290:
88:
486:
585:
753:)) is finite. There is a notorious problem about whether this subgroup can be effectively computed: there is a procedure for computing it that will terminate with the correct answer if there is some prime
657:
342:
1079:
1052:
285:{\displaystyle \operatorname {Sel} ^{(f)}(A/K)=\bigcap _{v}\ker(H^{1}(G_{K},\ker(f))\rightarrow H^{1}(G_{K_{v}},A_{v})/\operatorname {im} (\kappa _{v}))}
350:
1276:
974:
937:
491:
1261:
1123:
1240:
1306:
1204:
1072:
1245:
1230:
1266:
966:
1170:
1065:
804:
770:
762:
675:
921:
729:
The Selmer group in the middle of this exact sequence is finite and effectively computable. This implies the weak
730:
590:
1337:
1235:
1156:
1131:
991:
28:
1271:
1109:
796:
1199:
1114:
20:
1225:
1179:
320:
659:. Geometrically, the principal homogeneous spaces coming from elements of the Selmer group have
1291:
1281:
1027:
970:
958:
933:
892:
307:
79:
48:
1209:
1161:
1017:
1008:
925:
911:
884:
872:
36:
1039:
984:
947:
904:
1035:
980:
954:
943:
900:
784:
808:
683:
875:(1962), "Arithmetic on curves of genus 1. III. The Tate–Šafarevič and Selmer groups",
1331:
1140:
1095:
345:
1316:
915:
761:-component of the Tate–Shafarevich group is finite. It is conjectured that the
1057:
888:
1088:
481:{\displaystyle B_{v}(K_{v})/f(A_{v}(K_{v}))\rightarrow H^{1}(G_{K_{v}},A_{v})}
1031:
929:
896:
819:
More generally one can define the Selmer group of a finite Galois module
1022:
580:{\displaystyle H^{1}(G_{K_{v}},A_{v})/\operatorname {im} (\kappa _{v})}
64:
44:
674:. The Selmer group is finite. This implies that the part of the
1061:
957:(1994), "Iwasawa Theory and p-adic Deformation of Motives", in
791:) has generalized the notion of Selmer group to more general
920:, London Mathematical Society Student Texts, vol. 24,
823:(such as the kernel of an isogeny) as the elements of
840:) that have images inside certain given subgroups of
593:
494:
353:
323:
91:
1290:
1254:
1218:
1192:
1149:
1102:
651:
579:
480:
336:
284:
769:would work. However, if (as seems unlikely) the
78:of abelian varieties can be defined in terms of
877:Proceedings of the London Mathematical Society
1073:
8:
961:; Jannsen, Uwe; Kleiman, Steven L. (eds.),
765:is in fact finite, in which case any prime
1080:
1066:
1058:
815:The Selmer group of a finite Galois module
781:, then the procedure may never terminate.
1021:
788:
631:
616:
611:
598:
592:
568:
550:
532:
517:
512:
499:
493:
460:
445:
440:
427:
408:
395:
380:
371:
358:
352:
328:
322:
270:
252:
234:
219:
214:
201:
167:
154:
135:
117:
96:
90:
59:The Selmer group of an abelian variety
40:
1053:Wiles's proof of Fermat's Last Theorem
652:{\displaystyle H^{1}(G_{K_{v}},A_{v})}
32:
7:
1262:Birch and Swinnerton-Dyer conjecture
994:(1951), "The Diophantine equation
43:), is a group constructed from an
14:
1307:Main conjecture of Iwasawa theory
666:-rational points for all places
27:, named in honor of the work of
682:is finite due to the following
1241:Ramanujan–Petersson conjecture
1231:Generalized Riemann hypothesis
1127:-functions of Hecke characters
646:
640:
637:
604:
574:
561:
547:
544:
538:
505:
475:
472:
466:
433:
420:
417:
414:
401:
388:
377:
364:
279:
276:
263:
249:
246:
240:
207:
194:
191:
188:
182:
160:
147:
125:
111:
103:
97:
55:The Selmer group of an isogeny
37:John William Scott Cassels
1:
1200:Analytic class number formula
967:American Mathematical Society
1205:Riemann–von Mangoldt formula
917:Lectures on elliptic curves
912:Cassels, John William Scott
873:Cassels, John William Scott
777:-component for every prime
337:{\displaystyle \kappa _{v}}
1354:
922:Cambridge University Press
29:Ernst Sejersted Selmer
930:10.1017/CBO9781139172530
889:10.1112/plms/s3-12.1.259
16:Construct in mathematics
1157:Dedekind zeta functions
1006: = 0",
797:Galois representations
771:Tate–Shafarevich group
763:Tate–Shafarevich group
676:Tate–Shafarevich group
653:
581:
482:
338:
286:
1277:Bloch–Kato conjecture
1272:Beilinson conjectures
1255:Algebraic conjectures
1110:Riemann zeta function
654:
582:
483:
339:
287:
1282:Langlands conjecture
1267:Deligne's conjecture
1219:Analytic conjectures
965:, Providence, R.I.:
803:-adic variations of
731:Mordell–Weil theorem
591:
492:
351:
321:
89:
1236:Lindelöf hypothesis
785:Ralph Greenberg
63:with respect to an
21:arithmetic geometry
1226:Riemann hypothesis
1150:Algebraic examples
1023:10.1007/BF02395746
959:Serre, Jean-Pierre
807:in the context of
733:that its subgroup
649:
577:
478:
334:
282:
140:
1325:
1324:
1103:Analytic examples
976:978-0-8218-1637-0
939:978-0-521-41517-0
587:is isomorphic to
131:
80:Galois cohomology
49:abelian varieties
1345:
1246:Artin conjecture
1210:Weil conjectures
1082:
1075:
1068:
1059:
1042:
1025:
1009:Acta Mathematica
992:Selmer, Ernst S.
987:
955:Greenberg, Ralph
950:
907:
879:, Third Series,
773:has an infinite
658:
656:
655:
650:
636:
635:
623:
622:
621:
620:
603:
602:
586:
584:
583:
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573:
572:
554:
537:
536:
524:
523:
522:
521:
504:
503:
487:
485:
484:
479:
465:
464:
452:
451:
450:
449:
432:
431:
413:
412:
400:
399:
384:
376:
375:
363:
362:
343:
341:
340:
335:
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332:
291:
289:
288:
283:
275:
274:
256:
239:
238:
226:
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223:
206:
205:
172:
171:
159:
158:
139:
121:
107:
106:
1353:
1352:
1348:
1347:
1346:
1344:
1343:
1342:
1328:
1327:
1326:
1321:
1286:
1250:
1214:
1188:
1145:
1098:
1086:
1049:
990:
977:
953:
940:
910:
871:
868:
857:
856:
835:
817:
665:
627:
612:
607:
594:
589:
588:
564:
528:
513:
508:
495:
490:
489:
456:
441:
436:
423:
404:
391:
367:
354:
349:
348:
324:
319:
318:
316:
301:
266:
230:
215:
210:
197:
163:
150:
92:
87:
86:
57:
17:
12:
11:
5:
1351:
1349:
1341:
1340:
1330:
1329:
1323:
1322:
1320:
1319:
1314:
1309:
1303:
1301:
1288:
1287:
1285:
1284:
1279:
1274:
1269:
1264:
1258:
1256:
1252:
1251:
1249:
1248:
1243:
1238:
1233:
1228:
1222:
1220:
1216:
1215:
1213:
1212:
1207:
1202:
1196:
1194:
1190:
1189:
1187:
1186:
1177:
1168:
1159:
1153:
1151:
1147:
1146:
1144:
1143:
1138:
1129:
1121:
1112:
1106:
1104:
1100:
1099:
1087:
1085:
1084:
1077:
1070:
1062:
1056:
1055:
1048:
1045:
1044:
1043:
988:
975:
951:
938:
908:
867:
864:
852:
848:
831:
816:
813:
809:Iwasawa theory
757:such that the
727:
726:
684:exact sequence
663:
648:
645:
642:
639:
634:
630:
626:
619:
615:
610:
606:
601:
597:
576:
571:
567:
563:
560:
557:
553:
549:
546:
543:
540:
535:
531:
527:
520:
516:
511:
507:
502:
498:
477:
474:
471:
468:
463:
459:
455:
448:
444:
439:
435:
430:
426:
422:
419:
416:
411:
407:
403:
398:
394:
390:
387:
383:
379:
374:
370:
366:
361:
357:
331:
327:
314:
299:
293:
292:
281:
278:
273:
269:
265:
262:
259:
255:
251:
248:
245:
242:
237:
233:
229:
222:
218:
213:
209:
204:
200:
196:
193:
190:
187:
184:
181:
178:
175:
170:
166:
162:
157:
153:
149:
146:
143:
138:
134:
130:
127:
124:
120:
116:
113:
110:
105:
102:
99:
95:
56:
53:
15:
13:
10:
9:
6:
4:
3:
2:
1350:
1339:
1338:Number theory
1336:
1335:
1333:
1318:
1315:
1313:
1310:
1308:
1305:
1304:
1302:
1300:
1298:
1294:
1289:
1283:
1280:
1278:
1275:
1273:
1270:
1268:
1265:
1263:
1260:
1259:
1257:
1253:
1247:
1244:
1242:
1239:
1237:
1234:
1232:
1229:
1227:
1224:
1223:
1221:
1217:
1211:
1208:
1206:
1203:
1201:
1198:
1197:
1195:
1191:
1185:
1183:
1178:
1176:
1174:
1169:
1167:
1165:
1160:
1158:
1155:
1154:
1152:
1148:
1142:
1141:Selberg class
1139:
1137:
1135:
1130:
1128:
1126:
1122:
1120:
1118:
1113:
1111:
1108:
1107:
1105:
1101:
1097:
1096:number theory
1093:
1091:
1083:
1078:
1076:
1071:
1069:
1064:
1063:
1060:
1054:
1051:
1050:
1046:
1041:
1037:
1033:
1029:
1024:
1019:
1015:
1011:
1010:
1005:
1002: +
1001:
998: +
997:
993:
989:
986:
982:
978:
972:
968:
964:
960:
956:
952:
949:
945:
941:
935:
931:
927:
923:
919:
918:
913:
909:
906:
902:
898:
894:
890:
886:
882:
878:
874:
870:
869:
865:
863:
861:
855:
851:
847:
843:
839:
834:
830:
826:
822:
814:
812:
810:
806:
802:
798:
794:
790:
786:
782:
780:
776:
772:
768:
764:
760:
756:
752:
748:
744:
740:
736:
732:
724:
720:
716:
712:
708:
704:
700:
696:
692:
688:
687:
686:
685:
681:
677:
673:
669:
662:
643:
632:
628:
624:
617:
613:
608:
599:
595:
569:
565:
558:
555:
551:
541:
533:
529:
525:
518:
514:
509:
500:
496:
469:
461:
457:
453:
446:
442:
437:
428:
424:
409:
405:
396:
392:
385:
381:
372:
368:
359:
355:
347:
344:is the local
329:
325:
313:
309:
305:
298:
271:
267:
260:
257:
253:
243:
235:
231:
227:
220:
216:
211:
202:
198:
185:
179:
176:
173:
168:
164:
155:
151:
144:
141:
136:
132:
128:
122:
118:
114:
108:
100:
93:
85:
84:
83:
81:
77:
74: →
73:
70: :
69:
66:
62:
54:
52:
50:
46:
42:
38:
34:
30:
26:
22:
1317:Euler system
1312:Selmer group
1311:
1296:
1292:
1181:
1172:
1163:
1133:
1132:Automorphic
1124:
1116:
1089:
1013:
1007:
1003:
999:
995:
962:
916:
880:
876:
859:
853:
849:
845:
841:
837:
832:
828:
824:
820:
818:
800:
792:
783:
778:
774:
766:
758:
754:
750:
746:
742:
738:
734:
728:
722:
718:
714:
710:
706:
702:
698:
694:
690:
679:
671:
667:
660:
488:. Note that
311:
303:
302:denotes the
296:
294:
75:
71:
67:
60:
58:
25:Selmer group
24:
18:
1171:Hasse–Weil
1016:: 203–362,
883:: 259–296,
1299:-functions
1184:-functions
1175:-functions
1166:-functions
1136:-functions
1119:-functions
1115:Dirichlet
1092:-functions
866:References
678:killed by
346:Kummer map
1032:0001-5962
897:0024-6115
709:)) → Sel(
566:κ
559:
421:→
326:κ
268:κ
261:
195:→
180:
145:
133:⋂
109:
1332:Category
1193:Theorems
1180:Motivic
1047:See also
914:(1991),
1040:0041871
985:1265554
963:Motives
948:1144763
905:0163913
805:motives
799:and to
787: (
308:torsion
65:isogeny
45:isogeny
39: (
31: (
23:, the
1295:-adic
1162:Artin
1038:
1030:
983:
973:
946:
936:
903:
895:
795:-adic
725:) → 0.
717:) → Ш(
295:where
35:) by
1028:ISSN
971:ISBN
934:ISBN
893:ISSN
789:1994
689:0 →
317:and
41:1962
33:1951
1094:in
1018:doi
926:doi
885:doi
862:).
670:of
310:of
177:ker
142:ker
94:Sel
82:as
47:of
19:In
1334::
1036:MR
1034:,
1026:,
1014:85
1012:,
1004:cz
1000:by
996:ax
981:MR
979:,
969:,
944:MR
942:,
932:,
924:,
901:MR
899:,
891:,
881:12
811:.
741:)/
697:)/
556:im
258:im
51:.
1297:L
1293:p
1182:L
1173:L
1164:L
1134:L
1125:L
1117:L
1090:L
1081:e
1074:t
1067:v
1020::
928::
887::
860:M
858:,
854:v
850:K
846:G
844:(
842:H
838:M
836:,
833:K
829:G
827:(
825:H
821:M
801:p
793:p
779:p
775:p
767:p
759:p
755:p
751:K
749:(
747:A
745:(
743:f
739:K
737:(
735:B
723:K
721:/
719:A
715:K
713:/
711:A
707:K
705:(
703:A
701:(
699:f
695:K
693:(
691:B
680:f
672:K
668:v
664:v
661:K
647:]
644:f
641:[
638:)
633:v
629:A
625:,
618:v
614:K
609:G
605:(
600:1
596:H
575:)
570:v
562:(
552:/
548:)
545:]
542:f
539:[
534:v
530:A
526:,
519:v
515:K
510:G
506:(
501:1
497:H
476:)
473:]
470:f
467:[
462:v
458:A
454:,
447:v
443:K
438:G
434:(
429:1
425:H
418:)
415:)
410:v
406:K
402:(
397:v
393:A
389:(
386:f
382:/
378:)
373:v
369:K
365:(
360:v
356:B
330:v
315:v
312:A
306:-
304:f
300:v
297:A
280:)
277:)
272:v
264:(
254:/
250:)
247:]
244:f
241:[
236:v
232:A
228:,
221:v
217:K
212:G
208:(
203:1
199:H
192:)
189:)
186:f
183:(
174:,
169:K
165:G
161:(
156:1
152:H
148:(
137:v
129:=
126:)
123:K
119:/
115:A
112:(
104:)
101:f
98:(
76:B
72:A
68:f
61:A
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