697:
is a square whenever it is finite; this mistake originated in a paper by
Swinnerton-Dyer, who misquoted one of the results of Tate. Poonen and Stoll gave some examples where the order is twice a square, such as the Jacobian of a certain genus 2 curve over the rationals whose Tate–Shafarevich group
402:
647:
and the pairing is an alternating form. The kernel of this form is the subgroup of divisible elements, which is trivial if the Tate–Shafarevich conjecture is true. Tate extended the pairing to general abelian varieties, as a variation of
706:
is a square or twice a square (if it is finite), and if in addition the principal polarization comes from a rational divisor (as is the case for elliptic curves) then the form is alternating and the order of
267:
132:
203:
698:
has order 2, and Stein gave some examples where the power of an odd prime dividing the order is odd. If the abelian variety has a principal polarization then the form on
598:
1061:
1137:
758:
is finite in
Konstantinous' examples and these examples confirm a conjecture of Stein. Thus modulo squares any integer can be the order of
693:
is finite then it is a square. For more general abelian varieties it was sometimes incorrectly believed for many years that the order of
397:{\displaystyle \bigcap _{v}\mathrm {ker} \left(H^{1}\left(G_{K},A\right)\rightarrow H^{1}\left(G_{K_{v}},A_{v}\right)\right).}
1186:
1157:
1120:
886:
771:
548:
The special case of the Tate–Shafarevich group for the finite group scheme consisting of points of some given finite order
711:
is a square (if it is finite). On the other hand building on the results just presented
Konstantinous showed that for any
60:
463:
Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of
137:
586:
1019:
Cassels, John
William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups",
574:. Victor A. Kolyvagin extended this to modular elliptic curves over the rationals of analytic rank at most 1 (The
1560:
1531:
1225:
689:
For an elliptic curve, Cassels showed that the pairing is alternating, and a consequence is that if the order of
925:
1344:
Rubin, Karl (1987), "Tate–Shafarevich groups and L-functions of elliptic curves with complex multiplication",
1178:
1604:
Konstantinous, Alexandros (2024-04-25). "A note on the order of the Tate-Shafarevich group modulo squares".
1629:
1104:
55:
1527:
1506:
1220:
412:
1346:
1288:
Poonen, Bjorn; Stoll, Michael (1999), "The
Cassels-Tate pairing on polarized abelian varieties",
513:. Carl-Erik Lind gave an example of such a homogeneous space, by showing that the genus 1 curve
1634:
571:
686:, but unlike the case of elliptic curves this need not be alternating or even skew symmetric.
1290:
1056:
206:
1597:
1543:
1520:
1515:, Séminaire Bourbaki; 10e année: 1957/1958, vol. 13, Paris: Secrétariat Mathématique,
1499:
1475:
1451:
1423:
1383:
1355:
1337:
1280:
1262:
1130:
1090:
1048:
529:-adic fields, but has no rational points. Ernst S. Selmer gave many more examples, such as
8:
1272:
Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom
Geschlecht Eins
1195:
Kolyvagin, V. A. (1988), "Finiteness of E(Q) and SH(E,Q) for a subclass of Weil curves",
1141:
483:
237:
17:
1359:
1605:
1585:
1510:
1325:
1299:
1250:
575:
1577:
1439:
1430:
Shafarevich, I. R. (1959), "The group of principal homogeneous algebraic manifolds",
1411:
1371:
1317:
1242:
1204:
1182:
1170:
1153:
1116:
1078:
1036:
919:
882:
246:
1470:, Progr. Math., vol. 224, Basel, Boston, Berlin: Birkhäuser, pp. 277–289,
1569:
1460:
1401:
1392:
1363:
1309:
1234:
1108:
1070:
1028:
872:
420:
416:
1483:
566:
The Tate–Shafarevich conjecture states that the Tate–Shafarevich group is finite.
1593:
1539:
1516:
1495:
1491:
1471:
1447:
1419:
1379:
1333:
1276:
1270:
1258:
1166:
1149:
1126:
1098:
1086:
1044:
909:"THE SELMER GROUP, THE SHAFAREVICH-TATE GROUP, AND THE WEAK MORDELL-WEIL THEOREM"
504:
36:
1032:
632:
479:
444:
908:
864:
1623:
1581:
1536:
Proceedings of the
International Congress of Mathematicians (Stockholm, 1962)
1443:
1415:
1375:
1321:
1246:
1208:
1112:
1082:
1074:
1040:
582:
1555:
649:
555:
222:
45:
1589:
1406:
1367:
1254:
1216:
877:
712:
567:
408:
1547:
1329:
236:
by completing with respect to all its
Archimedean and non Archimedean
1304:
1573:
1238:
507:
fails to hold for rational equations with coefficients in the field
1610:
1390:
Selmer, Ernst S. (1951), "The
Diophantine equation ax³+by³+cz³=0",
1313:
503:-rational point. Thus, the group measures the extent to which the
440:
1148:, Graduate Texts in Mathematics, vol. 201, Berlin, New York:
1057:"Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung"
585:, thus the conjecture is equivalent to stating that the group is
1223:(1958), "Principal homogeneous spaces over abelian varieties",
1169:(1994), "Iwasawa Theory and p-adic Deformation of Motives", in
871:, vol. 1479, Springer Berlin Heidelberg, pp. 94–121,
1488:
Proceedings of a
Conference on Local Fields (Driebergen, 1966)
570:
proved this for some elliptic curves of rank at most 1 with
1484:"The conjectures of Birch and Swinnerton-Dyer, and of Tate"
1103:, London Mathematical Society Student Texts, vol. 24,
959:
578:
later showed that the modularity assumption always holds).
1532:"Duality theorems in Galois cohomology over number fields"
995:
458:
1455:
English translation in his collected mathematical papers
1275:(Thesis). Vol. 1940. University of Uppsala. 97 pp.
971:
221:(i.e., the real and complex completions as well as the
1558:(1955), "On algebraic groups and homogeneous spaces",
796:
1538:, Djursholm: Inst. Mittag-Leffler, pp. 288–295,
1197:
Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya
270:
140:
63:
844:
949:
947:
784:
215:, that become trivial in all of the completions of
932:
702:is skew symmetric which implies that the order of
396:
197:
126:
983:
820:
581:It is known that the Tate–Shafarevich group is a
447:", for Shafarevich, replacing the older notation
127:{\displaystyle \mathrm {WC} (A/K)=H^{1}(G_{K},A)}
1621:
944:
832:
554:of an abelian variety is closely related to the
1481:
1136:
1062:Journal für die reine und angewandte Mathematik
965:
808:
198:{\displaystyle G_{K}=\mathrm {Gal} (K^{alg}/K)}
1021:Proceedings of the London Mathematical Society
561:
1603:
1001:
865:"On the structure of shafarevich-tate groups"
1461:"Shafarevich–Tate groups of nonsquare order"
1429:
1287:
1173:; Jannsen, Uwe; Kleiman, Steven L. (eds.),
977:
802:
523:has solutions over the reals and over all
1609:
1405:
1303:
1194:
1165:
876:
862:
850:
631:is its dual. Cassels introduced this for
1215:
1054:
790:
1096:
1018:
938:
592:
1622:
1389:
906:
826:
668:, which induces a bilinear pairing on
459:Elements of the Tate–Shafarevich group
1458:
1343:
1146:Diophantine geometry: an introduction
1055:Cassels, John William Scott (1962b),
989:
838:
1554:
1526:
1505:
1468:Modular curves and abelian varieties
1268:
1097:Cassels, John William Scott (1991),
953:
814:
772:Birch and Swinnerton-Dyer conjecture
13:
288:
285:
282:
161:
158:
155:
68:
65:
14:
1646:
54:consists of the elements of the
1561:American Journal of Mathematics
1486:, in Springer, Tonny A. (ed.),
1226:American Journal of Mathematics
900:
856:
652:. A choice of polarization on
597:The Cassels–Tate pairing is a
333:
192:
165:
121:
102:
86:
72:
48:) defined over a number field
1:
1179:American Mathematical Society
1011:
407:This group was introduced by
1512:WC-groups over p-adic fields
1482:Swinnerton-Dyer, P. (1967),
907:Poonen, Bjorn (2024-09-01).
777:
721:there is an abelian variety
7:
1100:Lectures on elliptic curves
765:
562:Tate-Shafarevich conjecture
10:
1651:
1459:Stein, William A. (2004),
1432:Doklady Akademii Nauk SSSR
1105:Cambridge University Press
924:: CS1 maint: url-status (
625:is an abelian variety and
863:Kolyvagin, V. A. (1991),
1347:Inventiones Mathematicae
1269:Lind, Carl-Erik (1940).
1113:10.1017/CBO9781139172530
1075:10.1515/crll.1962.211.95
1033:10.1112/plms/s3-12.1.259
423:introduced the notation
1203:(3): 522–540, 670–671,
978:Poonen & Stoll 1999
641:can be identified with
572:complex multiplication
398:
199:
128:
22:Tate–Shafarevich group
1291:Annals of Mathematics
749: ⋅
399:
245:). Thus, in terms of
207:absolute Galois group
200:
129:
44:(or more generally a
1494:, pp. 132–157,
1490:, Berlin, New York:
1177:, Providence, R.I.:
1142:Silverman, Joseph H.
966:Swinnerton-Dyer 1967
791:Lang & Tate 1958
593:Cassels–Tate pairing
268:
138:
61:
1360:1987InMat..89..527R
56:Weil–Châtelet group
18:arithmetic geometry
1630:Algebraic geometry
1407:10.1007/BF02395746
1368:10.1007/BF01388984
1171:Serre, Jean-Pierre
1002:Konstantinous 2024
878:10.1007/bfb0086267
869:Algebraic Geometry
587:finitely generated
576:modularity theorem
394:
280:
261:can be defined as
195:
124:
1294:, Second Series,
1188:978-0-8218-1637-0
1159:978-0-387-98981-5
1122:978-0-521-41517-0
888:978-3-540-54456-2
656:gives a map from
271:
247:Galois cohomology
1642:
1615:
1613:
1600:
1551:
1546:, archived from
1523:
1502:
1478:
1465:
1454:
1426:
1409:
1393:Acta Mathematica
1386:
1340:
1307:
1298:(3): 1109–1149,
1284:
1265:
1212:
1191:
1167:Greenberg, Ralph
1162:
1133:
1093:
1051:
1023:, Third Series,
1005:
999:
993:
987:
981:
975:
969:
963:
957:
951:
942:
936:
930:
929:
923:
915:
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904:
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896:
895:
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860:
854:
848:
842:
836:
830:
824:
818:
812:
806:
803:Shafarevich 1959
800:
794:
788:
761:
757:
754:. In particular
753:
744:
738:
732:
726:
720:
710:
705:
701:
696:
692:
685:
675:
667:
661:
646:
640:
630:
624:
618:
599:bilinear pairing
553:
544:
528:
522:
512:
502:
496:
490:
477:
468:
454:
450:
438:
434:
417:Igor Shafarevich
403:
401:
400:
395:
390:
386:
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381:
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367:
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345:
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328:
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306:
305:
291:
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196:
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183:
182:
164:
150:
149:
133:
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130:
125:
114:
113:
101:
100:
82:
71:
53:
43:
34:
1650:
1649:
1645:
1644:
1643:
1641:
1640:
1639:
1620:
1619:
1618:
1574:10.2307/2372637
1492:Springer-Verlag
1463:
1239:10.2307/2372778
1189:
1160:
1150:Springer-Verlag
1123:
1069:(211): 95–112,
1014:
1009:
1008:
1000:
996:
988:
984:
976:
972:
964:
960:
952:
945:
937:
933:
917:
916:
911:
905:
901:
893:
891:
889:
861:
857:
849:
845:
837:
833:
825:
821:
813:
809:
801:
797:
789:
785:
780:
768:
759:
755:
742:
740:
734:
733:and an integer
728:
722:
716:
708:
703:
699:
694:
690:
677:
676:with values in
669:
663:
657:
642:
636:
633:elliptic curves
626:
620:
601:
595:
564:
549:
530:
524:
514:
508:
505:Hasse principle
498:
492:
486:
480:rational points
475:
470:
464:
461:
452:
448:
436:
424:
375:
371:
360:
356:
355:
351:
350:
346:
340:
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316:
312:
311:
307:
301:
297:
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275:
269:
266:
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240:
231:
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216:
210:
184:
172:
168:
154:
145:
141:
139:
136:
135:
109:
105:
96:
92:
78:
64:
62:
59:
58:
49:
39:
37:abelian variety
24:
12:
11:
5:
1648:
1638:
1637:
1632:
1617:
1616:
1601:
1568:(3): 493–512,
1552:
1524:
1503:
1479:
1456:
1434:(in Russian),
1427:
1387:
1354:(3): 527–559,
1341:
1314:10.2307/121064
1285:
1266:
1233:(3): 659–684,
1213:
1192:
1187:
1163:
1158:
1134:
1121:
1094:
1052:
1015:
1013:
1010:
1007:
1006:
994:
982:
970:
958:
943:
931:
899:
887:
855:
851:Kolyvagin 1988
843:
831:
819:
807:
795:
782:
781:
779:
776:
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767:
764:
594:
591:
563:
560:
473:
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457:
405:
404:
393:
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384:
378:
374:
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335:
331:
327:
324:
319:
315:
310:
304:
300:
295:
290:
287:
284:
278:
274:
230:obtained from
194:
191:
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178:
175:
171:
167:
163:
160:
157:
153:
148:
144:
123:
120:
117:
112:
108:
104:
99:
95:
91:
88:
85:
81:
77:
74:
70:
67:
9:
6:
4:
3:
2:
1647:
1636:
1635:Number theory
1633:
1631:
1628:
1627:
1625:
1612:
1607:
1602:
1599:
1595:
1591:
1587:
1583:
1579:
1575:
1571:
1567:
1563:
1562:
1557:
1553:
1550:on 2011-07-17
1549:
1545:
1541:
1537:
1533:
1529:
1525:
1522:
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1514:
1513:
1508:
1504:
1501:
1497:
1493:
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1462:
1457:
1453:
1449:
1445:
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1433:
1428:
1425:
1421:
1417:
1413:
1408:
1403:
1399:
1395:
1394:
1388:
1385:
1381:
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1373:
1369:
1365:
1361:
1357:
1353:
1349:
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1339:
1335:
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1327:
1323:
1319:
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1311:
1306:
1301:
1297:
1293:
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1139:
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1128:
1124:
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1101:
1095:
1092:
1088:
1084:
1080:
1076:
1072:
1068:
1064:
1063:
1058:
1053:
1050:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
1017:
1016:
1003:
998:
991:
986:
979:
974:
967:
962:
955:
950:
948:
940:
935:
927:
921:
910:
903:
890:
884:
879:
874:
870:
866:
859:
852:
847:
840:
835:
828:
823:
816:
811:
804:
799:
792:
787:
783:
773:
770:
769:
763:
752:
748:
737:
731:
727:defined over
725:
719:
714:
687:
684:
680:
673:
666:
660:
655:
651:
645:
639:
634:
629:
623:
617:
613:
609:
605:
600:
590:
588:
584:
583:torsion group
579:
577:
573:
569:
559:
557:
552:
546:
542:
538:
534:
527:
521:
517:
511:
506:
501:
495:
489:
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481:
476:
467:
456:
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442:
432:
428:
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418:
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376:
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329:
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322:
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308:
302:
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189:
185:
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118:
115:
110:
106:
97:
93:
89:
83:
79:
75:
57:
52:
47:
42:
38:
32:
28:
23:
19:
1565:
1559:
1548:the original
1535:
1511:
1487:
1467:
1435:
1431:
1397:
1391:
1351:
1345:
1305:math/9911267
1295:
1289:
1271:
1230:
1224:
1200:
1196:
1174:
1145:
1138:Hindry, Marc
1099:
1066:
1060:
1024:
1020:
997:
985:
973:
961:
939:Cassels 1962
934:
902:
892:, retrieved
868:
858:
846:
834:
822:
810:
798:
786:
750:
746:
735:
729:
723:
717:
688:
682:
678:
671:
664:
658:
653:
650:Tate duality
643:
637:
627:
621:
615:
611:
607:
603:
596:
580:
565:
556:Selmer group
550:
547:
540:
536:
532:
525:
519:
515:
509:
499:
493:
487:
471:
465:
462:
430:
426:
406:
256:
252:
241:
232:
228:-adic fields
224:
217:
211:
50:
46:group scheme
40:
30:
26:
21:
15:
1556:Weil, André
1400:: 203–362,
1217:Lang, Serge
1027:: 259–296,
827:Selmer 1951
1624:Categories
1611:2404.16785
1528:Tate, John
1507:Tate, John
1221:Tate, John
1012:References
990:Stein 2004
894:2024-09-01
839:Rubin 1987
713:squarefree
568:Karl Rubin
482:for every
469:that have
409:Serge Lang
238:valuations
1582:0002-9327
1444:0002-3264
1438:: 42–43,
1416:0001-5962
1376:0020-9910
1322:0003-486X
1247:0002-9327
1209:0373-2436
1083:0075-4102
1041:0024-6115
954:Tate 1963
815:Lind 1940
778:Citations
745:| =
497:, but no
413:John Tate
334:→
273:⋂
1530:(1963),
1509:(1958),
1211:, 954295
1144:(2000),
920:cite web
766:See also
619:, where
518:− 17 = 2
443:letter "
441:Cyrillic
435:, where
134:, where
1598:0074084
1590:2372637
1544:0175892
1521:0105420
1500:0230727
1476:2058655
1452:0106227
1424:0041871
1384:0903383
1356:Bibcode
1338:1740984
1281:0022563
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