708:
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
413:
658:
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
717:
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
278:
143:
214:
709:
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
609:
1072:
1148:
769:
is finite in
Konstantinous' examples and these examples confirm a conjecture of Stein. Thus modulo squares any integer can be the order of
704:
is finite then it is a square. For more general abelian varieties it was sometimes incorrectly believed for many years that the order of
17:
408:{\displaystyle \bigcap _{v}\mathrm {ker} \left(H^{1}\left(G_{K},A\right)\rightarrow H^{1}\left(G_{K_{v}},A_{v}\right)\right).}
1197:
1168:
1131:
897:
782:
559:
The special case of the Tate–Shafarevich group for the finite group scheme consisting of points of some given finite order
722:
is a square (if it is finite). On the other hand building on the results just presented
Konstantinous showed that for any
71:
474:
Geometrically, the non-trivial elements of the Tate–Shafarevich group can be thought of as the homogeneous spaces of
148:
597:
1030:
Cassels, John
William Scott (1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups",
585:. Victor A. Kolyvagin extended this to modular elliptic curves over the rationals of analytic rank at most 1 (The
1571:
1542:
1236:
700:
For an elliptic curve, Cassels showed that the pairing is alternating, and a consequence is that if the order of
936:
1355:
Rubin, Karl (1987), "Tate–Shafarevich groups and L-functions of elliptic curves with complex multiplication",
1189:
1615:
Konstantinous, Alexandros (2024-04-25). "A note on the order of the Tate-Shafarevich group modulo squares".
1640:
1115:
66:
1538:
1517:
1231:
423:
1357:
1299:
Poonen, Bjorn; Stoll, Michael (1999), "The
Cassels-Tate pairing on polarized abelian varieties",
524:. Carl-Erik Lind gave an example of such a homogeneous space, by showing that the genus 1 curve
1645:
582:
697:, but unlike the case of elliptic curves this need not be alternating or even skew symmetric.
1301:
1067:
217:
1608:
1554:
1531:
1526:, Séminaire Bourbaki; 10e année: 1957/1958, vol. 13, Paris: Secrétariat Mathématique,
1510:
1486:
1462:
1434:
1394:
1366:
1348:
1291:
1273:
1141:
1101:
1059:
540:-adic fields, but has no rational points. Ernst S. Selmer gave many more examples, such as
8:
1283:
Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom
Geschlecht Eins
1206:
Kolyvagin, V. A. (1988), "Finiteness of E(Q) and SH(E,Q) for a subclass of Weil curves",
1152:
494:
248:
28:
1370:
1616:
1596:
1521:
1336:
1310:
1261:
586:
1588:
1450:
1441:
Shafarevich, I. R. (1959), "The group of principal homogeneous algebraic manifolds",
1422:
1382:
1328:
1253:
1215:
1193:
1181:
1164:
1127:
1089:
1047:
930:
893:
257:
1481:, Progr. Math., vol. 224, Basel, Boston, Berlin: Birkhäuser, pp. 277–289,
1580:
1471:
1412:
1403:
1374:
1320:
1245:
1119:
1081:
1039:
883:
431:
427:
1494:
577:
The Tate–Shafarevich conjecture states that the Tate–Shafarevich group is finite.
1604:
1550:
1527:
1506:
1502:
1482:
1458:
1430:
1390:
1344:
1287:
1281:
1269:
1177:
1160:
1137:
1109:
1097:
1055:
920:"THE SELMER GROUP, THE SHAFAREVICH-TATE GROUP, AND THE WEAK MORDELL-WEIL THEOREM"
515:
47:
1043:
643:
490:
455:
919:
875:
1634:
1592:
1547:
Proceedings of the
International Congress of Mathematicians (Stockholm, 1962)
1454:
1426:
1386:
1332:
1257:
1219:
1123:
1093:
1085:
1051:
593:
1566:
660:
566:
233:
56:
1600:
1417:
1378:
1265:
1227:
888:
723:
578:
419:
1558:
1340:
247:
by completing with respect to all its
Archimedean and non Archimedean
1315:
1584:
1249:
518:
fails to hold for rational equations with coefficients in the field
1621:
1401:
Selmer, Ernst S. (1951), "The
Diophantine equation ax³+by³+cz³=0",
1324:
514:-rational point. Thus, the group measures the extent to which the
451:
1159:, Graduate Texts in Mathematics, vol. 201, Berlin, New York:
1068:"Arithmetic on curves of genus 1. IV. Proof of the Hauptvermutung"
596:, thus the conjecture is equivalent to stating that the group is
1234:(1958), "Principal homogeneous spaces over abelian varieties",
1180:(1994), "Iwasawa Theory and p-adic Deformation of Motives", in
882:, vol. 1479, Springer Berlin Heidelberg, pp. 94–121,
1499:
Proceedings of a
Conference on Local Fields (Driebergen, 1966)
581:
proved this for some elliptic curves of rank at most 1 with
1495:"The conjectures of Birch and Swinnerton-Dyer, and of Tate"
1114:, London Mathematical Society Student Texts, vol. 24,
970:
589:
later showed that the modularity assumption always holds).
1543:"Duality theorems in Galois cohomology over number fields"
1006:
469:
1466:
English translation in his collected mathematical papers
1286:(Thesis). Vol. 1940. University of Uppsala. 97 pp.
982:
232:(i.e., the real and complex completions as well as the
1569:(1955), "On algebraic groups and homogeneous spaces",
807:
1549:, Djursholm: Inst. Mittag-Leffler, pp. 288–295,
1208:
Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya
281:
151:
74:
855:
960:
958:
795:
226:, that become trivial in all of the completions of
943:
713:is skew symmetric which implies that the order of
407:
208:
137:
994:
831:
592:It is known that the Tate–Shafarevich group is a
458:", for Shafarevich, replacing the older notation
138:{\displaystyle \mathrm {WC} (A/K)=H^{1}(G_{K},A)}
1632:
955:
843:
565:of an abelian variety is closely related to the
1492:
1147:
1073:Journal für die reine und angewandte Mathematik
976:
819:
209:{\displaystyle G_{K}=\mathrm {Gal} (K^{alg}/K)}
1032:Proceedings of the London Mathematical Society
572:
1614:
1012:
876:"On the structure of shafarevich-tate groups"
1472:"Shafarevich–Tate groups of nonsquare order"
1440:
1298:
1184:; Jannsen, Uwe; Kleiman, Steven L. (eds.),
988:
813:
534:has solutions over the reals and over all
1620:
1416:
1314:
1205:
1176:
887:
873:
861:
642:is its dual. Cassels introduced this for
1226:
1065:
801:
1107:
1029:
949:
603:
14:
1633:
1400:
917:
837:
679:, which induces a bilinear pairing on
470:Elements of the Tate–Shafarevich group
1469:
1354:
1157:Diophantine geometry: an introduction
1066:Cassels, John William Scott (1962b),
1000:
849:
1565:
1537:
1516:
1479:Modular curves and abelian varieties
1279:
1108:Cassels, John William Scott (1991),
964:
825:
783:Birch and Swinnerton-Dyer conjecture
24:
299:
296:
293:
172:
169:
166:
79:
76:
25:
1657:
65:consists of the elements of the
1572:American Journal of Mathematics
1497:, in Springer, Tonny A. (ed.),
1237:American Journal of Mathematics
911:
867:
663:. A choice of polarization on
608:The Cassels–Tate pairing is a
344:
203:
176:
132:
113:
97:
83:
59:) defined over a number field
13:
1:
1190:American Mathematical Society
1022:
418:This group was introduced by
1523:WC-groups over p-adic fields
1493:Swinnerton-Dyer, P. (1967),
918:Poonen, Bjorn (2024-09-01).
788:
732:there is an abelian variety
7:
1111:Lectures on elliptic curves
776:
573:Tate-Shafarevich conjecture
10:
1662:
1470:Stein, William A. (2004),
1443:Doklady Akademii Nauk SSSR
1116:Cambridge University Press
935:: CS1 maint: url-status (
636:is an abelian variety and
874:Kolyvagin, V. A. (1991),
1358:Inventiones Mathematicae
1280:Lind, Carl-Erik (1940).
1124:10.1017/CBO9781139172530
1086:10.1515/crll.1962.211.95
1044:10.1112/plms/s3-12.1.259
434:introduced the notation
1214:(3): 522–540, 670–671,
989:Poonen & Stoll 1999
652:can be identified with
583:complex multiplication
409:
210:
139:
33:Tate–Shafarevich group
1302:Annals of Mathematics
760: ⋅
410:
256:). Thus, in terms of
218:absolute Galois group
211:
140:
55:(or more generally a
1505:, pp. 132–157,
1501:, Berlin, New York:
1188:, Providence, R.I.:
1153:Silverman, Joseph H.
977:Swinnerton-Dyer 1967
802:Lang & Tate 1958
604:Cassels–Tate pairing
279:
149:
72:
18:Cassels–Tate pairing
1371:1987InMat..89..527R
67:Weil–Châtelet group
29:arithmetic geometry
1641:Algebraic geometry
1418:10.1007/BF02395746
1379:10.1007/BF01388984
1182:Serre, Jean-Pierre
1013:Konstantinous 2024
889:10.1007/bfb0086267
880:Algebraic Geometry
598:finitely generated
587:modularity theorem
405:
291:
272:can be defined as
206:
135:
1305:, Second Series,
1199:978-0-8218-1637-0
1170:978-0-387-98981-5
1133:978-0-521-41517-0
899:978-3-540-54456-2
667:gives a map from
282:
258:Galois cohomology
16:(Redirected from
1653:
1626:
1624:
1611:
1562:
1557:, archived from
1534:
1513:
1489:
1476:
1465:
1437:
1420:
1404:Acta Mathematica
1397:
1351:
1318:
1309:(3): 1109–1149,
1295:
1276:
1223:
1202:
1178:Greenberg, Ralph
1173:
1144:
1104:
1062:
1034:, Third Series,
1016:
1010:
1004:
998:
992:
986:
980:
974:
968:
962:
953:
947:
941:
940:
934:
926:
924:
915:
909:
908:
907:
906:
891:
871:
865:
859:
853:
847:
841:
835:
829:
823:
817:
814:Shafarevich 1959
811:
805:
799:
772:
768:
765:. In particular
764:
755:
749:
743:
737:
731:
721:
716:
712:
707:
703:
696:
686:
678:
672:
657:
651:
641:
635:
629:
610:bilinear pairing
564:
555:
539:
533:
523:
513:
507:
501:
488:
479:
465:
461:
449:
445:
428:Igor Shafarevich
414:
412:
411:
406:
401:
397:
396:
392:
391:
390:
378:
377:
376:
375:
356:
355:
343:
339:
332:
331:
317:
316:
302:
290:
271:
255:
246:
238:
231:
225:
215:
213:
212:
207:
199:
194:
193:
175:
161:
160:
144:
142:
141:
136:
125:
124:
112:
111:
93:
82:
64:
54:
45:
21:
1661:
1660:
1656:
1655:
1654:
1652:
1651:
1650:
1631:
1630:
1629:
1585:10.2307/2372637
1503:Springer-Verlag
1474:
1250:10.2307/2372778
1200:
1171:
1161:Springer-Verlag
1134:
1080:(211): 95–112,
1025:
1020:
1019:
1011:
1007:
999:
995:
987:
983:
975:
971:
963:
956:
948:
944:
928:
927:
922:
916:
912:
904:
902:
900:
872:
868:
860:
856:
848:
844:
836:
832:
824:
820:
812:
808:
800:
796:
791:
779:
770:
766:
753:
751:
745:
744:and an integer
739:
733:
727:
719:
714:
710:
705:
701:
688:
687:with values in
680:
674:
668:
653:
647:
644:elliptic curves
637:
631:
612:
606:
575:
560:
541:
535:
525:
519:
516:Hasse principle
509:
503:
497:
491:rational points
486:
481:
475:
472:
463:
459:
447:
435:
386:
382:
371:
367:
366:
362:
361:
357:
351:
347:
327:
323:
322:
318:
312:
308:
307:
303:
292:
286:
280:
277:
276:
261:
251:
242:
234:
227:
221:
195:
183:
179:
165:
156:
152:
150:
147:
146:
120:
116:
107:
103:
89:
75:
73:
70:
69:
60:
50:
48:abelian variety
35:
23:
22:
15:
12:
11:
5:
1659:
1649:
1648:
1643:
1628:
1627:
1612:
1579:(3): 493–512,
1563:
1535:
1514:
1490:
1467:
1445:(in Russian),
1438:
1398:
1365:(3): 527–559,
1352:
1325:10.2307/121064
1296:
1277:
1244:(3): 659–684,
1224:
1203:
1198:
1174:
1169:
1145:
1132:
1105:
1063:
1026:
1024:
1021:
1018:
1017:
1005:
993:
981:
969:
954:
942:
910:
898:
866:
862:Kolyvagin 1988
854:
842:
830:
818:
806:
793:
792:
790:
787:
786:
785:
778:
775:
605:
602:
574:
571:
484:
471:
468:
416:
415:
404:
400:
395:
389:
385:
381:
374:
370:
365:
360:
354:
350:
346:
342:
338:
335:
330:
326:
321:
315:
311:
306:
301:
298:
295:
289:
285:
241:obtained from
205:
202:
198:
192:
189:
186:
182:
178:
174:
171:
168:
164:
159:
155:
134:
131:
128:
123:
119:
115:
110:
106:
102:
99:
96:
92:
88:
85:
81:
78:
9:
6:
4:
3:
2:
1658:
1647:
1646:Number theory
1644:
1642:
1639:
1638:
1636:
1623:
1618:
1613:
1610:
1606:
1602:
1598:
1594:
1590:
1586:
1582:
1578:
1574:
1573:
1568:
1564:
1561:on 2011-07-17
1560:
1556:
1552:
1548:
1544:
1540:
1536:
1533:
1529:
1525:
1524:
1519:
1515:
1512:
1508:
1504:
1500:
1496:
1491:
1488:
1484:
1480:
1473:
1468:
1464:
1460:
1456:
1452:
1448:
1444:
1439:
1436:
1432:
1428:
1424:
1419:
1414:
1410:
1406:
1405:
1399:
1396:
1392:
1388:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1359:
1353:
1350:
1346:
1342:
1338:
1334:
1330:
1326:
1322:
1317:
1312:
1308:
1304:
1303:
1297:
1293:
1289:
1285:
1284:
1278:
1275:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1243:
1239:
1238:
1233:
1229:
1225:
1221:
1217:
1213:
1209:
1204:
1201:
1195:
1191:
1187:
1183:
1179:
1175:
1172:
1166:
1162:
1158:
1154:
1150:
1146:
1143:
1139:
1135:
1129:
1125:
1121:
1117:
1113:
1112:
1106:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1075:
1074:
1069:
1064:
1061:
1057:
1053:
1049:
1045:
1041:
1037:
1033:
1028:
1027:
1014:
1009:
1002:
997:
990:
985:
978:
973:
966:
961:
959:
951:
946:
938:
932:
921:
914:
901:
895:
890:
885:
881:
877:
870:
863:
858:
851:
846:
839:
834:
827:
822:
815:
810:
803:
798:
794:
784:
781:
780:
774:
763:
759:
748:
742:
738:defined over
736:
730:
725:
698:
695:
691:
684:
677:
671:
666:
662:
656:
650:
645:
640:
634:
628:
624:
620:
616:
611:
601:
599:
595:
594:torsion group
590:
588:
584:
580:
570:
568:
563:
557:
553:
549:
545:
538:
532:
528:
522:
517:
512:
506:
500:
496:
492:
487:
478:
467:
457:
453:
443:
439:
433:
429:
425:
421:
402:
398:
393:
387:
383:
379:
372:
368:
363:
358:
352:
348:
340:
336:
333:
328:
324:
319:
313:
309:
304:
287:
283:
275:
274:
273:
269:
265:
259:
254:
250:
245:
240:
237:
230:
224:
219:
200:
196:
190:
187:
184:
180:
162:
157:
153:
129:
126:
121:
117:
108:
104:
100:
94:
90:
86:
68:
63:
58:
53:
49:
43:
39:
34:
30:
19:
1576:
1570:
1559:the original
1546:
1522:
1498:
1478:
1446:
1442:
1408:
1402:
1362:
1356:
1316:math/9911267
1306:
1300:
1282:
1241:
1235:
1211:
1207:
1185:
1156:
1149:Hindry, Marc
1110:
1077:
1071:
1035:
1031:
1008:
996:
984:
972:
950:Cassels 1962
945:
913:
903:, retrieved
879:
869:
857:
845:
833:
821:
809:
797:
761:
757:
746:
740:
734:
728:
699:
693:
689:
682:
675:
669:
664:
661:Tate duality
654:
648:
638:
632:
626:
622:
618:
614:
607:
591:
576:
567:Selmer group
561:
558:
551:
547:
543:
536:
530:
526:
520:
510:
504:
498:
482:
476:
473:
441:
437:
417:
267:
263:
252:
243:
239:-adic fields
235:
228:
222:
61:
57:group scheme
51:
41:
37:
32:
26:
1567:Weil, André
1411:: 203–362,
1228:Lang, Serge
1038:: 259–296,
838:Selmer 1951
1635:Categories
1622:2404.16785
1539:Tate, John
1518:Tate, John
1232:Tate, John
1023:References
1001:Stein 2004
905:2024-09-01
850:Rubin 1987
724:squarefree
579:Karl Rubin
493:for every
480:that have
420:Serge Lang
249:valuations
1593:0002-9327
1455:0002-3264
1449:: 42–43,
1427:0001-5962
1387:0020-9910
1333:0003-486X
1258:0002-9327
1220:0373-2436
1094:0075-4102
1052:0024-6115
965:Tate 1963
826:Lind 1940
789:Citations
756:| =
508:, but no
424:John Tate
345:→
284:⋂
1541:(1963),
1520:(1958),
1222:, 954295
1155:(2000),
931:cite web
777:See also
630:, where
529:− 17 = 2
454:letter "
452:Cyrillic
446:, where
145:, where
1609:0074084
1601:2372637
1555:0175892
1532:0105420
1511:0230727
1487:2058655
1463:0106227
1435:0041871
1395:0903383
1367:Bibcode
1349:1740984
1292:0022563
1274:0106226
1266:2372778
1186:Motives
1142:1144763
1102:0163915
1060:0163913
726:number
646:, when
450:is the
432:Cassels
216:is the
31:, the
1607:
1599:
1591:
1553:
1530:
1509:
1485:
1461:
1453:
1433:
1425:
1393:
1385:
1347:
1341:121064
1339:
1331:
1290:
1272:
1264:
1256:
1218:
1196:
1167:
1140:
1130:
1100:
1092:
1058:
1050:
896:
752:|
617:) × Ш(
46:of an
1617:arXiv
1597:JSTOR
1475:(PDF)
1337:JSTOR
1311:arXiv
1262:JSTOR
923:(PDF)
750:with
495:place
1589:ISSN
1451:ISSN
1423:ISSN
1383:ISSN
1329:ISSN
1254:ISSN
1216:ISSN
1194:ISBN
1165:ISBN
1128:ISBN
1090:ISSN
1048:ISSN
937:link
894:ISBN
621:) →
426:and
422:and
1581:doi
1447:124
1413:doi
1375:doi
1321:doi
1307:150
1246:doi
1120:doi
1082:doi
1078:211
1040:doi
884:doi
673:to
554:= 0
550:+ 5
546:+ 4
502:of
462:or
456:Sha
220:of
27:In
1637::
1605:MR
1603:,
1595:,
1587:,
1577:77
1575:,
1551:MR
1545:,
1528:MR
1507:MR
1483:MR
1477:,
1459:MR
1457:,
1431:MR
1429:,
1421:,
1409:85
1407:,
1391:MR
1389:,
1381:,
1373:,
1363:89
1361:,
1345:MR
1343:,
1335:,
1327:,
1319:,
1288:MR
1270:MR
1268:,
1260:,
1252:,
1242:80
1240:,
1230:;
1212:52
1210:,
1192:,
1163:,
1151:;
1138:MR
1136:,
1126:,
1118:,
1098:MR
1096:,
1088:,
1076:,
1070:,
1056:MR
1054:,
1046:,
1036:12
957:^
933:}}
929:{{
892:,
878:,
773:.
681:Ш(
613:Ш(
600:.
569:.
556:.
466:.
464:TŠ
460:TS
436:Ш(
430:.
262:Ш(
260:,
36:Ш(
1625:.
1619::
1583::
1415::
1377::
1369::
1323::
1313::
1294:.
1248::
1122::
1084::
1042::
1015:.
1003:.
991:.
979:.
967:.
952:.
939:)
925:.
886::
864:.
852:.
840:.
828:.
816:.
804:.
771:Ш
767:Ш
762:m
758:n
754:Ш
747:m
741:Q
735:A
729:n
720:Ш
715:Ш
711:Ш
706:Ш
702:Ш
694:Z
692:/
690:Q
685:)
683:A
676:Â
670:A
665:A
655:Â
649:A
639:Â
633:A
627:Z
625:/
623:Q
619:Â
615:A
562:n
552:z
548:y
544:x
542:3
537:p
531:y
527:x
521:K
511:K
505:K
499:v
489:-
485:v
483:K
477:A
448:Ш
444:)
442:K
440:/
438:A
403:.
399:)
394:)
388:v
384:A
380:,
373:v
369:K
364:G
359:(
353:1
349:H
341:)
337:A
334:,
329:K
325:G
320:(
314:1
310:H
305:(
300:r
297:e
294:k
288:v
270:)
268:K
266:/
264:A
253:v
244:K
236:p
229:K
223:K
204:)
201:K
197:/
191:g
188:l
185:a
181:K
177:(
173:l
170:a
167:G
163:=
158:K
154:G
133:)
130:A
127:,
122:K
118:G
114:(
109:1
105:H
101:=
98:)
95:K
91:/
87:A
84:(
80:C
77:W
62:K
52:A
44:)
42:K
40:/
38:A
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.