317:. By a result of Artin, the local conductor is an integer. Heuristically, the Artin conductor measures how far the action of the higher ramification groups is from being trivial. In particular, if χ is unramified, then its Artin conductor is zero. Thus if
244:
541:
1194:
Algebraic number theory. Proceedings of an instructional conference organized by the London
Mathematical Society (a NATO Advanced Study Institute) with the support of the International Mathematical Union
412:
696:
838:
580:
451:
306:
129:
105:
852:
is the character of the regular representation and 1 is the character of the trivial representation. The Swan character is the character of a representation of
137:
746:
showed that it can be realized over the corresponding ring of Witt vectors. It cannot in general be realized over the rationals or over the local field
475:
590:. Since the local Artin conductor is zero at unramified primes, the above product only need be taken over primes that ramify in
1117:
962:
342:
646:
1225:
1317:
909:
888:
1322:
781:
895:
1217:
607:
1312:
1137:
1063:
1296:
1273:
1174:
1125:
565:
436:
291:
114:
90:
1235:
1201:
1189:
1103:
1044:
8:
755:, suggesting that there is no easy way to construct the Artin representation explicitly.
56:
40:
25:
1244:
1210:
1162:
1091:
1048:
957:. Encyclopaedia of Mathematical Sciences. Vol. 49 (Second ed.). p. 329.
263:
1261:
1257:
1221:
1181:
1154:
1132:
1095:
1083:
1052:
968:
958:
865:
239:{\displaystyle f(\chi )=\sum _{i\geq 0}{\frac {g_{i}}{g_{0}}}(\chi (1)-\chi (G_{i}))}
1111:
1253:
1231:
1197:
1185:
1146:
1099:
1075:
1040:
1032:
902:
76:
60:
1292:
1269:
1170:
1121:
267:
1064:"Die gruppentheoretische Struktur der Diskriminanten algebraischer Zahlkörper."
873:
1306:
1265:
1158:
1087:
972:
1280:
1079:
725:) showed that the Artin representation can be realized over the local field
857:
37:
29:
1212:
Explicit Brauer
Induction: With Applications to Algebra and Number Theory
721:
asked for a direct construction of the Artin representation. Serre (
33:
17:
1023:(1930), "Zur Theorie der L-Reihen mit allgemeinen Gruppencharakteren.",
1166:
1059:
1036:
1020:
44:
1116:, Mémoires de la Société Mathématique de France, vol. 25, Paris:
1242:
Swan, Richard G. (1963), "The
Grothendieck ring of a finite group",
1150:
1025:
1113:
Colloque de Théorie des
Nombres (Univ. Bordeaux, Bordeaux, 1969)
1110:
Fontaine, Jean-Marc (1971), "Sur les représentations d'Artin",
536:{\displaystyle {\mathfrak {f}}(\chi )=\prod _{p}p^{f(\chi ,p)}}
901:
The Artin conductor appears in the functional equation of the
1216:, Cambridge Studies in Advanced Mathematics, vol. 40,
908:
The Artin and Swan representations are used to defined the
601:
1135:(1960), "Sur la rationalité des représentations d'Artin",
417:
in other words, the sum of the higher order terms with
784:
649:
568:
562:) is the local Artin conductor of the restriction of
478:
439:
345:
294:
140:
117:
93:
1209:
832:
690:
574:
535:
445:
406:
300:
238:
123:
99:
407:{\displaystyle f(\chi )-(\chi (1)-\chi (G_{0})),}
1304:
952:
691:{\displaystyle a_{G}=\sum _{\chi }f(\chi )\chi }
616:is a finite Galois extension of the local field
1068:Journal fĂĽr die Reine und Angewandte Mathematik
325:, then the Artin conductors of all χ are zero.
898:is expressed in terms of the Artin conductor.
1196:, London: Academic Press, pp. 128–161,
582:to the decomposition group of some prime of
1184:(1967), "VI. Local class field theory", in
953:Manin, Yu. I.; Panchishkin, A. A. (2007).
608:Galois module § Artin representations
424:
66:
1109:
993:
991:
936:
934:
891:for the discriminant of a global field.
743:
738:not equal to the residue characteristic
713:is the complex linear representation of
602:Artin representation and Artin character
948:
946:
1305:
1283:(1946), "L'avenir des mathématiques",
1207:
758:
1180:
1131:
1058:
1019:
988:
931:
922:
722:
546:where the product is over the primes
52:
48:
1279:
1241:
1000:
979:
955:Introduction to Modern Number Theory
943:
861:
833:{\displaystyle sw_{G}=a_{G}-r_{G}+1}
718:
55:) as an expression appearing in the
887:The Artin conductor appears in the
879:with character the Swan character.
481:
13:
14:
1334:
864:) showed that there is a unique
465:of global fields is an ideal of
882:
28:associated to a character of a
1118:Société Mathématique de France
910:conductor of an elliptic curve
889:conductor-discriminant formula
682:
676:
528:
516:
492:
486:
398:
395:
382:
373:
367:
361:
355:
349:
233:
230:
217:
208:
202:
196:
150:
144:
111:, then the Artin conductor of
1:
1013:
1258:10.1016/0040-9383(63)90025-9
7:
896:Serre modularity conjecture
10:
1339:
1218:Cambridge University Press
605:
288:) is the average value of
894:The optimal level in the
620:, with Galois group
1285:Bol. Soc. Mat. SĂŁo Paulo
915:
1080:10.1515/crll.1931.164.1
425:Global Artin conductors
1208:Snaith, V. P. (1994),
834:
717:with this character.
692:
576:
537:
457:of a finite extension
447:
431:global Artin conductor
408:
302:
240:
125:
101:
67:Local Artin conductors
1318:Representation theory
1138:Annals of Mathematics
835:
693:
606:Further information:
577:
575:{\displaystyle \chi }
538:
448:
446:{\displaystyle \chi }
409:
303:
301:{\displaystyle \chi }
241:
126:
124:{\displaystyle \chi }
102:
100:{\displaystyle \chi }
1323:Zeta and L-functions
912:or abelian variety.
782:
703:Artin representation
647:
566:
476:
453:of the Galois group
437:
433:of a representation
343:
336:of the character is
292:
138:
115:
91:
83:, with Galois group
1006:Snaith (1994) p.248
985:Snaith (1994) p.249
759:Swan representation
321:is unramified over
79:of the local field
57:functional equation
1182:Serre, Jean-Pierre
1133:Serre, Jean-Pierre
1120:, pp. 71–81,
1037:10.1007/BF02941010
997:Serre (1967) p.160
940:Serre (1967) p.159
928:Serre (1967) p.158
868:representation of
830:
688:
672:
572:
533:
507:
443:
404:
298:
264:ramification group
236:
171:
121:
107:is a character of
97:
1141:, Second Series,
964:978-3-540-20364-3
663:
640:is the character
498:
194:
156:
1330:
1299:
1276:
1238:
1215:
1204:
1177:
1128:
1106:
1055:
1007:
1004:
998:
995:
986:
983:
977:
976:
950:
941:
938:
929:
926:
903:Artin L-function
839:
837:
836:
831:
823:
822:
810:
809:
797:
796:
734:, for any prime
697:
695:
694:
689:
671:
659:
658:
581:
579:
578:
573:
542:
540:
539:
534:
532:
531:
506:
485:
484:
469:, defined to be
452:
450:
449:
444:
413:
411:
410:
405:
394:
393:
307:
305:
304:
299:
245:
243:
242:
237:
229:
228:
195:
193:
192:
183:
182:
173:
170:
130:
128:
127:
122:
106:
104:
103:
98:
77:Galois extension
61:Artin L-function
43:, introduced by
1338:
1337:
1333:
1332:
1331:
1329:
1328:
1327:
1303:
1302:
1252:(1–2): 85–110,
1228:
1186:Cassels, J.W.S.
1151:10.2307/1970142
1016:
1011:
1010:
1005:
1001:
996:
989:
984:
980:
965:
951:
944:
939:
932:
927:
923:
918:
885:
851:
818:
814:
805:
801:
792:
788:
783:
780:
779:
774:
761:
754:
744:Fontaine (1971)
733:
712:
667:
654:
650:
648:
645:
644:
635:
626:Artin character
610:
604:
567:
564:
563:
512:
508:
502:
480:
479:
477:
474:
473:
438:
435:
434:
427:
389:
385:
344:
341:
340:
316:
293:
290:
289:
287:
278:
268:lower numbering
257:
224:
220:
188:
184:
178:
174:
172:
160:
139:
136:
135:
116:
113:
112:
92:
89:
88:
69:
24:is a number or
22:Artin conductor
12:
11:
5:
1336:
1326:
1325:
1320:
1315:
1301:
1300:
1277:
1239:
1226:
1205:
1178:
1145:(2): 405–420,
1129:
1107:
1056:
1015:
1012:
1009:
1008:
999:
987:
978:
963:
942:
930:
920:
919:
917:
914:
884:
881:
877:-adic integers
847:
841:
840:
829:
826:
821:
817:
813:
808:
804:
800:
795:
791:
787:
770:
765:Swan character
760:
757:
750:
729:
708:
699:
698:
687:
684:
681:
678:
675:
670:
666:
662:
657:
653:
631:
603:
600:
571:
544:
543:
530:
527:
524:
521:
518:
515:
511:
505:
501:
497:
494:
491:
488:
483:
442:
426:
423:
415:
414:
403:
400:
397:
392:
388:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
351:
348:
334:Swan conductor
330:wild invariant
312:
297:
283:
274:
253:
247:
246:
235:
232:
227:
223:
219:
216:
213:
210:
207:
204:
201:
198:
191:
187:
181:
177:
169:
166:
163:
159:
155:
152:
149:
146:
143:
131:is the number
120:
96:
68:
65:
45:Emil Artin
9:
6:
4:
3:
2:
1335:
1324:
1321:
1319:
1316:
1314:
1313:Number theory
1311:
1310:
1308:
1298:
1294:
1290:
1286:
1282:
1278:
1275:
1271:
1267:
1263:
1259:
1255:
1251:
1247:
1246:
1240:
1237:
1233:
1229:
1227:0-521-46015-8
1223:
1219:
1214:
1213:
1206:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1176:
1172:
1168:
1164:
1160:
1156:
1152:
1148:
1144:
1140:
1139:
1134:
1130:
1127:
1123:
1119:
1115:
1114:
1108:
1105:
1101:
1097:
1093:
1089:
1085:
1081:
1077:
1074:(164): 1–11,
1073:
1070:(in German),
1069:
1065:
1061:
1057:
1054:
1050:
1046:
1042:
1038:
1034:
1030:
1027:(in German),
1026:
1022:
1018:
1017:
1003:
994:
992:
982:
974:
970:
966:
960:
956:
949:
947:
937:
935:
925:
921:
913:
911:
906:
904:
899:
897:
892:
890:
880:
878:
876:
871:
867:
863:
859:
855:
850:
846:
827:
824:
819:
815:
811:
806:
802:
798:
793:
789:
785:
778:
777:
776:
773:
769:
766:
756:
753:
749:
745:
741:
737:
732:
728:
724:
720:
716:
711:
707:
704:
685:
679:
673:
668:
664:
660:
655:
651:
643:
642:
641:
639:
634:
630:
627:
623:
619:
615:
612:Suppose that
609:
599:
597:
593:
589:
585:
569:
561:
557:
553:
549:
525:
522:
519:
513:
509:
503:
499:
495:
489:
472:
471:
470:
468:
464:
460:
456:
440:
432:
422:
420:
401:
390:
386:
379:
376:
370:
364:
358:
352:
346:
339:
338:
337:
335:
331:
326:
324:
320:
315:
311:
295:
286:
282:
277:
273:
269:
265:
261:
256:
252:
225:
221:
214:
211:
205:
199:
189:
185:
179:
175:
167:
164:
161:
157:
153:
147:
141:
134:
133:
132:
118:
110:
94:
86:
82:
78:
74:
71:Suppose that
64:
62:
58:
54:
50:
46:
42:
39:
35:
31:
27:
23:
19:
1288:
1284:
1249:
1243:
1211:
1193:
1190:Fröhlich, A.
1142:
1136:
1112:
1071:
1067:
1028:
1024:
1002:
981:
954:
924:
907:
900:
893:
886:
883:Applications
874:
869:
853:
848:
844:
842:
775:is given by
771:
767:
764:
762:
751:
747:
739:
735:
730:
726:
714:
709:
705:
702:
700:
637:
632:
628:
625:
621:
617:
613:
611:
595:
591:
587:
583:
559:
555:
551:
547:
545:
466:
462:
458:
454:
430:
428:
418:
416:
333:
329:
327:
322:
318:
313:
309:
284:
280:
275:
271:
270:), of order
259:
254:
250:
248:
108:
84:
80:
75:is a finite
72:
70:
30:Galois group
21:
15:
1281:Weil, André
1060:Artin, Emil
1031:: 292–306,
1021:Artin, Emil
719:Weil (1946)
586:lying over
18:mathematics
1307:Categories
1236:0991.20005
1202:0153.07403
1104:0001.00801
1045:56.0173.02
1014:References
866:projective
1291:: 55–68,
1266:0040-9383
1159:0003-486X
1096:117731518
1088:0075-4102
1053:120987633
973:0938-0396
872:over the
812:−
686:χ
680:χ
669:χ
665:∑
570:χ
520:χ
500:∏
490:χ
441:χ
380:χ
377:−
365:χ
359:−
353:χ
296:χ
215:χ
212:−
200:χ
165:≥
158:∑
148:χ
119:χ
95:χ
1245:Topology
1192:(eds.),
1062:(1931),
701:and the
421:> 0.
279:, and χ(
1297:0020961
1274:0153722
1175:0171775
1167:1970142
1126:0374106
860: (
258:is the
47: (
1295:
1272:
1264:
1234:
1224:
1200:
1173:
1165:
1157:
1124:
1102:
1094:
1086:
1051:
1043:
971:
961:
843:where
624:. The
554:, and
249:where
59:of an
38:global
20:, the
1163:JSTOR
1092:S2CID
1049:S2CID
916:Notes
87:. If
41:field
34:local
32:of a
26:ideal
1262:ISSN
1222:ISBN
1155:ISSN
1084:ISSN
1072:1931
969:ISSN
959:ISBN
862:1963
858:Swan
763:The
723:1960
429:The
328:The
266:(in
262:-th
53:1931
49:1930
1254:doi
1232:Zbl
1198:Zbl
1147:doi
1100:Zbl
1076:doi
1041:JFM
1033:doi
636:of
558:(χ,
550:of
332:or
308:on
36:or
16:In
1309::
1293:MR
1287:,
1270:MR
1268:,
1260:,
1248:,
1230:,
1220:,
1188:;
1171:MR
1169:,
1161:,
1153:,
1143:72
1122:MR
1098:,
1090:,
1082:,
1066:,
1047:,
1039:,
990:^
967:.
945:^
933:^
905:.
856:.
768:sw
742:.
598:.
63:.
51:,
1289:1
1256::
1250:2
1149::
1078::
1035::
1029:8
975:.
875:l
870:G
854:G
849:g
845:r
828:1
825:+
820:G
816:r
807:G
803:a
799:=
794:G
790:w
786:s
772:G
752:p
748:Q
740:p
736:l
731:l
727:Q
715:G
710:G
706:A
683:)
677:(
674:f
661:=
656:G
652:a
638:G
633:G
629:a
622:G
618:K
614:L
596:K
594:/
592:L
588:p
584:L
560:p
556:f
552:K
548:p
529:)
526:p
523:,
517:(
514:f
510:p
504:p
496:=
493:)
487:(
482:f
467:K
463:K
461:/
459:L
455:G
419:i
402:,
399:)
396:)
391:0
387:G
383:(
374:)
371:1
368:(
362:(
356:)
350:(
347:f
323:K
319:L
314:i
310:G
285:i
281:G
276:i
272:g
260:i
255:i
251:G
234:)
231:)
226:i
222:G
218:(
209:)
206:1
203:(
197:(
190:0
186:g
180:i
176:g
168:0
162:i
154:=
151:)
145:(
142:f
109:G
85:G
81:K
73:L
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