487:
471:
468:
467:
462:
456:
455:
454:
451:
450:
445:
444:
443:
442:
437:
432:
429:
423:
422:
416:
415:
414:
413:
407:
406:
400:
399:
394:
385:
384:
382:Hall's theorem
379:
377:Sylow theorems
373:
372:
367:
359:
358:
357:
356:
350:
345:
342:Dihedral group
338:
337:
332:
326:
321:
315:
310:
299:
294:
293:
290:
289:
284:
283:
282:
281:
276:
268:
267:
266:
265:
260:
255:
250:
245:
240:
235:
233:multiplicative
230:
225:
220:
215:
207:
206:
205:
204:
199:
191:
190:
182:
181:
180:
179:
177:Wreath product
174:
169:
164:
162:direct product
156:
154:Quotient group
148:
147:
146:
145:
140:
135:
125:
122:
121:
118:
117:
109:
108:
88:
87:
42:
40:
33:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
4408:
4397:
4394:
4392:
4389:
4387:
4384:
4382:
4379:
4377:
4374:
4373:
4371:
4356:
4355:
4349:
4346:
4342:
4341:
4336:
4331:
4328:
4324:
4320:
4316:
4312:
4306:
4302:
4298:
4294:
4290:
4279:
4278:
4272:
4269:
4267:3-540-52015-5
4263:
4259:
4255:
4251:
4247:
4244:
4240:
4239:
4234:
4230:
4226:
4222:
4216:
4212:
4208:
4203:
4202:
4198:
4186:
4179:
4173:
4170:
4165:
4161:
4157:
4153:
4149:
4145:
4141:
4137:
4133:
4129:
4128:
4101:
4095:
4092:
4087:
4083:
4079:
4075:
4071:
4067:
4066:
4061:
4059:
4050:
4047:
4043:
4038:
4035:
4017:
4013:
4004:
3988:
3981:
3978:
3971:
3967:
3964:
3962:
3959:
3957:
3954:
3952:
3949:
3947:
3944:
3943:
3939:
3937:
3935:
3931:
3923:
3921:
3905:
3901:
3875:
3872:
3842:
3836:
3831:
3825:Non-existence
3824:
3822:
3803:
3800:
3775:
3772:
3747:
3744:
3736:
3720:
3717:
3714:
3689:
3685:
3681:
3678:
3673:
3669:
3665:
3662:
3659:
3654:
3650:
3646:
3643:
3638:
3634:
3626:
3619:
3616:
3613:
3610:
3607:
3601:
3598:
3595:
3570:
3567:
3547:
3544:
3541:
3514:
3510:
3499:
3496:
3471:
3467:
3463:
3460:
3455:
3451:
3447:
3441:
3438:
3435:
3412:
3409:
3406:
3403:
3400:
3395:
3391:
3383:be such that
3365:
3362:
3359:
3356:
3344:
3342:
3340:
3336:
3332:
3328:
3324:
3306:
3302:
3293:
3289:
3273:
3269:
3265:
3256:
3254:
3250:
3246:
3242:
3238:
3234:
3231:
3227:
3223:
3219:
3215:
3211:
3207:
3203:
3199:
3191:
3189:
3187:
3183:
3179:
3175:
3159:
3153:
3145:
3141:
3137:
3119:
3115:
3111:
3108:
3103:
3099:
3095:
3087:
3083:
3079:
3061:
3057:
3034:
3030:
3021:
3018:
3014:
3010:
3002:
3000:
2998:
2977:
2971:
2950:
2946:
2942:
2926:
2923:
2914:
2910:
2905:
2901:
2900:
2895:
2888:Polarisations
2887:
2885:
2883:
2879:
2878:Cartier duals
2875:
2874:group schemes
2871:
2867:
2863:
2859:
2855:
2851:
2835:
2813:
2803:
2799:
2787:
2769:
2765:
2756:
2752:
2748:
2743:
2739:
2718:
2712:
2709:
2706:
2698:
2693:
2677:
2673:
2664:
2646:
2642:
2633:
2615:
2611:
2607:
2604:
2598:
2595:
2592:
2589:
2586:
2583:
2578:
2574:
2565:
2561:
2543:
2539:
2532:
2529:
2526:
2518:
2514:
2496:
2492:
2483:
2465:
2461:
2448:
2432:
2429:
2423:
2412:
2408:
2390:
2384:
2381:
2373:
2369:
2365:
2361:
2360:
2359:
2345:
2342:
2339:
2331:
2328:
2324:
2320:
2316:
2298:
2294:
2286:
2282:
2279:over a field
2278:
2272:
2264:
2259:
2257:
2255:
2252:if it is not
2251:
2235:
2232:
2229:
2221:
2218:of dimension
2217:
2213:
2210:of dimension
2209:
2201:
2199:
2197:
2193:
2189:
2171:
2157:
2153:
2149:
2145:
2141:
2138:
2134:
2132:
2128:The group of
2126:
2112:
2109:
2106:
2098:
2094:
2090:
2086:
2082:
2078:
2074:
2070:
2066:
2062:
2044:
2041:
2028:
2024:
2007:
2003:
1998:
1996:
1992:
1988:
1970:
1967:
1954:
1950:
1933:
1930:. Hence, its
1915:
1912:
1898:
1881:
1877:
1873:
1872:torsion group
1869:
1865:
1861:
1834:
1832:
1824:
1822:
1820:
1816:
1812:
1811:finite fields
1808:
1804:
1798:
1796:
1766:
1762:
1759:
1755:
1751:
1749:
1745:
1742:
1738:
1734:
1733:
1732:
1730:
1722:
1720:
1706:
1686:
1680:
1660:
1652:
1644:
1641:
1636:
1634:
1630:
1626:
1622:
1618:
1614:
1610:
1602:
1600:
1584:
1580:
1571:
1567:
1563:
1559:
1555:
1551:
1547:
1543:
1539:
1535:
1534:
1529:
1525:
1521:
1518:comes from a
1517:
1499:
1495:
1486:
1482:
1478:
1474:
1470:
1466:
1462:
1458:
1455:of dimension
1454:
1438:
1435:
1432:
1425:
1421:
1413:
1411:
1409:
1405:
1401:
1397:
1381:
1378:
1375:
1367:
1363:
1359:
1355:
1352:
1348:
1344:
1340:
1336:
1332:
1316:
1312:
1308:
1305:
1302:
1294:
1290:
1286:
1285:complex torus
1278:
1276:
1274:
1258:
1255:
1252:
1244:
1228:
1225:
1222:
1213:
1211:
1210:
1205:
1201:
1197:
1193:
1189:
1185:
1181:
1177:
1173:
1169:
1165:
1161:
1157:
1153:
1149:
1145:
1141:
1137:
1133:
1130:of dimension
1129:
1128:complex torus
1121:
1116:
1114:
1112:
1108:
1104:
1100:
1095:
1093:
1089:
1084:
1082:
1078:
1074:
1070:
1066:
1061:
1059:
1057:
1051:
1047:
1043:
1039:
1035:
1030:
1028:
1024:
1020:
1016:
1012:
1008:
1002:
994:
992:
990:
986:
982:
978:
974:
970:
966:
961:
959:
955:
951:
950:finite fields
947:
943:
938:
936:
932:
928:
924:
920:
916:
911:
909:
905:
901:
897:
893:
889:
885:
881:
877:
865:
860:
858:
853:
851:
846:
845:
843:
842:
835:
832:
831:
828:
825:
824:
821:
818:
817:
814:
811:
810:
807:
802:
801:
791:
788:
785:
784:
782:
776:
773:
771:
768:
767:
764:
761:
759:
756:
754:
751:
750:
747:
741:
739:
733:
731:
725:
723:
717:
715:
709:
708:
704:
700:
697:
696:
692:
688:
685:
684:
680:
676:
673:
672:
668:
664:
661:
660:
656:
652:
649:
648:
644:
640:
637:
636:
632:
628:
625:
624:
620:
616:
613:
612:
609:
606:
604:
601:
600:
597:
593:
588:
587:
580:
577:
575:
572:
570:
567:
566:
538:
513:
512:
510:
504:
501:
476:
473:
472:
466:
463:
461:
458:
457:
453:
452:
441:
438:
436:
433:
430:
427:
426:
425:
424:
421:
417:
412:
409:
408:
405:
402:
401:
398:
395:
393:
391:
387:
386:
383:
380:
378:
375:
374:
371:
368:
366:
363:
362:
361:
360:
354:
351:
348:
343:
340:
339:
335:
330:
327:
324:
319:
316:
313:
308:
305:
304:
303:
302:
297:
296:Finite groups
292:
291:
280:
277:
275:
272:
271:
270:
269:
264:
261:
259:
256:
254:
251:
249:
246:
244:
241:
239:
236:
234:
231:
229:
226:
224:
221:
219:
216:
214:
211:
210:
209:
208:
203:
200:
198:
195:
194:
193:
192:
189:
188:
183:
178:
175:
173:
170:
168:
165:
163:
160:
157:
155:
152:
151:
150:
149:
144:
141:
139:
136:
134:
131:
130:
129:
128:
123:Basic notions
120:
119:
115:
111:
110:
107:
102:
98:
93:
84:
81:
73:
70:February 2013
63:
59:
53:
52:
46:
41:
32:
31:
19:
4353:
4338:
4296:
4281:, retrieved
4276:
4253:
4236:
4206:
4172:
4131:
4125:
4094:
4069:
4063:
4057:
4049:
4037:
4002:
3994:. Retrieved
3980:
3929:
3927:
3828:
3348:
3338:
3331:Deformations
3322:
3287:
3257:
3252:
3248:
3236:
3233:group scheme
3221:
3217:
3213:
3205:
3195:
3185:
3181:
3177:
3143:
3140:polarisation
3139:
3138:is called a
3135:
3085:
3081:
3019:
3017:Riemann form
3012:
3008:
3006:
2996:
2912:
2903:
2897:
2894:polarisation
2893:
2891:
2882:Weil pairing
2869:
2868:— the
2865:
2861:
2853:
2849:
2694:
2662:
2631:
2563:
2559:
2516:
2512:
2481:
2452:
2446:
2410:
2371:
2367:
2363:
2329:
2322:
2318:
2284:
2280:
2276:
2274:
2249:
2219:
2215:
2211:
2207:
2205:
2195:
2191:
2187:
2139:
2137:global field
2130:
2127:
2096:
2092:
2088:
2084:
2080:
2076:
2072:
2064:
2060:
2005:
2001:
1999:
1994:
1991:cyclic group
1986:
1931:
1875:
1835:
1828:
1799:
1770:
1764:
1747:
1728:
1726:
1650:
1648:
1635:an isogeny.
1628:
1620:
1616:
1608:
1606:
1569:
1553:
1549:
1545:
1541:
1537:
1531:
1527:
1523:
1519:
1515:
1484:
1480:
1476:
1468:
1464:
1460:
1456:
1452:
1419:
1417:
1407:
1403:
1400:Riemann form
1395:
1357:
1346:
1342:
1338:
1334:
1330:
1292:
1288:
1282:
1214:
1207:
1191:
1187:
1171:
1167:
1163:
1156:vector space
1151:
1139:
1131:
1125:
1096:
1085:
1062:
1053:
1049:
1045:
1031:
1015:square roots
1004:
981:non-singular
962:
954:local fields
952:and various
946:Localization
939:
929:that can be
927:complex tori
918:
912:
891:
873:
826:
702:
690:
678:
666:
654:
642:
630:
618:
389:
346:
333:
322:
311:
307:Cyclic group
185:
172:Free product
143:Group action
106:Group theory
101:Group theory
100:
76:
67:
48:
4042:Milne, J.S.
3833: [
3735:Néron model
3076:are called
2327:line bundle
2190:called the
2099:-rank when
1831:commutative
1069:Weierstrass
1038:Carl Jacobi
977:commutative
876:mathematics
592:Topological
431:alternating
62:introducing
4370:Categories
4164:0612.14043
4086:0593.14029
3996:14 January
3985:Bruin, N.
3972:References
3325:, forms a
3088:such that
3078:equivalent
2358:such that
1880:isomorphic
1870:zero, the
1862:for every
1821:article).
1758:projective
1638:See also:
1479:generates
1368:values on
1122:Definition
1092:André Weil
1058:of genus 2
1034:Niels Abel
699:Symplectic
639:Orthogonal
596:Lie groups
503:Free group
228:continuous
167:Direct sum
45:references
4345:EMS Press
4295:(2008) ,
4283:6 October
4243:EMS Press
4231:(2001) ,
3876:
3804:
3776:
3748:
3718:
3679:−
3660:−
3647:−
3599:
3571:
3545:
3519:Δ
3439:−
3433:Δ
3366:∈
3245:connected
3157:→
2978:⊗
2872:-torsion
2814:∨
2804:∨
2770:∨
2762:→
2757:∨
2749::
2744:∨
2716:→
2710::
2678:∨
2647:∨
2616:∨
2608:×
2602:→
2596:×
2590::
2584:×
2544:∨
2536:→
2530::
2497:∨
2466:∨
2430:×
2385:×
2343:×
2321:-variety
2299:∨
2254:isogenous
1993:of order
1754:connected
1737:connected
1684:→
1436:≥
1379:×
1162:of rank 2
1088:Lefschetz
1073:Frobenius
904:group law
763:Conformal
651:Euclidean
258:nilpotent
4185:Archived
3940:See also
3588:. Then
3174:pullback
2558:so that
2362:for all
2202:Products
1741:complete
1366:integral
1200:morphism
1148:quotient
1077:Poincaré
1027:quintics
758:Poincaré
603:Solenoid
475:Integers
465:Lattices
440:sporadic
435:Lie type
263:solvable
253:dihedral
238:additive
223:infinite
133:Subgroup
4319:0282985
4199:Sources
4156:0807070
4136:Bibcode
4078:0802862
3946:Motives
3345:Example
2947:on the
2899:isogeny
2146:by the
2071:. When
2069:coprime
1345:. Then
1273:Riemann
1209:isogeny
1160:lattice
1065:Riemann
1046:periods
753:Lorentz
675:Unitary
574:Lattice
514:PSL(2,
248:abelian
159:(Semi-)
58:improve
4327:138290
4325:
4317:
4307:
4264:
4217:
4162:
4154:
4084:
4076:
3239:whose
3230:smooth
3226:proper
3198:scheme
2250:simple
2135:for a
2004:, the
1807:curves
1699:where
1560:, its
1364:takes
1360:whose
1329:where
1081:Picard
1079:, and
971:) and
608:Circle
539:SL(2,
428:cyclic
392:-group
243:cyclic
218:finite
213:simple
197:kernel
47:, but
4358:(PDF)
4188:(PDF)
4181:(PDF)
3990:(PDF)
3934:torus
3837:]
3737:over
3534:, so
3235:over
3224:is a
2059:when
1809:over
1763:over
1746:over
1633:up to
1611:is a
1473:group
1463:into
1424:genus
1291:be a
1196:group
1158:by a
1150:of a
1136:torus
1134:is a
1019:cubic
983:. An
915:field
894:is a
890:, an
792:Sp(∞)
789:SU(∞)
202:image
4323:OCLC
4305:ISBN
4285:2016
4262:ISBN
4215:ISBN
3998:2015
3873:Spec
3839:and
3801:Spec
3773:Spec
3745:Spec
3715:Spec
3596:Proj
3568:Spec
3542:Spec
3349:Let
3243:are
3084:and
3049:and
2924:>
2911:. A
2858:dual
2828:and
2192:rank
2075:and
2067:are
2063:and
1805:for
1756:and
1739:and
1406:and
1337:and
1256:>
1198:. A
1182:and
1109:and
1036:and
1021:and
919:over
886:and
786:O(∞)
775:Loop
594:and
4160:Zbl
4144:doi
4124:".
4082:Zbl
4070:283
3180:to
3142:of
2995:of
2515:on
2413:to
2374:to
2366:in
2332:on
2142:is
2125:).
1882:to
1878:is
1866:of
1607:An
1568:on
1548:of
1422:of
1356:on
1113:).
1017:of
991:0.
937:.
874:In
701:Sp(
689:SU(
665:SO(
629:SL(
617:GL(
4372::
4343:,
4337:,
4321:,
4315:MR
4313:,
4303:,
4260:,
4256:,
4241:,
4235:,
4213:,
4183:.
4158:.
4152:MR
4150:.
4142:.
4132:81
4130:.
4080:.
4074:MR
4068:.
4062:.
3936:.
3928:A
3835:ru
3821:.
3464:27
3442:16
3329:.
3255:.
3228:,
3188:.
2999:.
2892:A
2449:).
2198:.
1997:.
1833:.
1752:a
1735:a
1599:.
1126:A
1075:,
1071:,
1067:,
1060:.
882:,
677:U(
653:E(
641:O(
99:→
4166:.
4146::
4138::
4111:Z
4088:.
4058:p
4032::
4018:g
4014:C
4003:J
4000:.
3906:n
3902:p
3880:Z
3852:Q
3808:Z
3780:Z
3752:Z
3721:R
3695:)
3690:3
3686:z
3682:B
3674:2
3670:z
3666:x
3663:A
3655:3
3651:x
3644:z
3639:2
3635:y
3631:(
3627:/
3623:]
3620:z
3617:,
3614:y
3611:,
3608:x
3605:[
3602:R
3575:Z
3548:R
3522:]
3515:/
3511:1
3508:[
3504:Z
3500:=
3497:R
3477:)
3472:2
3468:B
3461:+
3456:3
3452:A
3448:4
3445:(
3436:=
3413:B
3410:+
3407:x
3404:A
3401:+
3396:3
3392:x
3370:Z
3363:B
3360:,
3357:A
3339:p
3323:n
3307:n
3303:p
3288:n
3274:S
3270:/
3266:A
3253:S
3249:g
3237:S
3222:g
3218:S
3206:p
3186:A
3182:A
3178:B
3160:B
3154:A
3144:A
3136:A
3120:2
3116:H
3112:m
3109:=
3104:1
3100:H
3096:n
3086:m
3082:n
3062:2
3058:H
3035:1
3031:H
3020:H
3013:A
2997:A
2982:Q
2975:)
2972:A
2969:(
2965:d
2962:n
2959:E
2927:1
2870:n
2866:n
2862:n
2854:n
2850:n
2836:A
2810:)
2800:A
2796:(
2766:A
2753:B
2740:f
2719:B
2713:A
2707:f
2674:A
2663:A
2643:A
2632:T
2612:A
2605:A
2599:T
2593:A
2587:f
2579:A
2575:1
2564:P
2560:L
2540:A
2533:T
2527:f
2517:T
2513:L
2493:A
2482:P
2462:A
2447:A
2433:T
2427:}
2424:0
2421:{
2411:L
2394:}
2391:t
2388:{
2382:A
2372:L
2368:T
2364:t
2346:T
2340:A
2330:L
2323:T
2319:k
2295:A
2281:k
2277:A
2236:n
2233:+
2230:m
2220:n
2216:B
2212:m
2208:A
2196:k
2188:r
2172:r
2167:Z
2140:k
2131:k
2113:p
2110:=
2107:n
2097:p
2093:p
2089:n
2085:g
2081:n
2077:p
2073:n
2065:p
2061:n
2045:g
2042:2
2038:)
2033:Z
2029:n
2025:/
2020:Z
2016:(
2006:n
2002:p
1995:n
1987:g
1971:g
1968:2
1964:)
1959:Z
1955:n
1951:/
1946:Z
1942:(
1932:n
1916:g
1913:2
1909:)
1904:Z
1899:/
1894:Q
1890:(
1876:g
1845:C
1780:C
1767:.
1765:k
1748:k
1729:k
1707:J
1687:A
1681:J
1661:A
1629:J
1621:n
1617:n
1585:g
1581:C
1570:g
1554:C
1550:C
1542:J
1538:J
1528:C
1524:C
1520:g
1516:J
1500:g
1496:C
1485:J
1481:J
1477:C
1469:J
1465:J
1461:C
1457:g
1453:J
1439:1
1433:g
1420:C
1408:L
1404:V
1396:X
1382:L
1376:L
1358:V
1347:X
1343:V
1339:L
1335:g
1331:V
1317:L
1313:/
1309:V
1306:=
1303:X
1293:g
1289:X
1259:1
1253:g
1229:1
1226:=
1223:g
1192:g
1188:g
1172:g
1168:g
1164:g
1152:g
1140:g
1132:g
863:e
856:t
849:v
745:8
743:E
737:7
735:E
729:6
727:E
721:4
719:F
713:2
711:G
705:)
703:n
693:)
691:n
681:)
679:n
669:)
667:n
657:)
655:n
645:)
643:n
633:)
631:n
621:)
619:n
561:)
548:Z
536:)
523:Z
499:)
486:Z
477:(
390:p
355:Q
347:n
344:D
334:n
331:A
323:n
320:S
312:n
309:Z
83:)
77:(
72:)
68:(
54:.
20:)
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