407:
468:
refers to the elements in an Euler system as "arithmetic incarnations of zeta" and describes the property of being an Euler system as "an arithmetic reflection of the fact that these incarnations are related to special values of Euler products".
107:
Although there are several definitions of special sorts of Euler system, there seems to be no published definition of an Euler system that covers all known cases. But it is possible to say roughly what an Euler system is, as follows:
655:
198:
699:
970:-adic Hodge theory and values of zeta functions of modular forms", in Pierre Berthelot; Jean-Marc Fontaine; Luc Illusie; Kazuya Kato; Michael Rapoport (eds.),
540:
402:{\displaystyle {\rm {cor}}_{G/F}(c_{G})=\prod _{q\in \Sigma (G/F)}P(\mathrm {Fr} _{q}^{-1}|{\rm {Hom}}_{O}(T,O(1));\mathrm {Fr} _{q}^{-1})c_{F}}
1288:
1485:
1256:
1179:
1117:
1066:
1470:
1332:
1449:
1515:
954:
929:
869:
817:
88:
661:
1546:
1413:
1281:
1454:
1439:
1475:
879:
79:, thus giving bounds on their orders, which in turn has led to deep theorems such as the finiteness of some
1379:
1274:
938:
905:
874:
1171:
1149:
1021:
Kolyvagin, V. A. (1988), "The
Mordell-Weil and Shafarevich-Tate groups for Weil elliptic curves",
1099:
1087:
1365:
1340:
1197:
778:
80:
45:
1444:
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1202:
155:
1408:
1323:
1239:
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1007:
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898:
8:
886:
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1227:
909:
793:
990:
974:, Astérisque, vol. 295, Paris: Société Mathématique de France, pp. 117–290,
1500:
1490:
1219:
1175:
1123:
1113:
1062:
1030:
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925:
72:
25:
1253:
1165:
650:{\displaystyle N_{Q(\zeta _{nl})/Q(\zeta _{l})}(\alpha _{nl})=\alpha _{n}^{F_{l}-1}}
1418:
1370:
1211:
1103:
1054:
942:
889:(1984), "Higher regulators and values of L-functions", in R. V. Gamkrelidze (ed.),
33:
994:
1260:
1235:
1185:
1153:
1143:
1131:
1072:
1058:
1038:
1003:
975:
921:
894:
752:
60:
because the factors relating different elements of an Euler system resemble the
29:
1266:
1053:, Progr. Math., vol. 87, Boston, MA: Birkhäuser Boston, pp. 435–483,
821:
57:
1540:
1349:
1304:
1223:
1127:
1034:
797:
774:
129:, or by something closely related such as square-free integers. The elements
65:
41:
1520:
96:
76:
61:
1083:
986:
963:
465:
92:
17:
1108:
1002:, vol. I, Zürich: European Mathematical Society, pp. 335–357,
1297:
1231:
949:, Springer Monographs in Mathematics, Springer-Verlag, pp. 71–87,
913:
84:
1167:
Galois representations in arithmetic algebraic geometry (Durham, 1996)
751:. Kolyvagin used this Euler system to give an elementary proof of the
1215:
777:
of an elliptic curve, and used this to show that in some cases the
522:
coprime. Then the cyclotomic Euler system is the set of numbers α
138:
are typically elements of some Galois cohomology group such as H(
1164:
Scholl, A. J. (1998), "An introduction to Kato's Euler systems",
1017:. Proceedings of the congress held in Madrid, August 22–30, 2006
121:. These elements are often indexed by certain number fields
972:
Cohomologies p-adiques et applications arithmétiques. III.
1198:"On the ideal class groups of real abelian number fields"
904:
71:
Euler systems can be used to construct annihilators of
1170:, London Math. Soc. Lecture Note Ser., vol. 254,
993:; Javier Soria; Juan Luis Varona; et al. (eds.),
1252:
Several papers on
Kolyvagin systems are available at
1023:
Izvestiya
Akademii Nauk SSSR. Seriya Matematicheskaya
664:
543:
201:
91:, considered simpler than the original proof due to
112:An Euler system is given by collection of elements
693:
649:
401:
165:The most important condition is that the elements
989:(2007), "Iwasawa theory and generalizations", in
1538:
1296:
1148:, Annals of Mathematics Studies, vol. 147,
920:, Lecture Notes in Mathematics, vol. 1716,
773:Kolyvagin constructed an Euler system from the
461:have to satisfy, such as congruence conditions.
893:(in Russian), vol. 24, pp. 181–238,
792:consists of certain elements occurring in the
694:{\displaystyle \alpha _{nl}\equiv \alpha _{n}}
1282:
937:
1092:Memoirs of the American Mathematical Society
431:) considered as an element of O, which when
816:to prove one divisibility in Barry Mazur's
423:) is defined to be the element det(1-τ
48:, which was motivated by his earlier paper
1289:
1275:
1082:
1049:Kolyvagin, V. A. (1990), "Euler systems",
24:is a collection of compatible elements of
1107:
1048:
1020:
885:
809:
49:
37:
996:International Congress of Mathematicians
867:
191:are related by a simple formula, such as
482:For every square-free positive integer
452:There may be other conditions that the
1539:
1195:
1163:
784:
53:
1270:
1141:
1051:The Grothendieck Festschrift, Vol. II
1471:Birch and Swinnerton-Dyer conjecture
985:
962:
918:Arithmetic Theory of Elliptic Curves
852:
840:
813:
812:—were used by Kazuya Kato in
477:
125:containing some fixed number field
13:
368:
365:
325:
322:
319:
293:
290:
260:
211:
208:
205:
14:
1558:
1516:Main conjecture of Iwasawa theory
1246:
947:Cyclotomic Fields and Zeta Values
870:"Euler systems for number fields"
818:main conjecture of Iwasawa theory
768:
763:
725:is a Frobenius automorphism with
89:main conjecture of Iwasawa theory
56:. Euler systems are named after
447:) considered as an element of O.
891:Current problems in mathematics
439:is not the same as det(1-τ
1450:Ramanujan–Petersson conjecture
1440:Generalized Riemann hypothesis
1336:-functions of Hecke characters
846:
834:
613:
597:
592:
579:
568:
552:
534:. These satisfy the relations
386:
357:
354:
348:
336:
312:
285:
277:
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243:
230:
1:
1409:Analytic class number formula
861:
800:. These elements—named
758:
102:
1414:Riemann–von Mangoldt formula
1059:10.1007/978-0-8176-4575-5_11
868:Banaszak, Grzegorz (2001) ,
154:-adic representation of the
7:
875:Encyclopedia of Mathematics
472:
32:. They were introduced by
10:
1563:
1196:Thaine, Francisco (1988),
1172:Cambridge University Press
1150:Princeton University Press
1499:
1463:
1427:
1401:
1358:
1311:
945:(2006), "Euler systems",
183:for two different fields
827:
712:is a prime not dividing
701:modulo all primes above
411:Here the "Euler factor"
1547:Algebraic number theory
1366:Dedekind zeta functions
808:who introduced them in
81:Tate-Shafarevich groups
46:modular elliptic curves
1254:Barry Mazur's web page
1086:; Rubin, Karl (2004),
779:Tate-Shafarevich group
695:
651:
403:
1486:Bloch–Kato conjecture
1481:Beilinson conjectures
1464:Algebraic conjectures
1319:Riemann zeta function
1203:Annals of Mathematics
696:
652:
404:
156:absolute Galois group
1491:Langlands conjecture
1476:Deligne's conjecture
1428:Analytic conjectures
1174:, pp. 379–460,
1142:Rubin, Karl (2000),
887:Beilinson, Alexander
662:
541:
199:
87:'s new proof of the
1445:Lindelöf hypothesis
1088:"Kolyvagin systems"
806:Alexander Beilinson
790:Kato's Euler system
785:Kato's Euler system
646:
385:
310:
1435:Riemann hypothesis
1359:Algebraic examples
1263:(as of July 2005).
1259:2011-05-17 at the
802:Beilinson elements
794:algebraic K-theory
691:
647:
619:
435:happens to act on
399:
363:
288:
281:
73:ideal class groups
28:groups indexed by
1534:
1533:
1312:Analytic examples
1206:, Second Series,
1181:978-0-521-64419-8
1119:978-0-8218-3512-8
1109:10.1090/memo/0799
1068:978-0-8176-3428-5
908:; Greenberg, R.;
249:
52:and the work of
40:) in his work on
26:Galois cohomology
1554:
1455:Artin conjecture
1419:Weil conjectures
1291:
1284:
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1261:Wayback Machine
1249:
1216:10.2307/1971460
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991:Marta Sanz-Solé
957:
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922:Springer-Verlag
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753:Gras conjecture
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769:Heegner points
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764:Elliptic units
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83:. This led to
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42:Heegner points
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1145:Euler systems
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931:3-540-66546-3
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528:= 1 − ζ
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77:Selmer groups
74:
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67:
66:Euler product
63:
62:Euler factors
59:
55:
54:Thaine (1988)
51:
47:
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27:
23:
19:
1526:Euler system
1525:
1521:Selmer group
1505:
1501:
1390:
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1372:
1342:
1341:Automorphic
1333:
1325:
1298:
1207:
1201:
1166:
1144:
1095:
1091:
1084:Mazur, Barry
1050:
1026:
1022:
1011:, retrieved
995:
987:Kato, Kazuya
971:
967:
964:Kato, Kazuya
946:
917:
906:Coates, J.H.
890:
873:
848:
836:
801:
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788:
772:
745:
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726:
721:
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713:
709:
707:
702:
530:
524:
519:
515:
510:
504:
498:
496:of 1, with ζ
492:
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117:
113:
106:
97:Andrew Wiles
70:
22:Euler system
21:
15:
1380:Hasse–Weil
1210:(1): 1–18,
943:Sujatha, R.
910:Ribet, K.A.
814:Kato (2004)
781:is finite.
466:Kazuya Kato
93:Barry Mazur
18:mathematics
1508:-functions
1393:-functions
1384:-functions
1375:-functions
1345:-functions
1328:-functions
1324:Dirichlet
1301:-functions
1013:2010-08-12
939:Coates, J.
862:References
759:Gauss sums
490:-th root ζ
103:Definition
85:Karl Rubin
1224:0003-486X
1128:0065-9266
1035:0373-2436
966:(2004), "
914:Rubin, K.
880:EMS Press
853:Kato 2007
841:Kato 2007
683:α
679:≡
667:α
640:−
621:α
602:α
584:ζ
557:ζ
379:−
304:−
261:Σ
258:∈
251:∏
34:Kolyvagin
1541:Category
1402:Theorems
1389:Motivic
1257:Archived
916:(1999),
843:, §2.5.1
486:pick an
473:Examples
415:(τ|
146:) where
1240:0951505
1232:1971460
1190:1696501
1158:1749177
1136:2031496
1100:viii+96
1098:(799):
1077:1106906
1043:0984214
1008:2334196
980:2104361
899:0760999
36: (
1504:-adic
1371:Artin
1238:
1230:
1222:
1188:
1178:
1156:
1134:
1126:
1116:
1075:
1065:
1041:
1033:
1006:
978:
953:
928:
897:
804:after
708:where
64:of an
30:fields
1228:JSTOR
1000:(PDF)
828:Notes
740:) = ζ
150:is a
20:, an
1220:ISSN
1176:ISBN
1124:ISSN
1114:ISBN
1063:ISBN
1031:ISSN
951:ISBN
926:ISBN
820:for
716:and
514:for
174:and
95:and
38:1990
1303:in
1212:doi
1208:128
1104:doi
1096:168
1055:doi
796:of
502:= ζ
158:of
75:or
44:on
16:In
1543::
1236:MR
1234:,
1226:,
1218:,
1200:,
1186:MR
1184:,
1154:MR
1152:,
1132:MR
1130:,
1122:,
1112:,
1102:,
1094:,
1090:,
1073:MR
1071:,
1061:,
1039:MR
1037:,
1027:52
1025:,
1004:MR
976:MR
941:;
924:,
912:;
895:MR
878:,
872:,
824:.
755:.
734:(ζ
499:mn
142:,
99:.
68:.
1506:L
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114:c
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