409:
Strictly speaking, Hecke made the stipulation about behavior on principal ideals for those admitting a totally positive generator. So, in terms of the definition given above, he really only worked with moduli where all real places appeared. The role of the infinite part
729:
is a
Dirichlet character of conductor 1. The number of Hilbert characters is the order of the class group of the field. Class field theory identifies the Hilbert characters with the characters of the Galois group of the Hilbert class
656:
The characters are 'big' in the sense that the infinity-type when present non-trivially means these characters are not of finite order. The finite-order Hecke characters are all, in a sense, accounted for by
549:
123:("unitary"). Any quasicharacter (of the idele class group) can be written uniquely as a unitary character times a real power of the norm, so there is no big difference between the two definitions.
425:
Both are essentially same notion which has 1 to 1 correspondence. The ideal definition is much more complicated than the idelic one, and Hecke's motivation for his definition was to construct
770:
583:= 1 when the character is trivial. For primitive Größencharakter (defined relative to a modulus in a similar manner to primitive Dirichlet characters), Hecke showed these
716:
is a Hecke character of finite order. It is determined by values on the set of totally positive principal ideals which are 1 with respect to some modulus
598:
Consider a character ψ of the idele class group, taken to be a map into the unit circle which is 1 on principal ideles and on an exceptional finite set
112:
This definition depends on the definition of a character, which varies slightly between authors: It may be defined as a homomorphism to the non-zero
178:(often written Grössencharakter, Grossencharacter, etc.), origin of a Hecke character, going back to Hecke, is defined in terms of a character on
1270:
364:
of a continuous homomorphism to the nonzero complex numbers from the product of the multiplicative groups of all
Archimedean completions of
1354:
881:
1230:
1187:
1144:
213:
1262:
1218:
630:
coordinate and 1 everywhere else. Let χ be the composite of Π and ψ. Then χ is well-defined as a character on the ideal group.
454:
158:(considered as a character on the idele group) is trivial on the group of finite ideles whose every v-adic component lies in 1 +
66:
579:-functions have a meromorphic continuation to the whole complex plane, being analytic except for a simple pole of order 1 at
1381:
1167:
987:
698:
877:
686:
1322:
979:
853:
681:
has Hecke characters that go beyond finite order in a serious way (see the example below). Later developments in
776:-adic integers. So a quasicharacter can be written as product of a power of the norm with a Dirichlet character.
864:
systematically, to remove the need for any special functions. A similar theory was independently developed by
86:
1376:
983:
920:
682:
641:
there corresponds a unique idele class character ψ. Here admissible refers to the existence of a modulus
62:
368:
where each local component of the homomorphism has the same real part (in the exponent). (Here we embed
98:
70:
575:
implies these
Dirichlet series are absolutely convergent in some right half-plane. Hecke proved these
746:
1330:
1308:
1119:
1091:
54:
712:
694:
237:
39:
916:
861:
658:
1248:
1222:
1266:
1226:
1183:
1140:
725:
678:
674:
90:
566:. The common real part condition governing the behavior of Größencharakter on the subgroups
1338:
1326:
1312:
1304:
1284:
1236:
1193:
1175:
1150:
1115:
1099:
1087:
905:
869:
865:
835:
666:
446:
179:
1280:
966:
together with the infinite places. This character has the property that for a prime ideal
1342:
1316:
1288:
1276:
1240:
1214:
1197:
1171:
1154:
1136:
1103:
825:, so the factor of 4 in the exponent ensures that the character is well defined on ideals.
381:
740:
For the field of rational numbers, the idele class group is isomorphic to the product of
441:-function from the rationals to other number fields. For a Größencharakter χ, its
1207:
1111:
971:
931:
849:
741:
113:
1370:
1350:
959:
31:
1135:. Graduate Texts in Mathematics. Vol. 111. With an appendix by Ruth Lawrence.
602:
containing all infinite places. Then ψ generates a character χ of the ideal group
183:
117:
94:
685:
theory indicated that the proper place of the 'big' characters was to provide the
885:
102:
43:
17:
857:
563:
106:
47:
287:
in accordance with the multiplicities of its factors: for each finite place
892:
transforming under the action of the ideles by a given χ has dimension 1.
856:'s 1950 Princeton doctoral dissertation, written under the supervision of
779:
A Hecke character χ of the
Gaussian integers of conductor 1 is of the form
994:
is a product of two
Dirichlet series, for χ and its complex conjugate.
868:
which was the subject of his 1950 ICM talk. A later reformulation in a
587:-functions satisfy a functional equation relating the values of the
376:
using embeddings corresponding to the various
Archimedean places on
1179:
554:
carried out over integral ideals relatively prime to the modulus
116:(also called a "quasicharacter"), or as a homomorphism to the
938:
with complex multiplication by the imaginary quadratic field
649:
such that the character χ is 1 on the ideals which are 1 mod
1297:
Fourier analysis in number fields and Hecke's zeta functions
908:
values: they were introduced by Weil in 1947 under the name
544:{\displaystyle \sum _{(I,m)=1}\chi (I)N(I)^{-s}=L(s,\chi )}
876:
showed that parts of Tate's proof could be expressed by
421:
Relationship between Größencharakter and Hecke character
236:, the "infinite part", being a (formal) product of real
840:
Hecke's original proof of the functional equation for
417:
is now subsumed under the notion of an infinity-type.
347:
into the nonzero complex numbers such that on ideals (
749:
457:
606:, the free abelian group on the prime ideals not in
275:
denote the subgroup of principal fractional ideals (
1209:
Elementary and analytic theory of algebraic numbers
1114:(1967). "VIII. Zeta-functions and L-functions". In
950:. Then there is an algebraic Hecke character χ for
1206:
764:
543:
1073:Husemoller (1987) pp. 302–303; (2002) pp. 321–322
380:.) Thus a Größencharakter may be defined on the
610:. Take a uniformising element π for each prime
225:, the "finite part", being an integral ideal of
101:. It corresponds uniquely to a character of the
372:into the product of Archimedean completions of
595:-function of its complex conjugate character.
310:− 1) is at least as large as the exponent for
1325:(1967). "VII. Global class field theory". In
1258:Grundlehren der mathematischen Wissenschaften
8:
1256:
626:to the class of the idele which is π in the
1064:Husemoller (1987) pp. 299–300; (2002) p.320
109:, via composition with the projection map.
1011:
756:
752:
751:
748:
677:shows. But even a field as simple as the
511:
462:
456:
327:is positive under each real embedding in
253:denote the group of fractional ideals of
437:) that extend the notion of a Dirichlet
1003:
1051:
1049:
1047:
558:of the Größencharakter. The notation
429:-functions (sometimes referred to as
7:
1337:. Academic Press. pp. 162–203.
873:
633:In the opposite direction, given an
1362:, vol. 312, Séminaire Bourbaki
1299:(Tate's 1950 thesis), reprinted in
1170:. Vol. 111 (second ed.).
1126:. Academic Press. pp. 204–230.
360:its value is equal to the value at
880:: the space of distributions (for
334:. A Größencharakter with modulus
27:Type of character in number theory
25:
622:to idele classes by mapping each
591:-function of a character and the
1356:Functions Zetas et Distributions
1219:Polish Scientific Publishers PWN
821:). The only units are powers of
772:with all the unit groups of the
765:{\displaystyle \mathbb {R} ^{+}}
61:, and a natural setting for the
445:-function is defined to be the
882:Schwartz–Bruhat test functions
538:
526:
508:
501:
495:
489:
475:
463:
65:and certain others which have
1:
1168:Graduate Texts in Mathematics
817:is a generator of the ideal (
338:is a group homomorphism from
1131:Husemöller, Dale H. (1987).
934:defined over a number field
915:. Such characters occur in
904:is a Hecke character taking
283:is near 1 at each place of
1398:
1311:(1967) pp. 305–347.
1253:Algebraische Zahlentheorie
896:Algebraic Hecke characters
833:
693:for an important class of
1261:. Vol. 322. Berlin:
1162:Husemöller, Dale (2002).
986:. As a consequence, the
980:characteristic polynomial
902:algebraic Hecke character
150:is a Hecke character mod
142:is a Hecke character mod
69:analogous to that of the
38:is a generalisation of a
988:Hasse–Weil zeta function
618:and define a map Π from
388:, which is the quotient
46:to construct a class of
1335:Algebraic Number Theory
1301:Algebraic Number Theory
1124:Algebraic Number Theory
1096:Algebraic Number Theory
1032:Heilbronn (1967) p. 205
954:, with exceptional set
63:Dedekind zeta-functions
1257:
1205:W. Narkiewicz (1990).
1055:Heilbronn (1967) p.207
1023:Heilbronn (1967) p.204
984:Frobenius endomorphism
921:complex multiplication
829:
766:
683:complex multiplication
545:
958:the set of primes of
848:,χ) used an explicit
767:
546:
134:is the largest ideal
130:of a Hecke character
99:global function field
71:Riemann zeta-function
1382:Zeta and L-functions
747:
455:
257:relatively prime to
105:which is trivial on
67:functional equations
978:) is a root of the
942:, and suppose that
878:distribution theory
713:Dirichlet character
695:algebraic varieties
146:. Here we say that
40:Dirichlet character
1331:Fröhlich, Albrecht
1120:Fröhlich, Albrecht
1098:. Academic Press.
1092:Fröhlich, Albrecht
919:and the theory of
917:class field theory
862:Pontryagin duality
813:an integer, where
762:
659:class field theory
541:
485:
1272:978-3-540-65399-8
1041:Tate (1967) p.169
726:Hilbert character
675:Artin reciprocity
645:based on the set
458:
180:fractional ideals
91:idele class group
16:(Redirected from
1389:
1363:
1361:
1346:
1305:J. W. S. Cassels
1292:
1260:
1249:Neukirch, Jürgen
1244:
1213:(2nd ed.).
1212:
1201:
1158:
1127:
1107:
1074:
1071:
1065:
1062:
1056:
1053:
1042:
1039:
1033:
1030:
1024:
1021:
1015:
1008:
946:is contained in
870:Bourbaki seminar
866:Kenkichi Iwasawa
771:
769:
768:
763:
761:
760:
755:
550:
548:
547:
542:
519:
518:
484:
447:Dirichlet series
107:principal ideles
42:, introduced by
21:
18:Grössencharacter
1397:
1396:
1392:
1391:
1390:
1388:
1387:
1386:
1367:
1366:
1359:
1349:
1327:Cassels, J.W.S.
1321:
1273:
1263:Springer-Verlag
1247:
1233:
1215:Springer-Verlag
1204:
1190:
1172:Springer-Verlag
1164:Elliptic curves
1161:
1147:
1137:Springer-Verlag
1133:Elliptic curves
1130:
1116:Cassels, J.W.S.
1110:
1094:, eds. (1967).
1088:Cassels, J.W.S.
1086:
1083:
1078:
1077:
1072:
1068:
1063:
1059:
1054:
1045:
1040:
1036:
1031:
1027:
1022:
1018:
1012:Husemöller 2002
1009:
1005:
1000:
913:
898:
838:
832:
750:
745:
744:
737:
707:
665:-functions are
637:character χ of
574:
507:
453:
452:
423:
416:
405:
396:
382:ray class group
359:
346:
333:
322:
305:
299:
274:
265:
252:
235:
224:
207:
201:
176:Größencharakter
172:
170:Größencharakter
165:
118:unit circle in
114:complex numbers
83:Hecke character
79:
36:Hecke character
28:
23:
22:
15:
12:
11:
5:
1395:
1393:
1385:
1384:
1379:
1369:
1368:
1365:
1364:
1347:
1319:
1293:
1271:
1245:
1231:
1202:
1188:
1180:10.1007/b97292
1159:
1145:
1128:
1108:
1082:
1079:
1076:
1075:
1066:
1057:
1043:
1034:
1025:
1016:
1002:
1001:
999:
996:
974:, the value χ(
972:good reduction
932:elliptic curve
911:
897:
894:
850:theta-function
834:Main article:
831:
828:
827:
826:
809:imaginary and
802:
801:
781:
780:
777:
759:
754:
742:positive reals
736:
733:
732:
731:
721:
706:
703:
679:Gaussian field
570:
552:
551:
540:
537:
534:
531:
528:
525:
522:
517:
514:
510:
506:
503:
500:
497:
494:
491:
488:
483:
480:
477:
474:
471:
468:
465:
461:
422:
419:
414:
401:
392:
355:
342:
331:
318:
301:
295:
270:
261:
248:
233:
220:
205:
197:
171:
168:
163:
78:
75:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1394:
1383:
1380:
1378:
1377:Number theory
1375:
1374:
1372:
1358:
1357:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1320:
1318:
1314:
1310:
1306:
1302:
1298:
1294:
1290:
1286:
1282:
1278:
1274:
1268:
1264:
1259:
1254:
1250:
1246:
1242:
1238:
1234:
1232:3-540-51250-0
1228:
1224:
1220:
1216:
1211:
1210:
1203:
1199:
1195:
1191:
1189:0-387-95490-2
1185:
1181:
1177:
1173:
1169:
1165:
1160:
1156:
1152:
1148:
1146:0-387-96371-5
1142:
1138:
1134:
1129:
1125:
1121:
1117:
1113:
1112:Heilbronn, H.
1109:
1105:
1101:
1097:
1093:
1089:
1085:
1084:
1080:
1070:
1067:
1061:
1058:
1052:
1050:
1048:
1044:
1038:
1035:
1029:
1026:
1020:
1017:
1013:
1007:
1004:
997:
995:
993:
989:
985:
981:
977:
973:
969:
965:
961:
960:bad reduction
957:
953:
949:
945:
941:
937:
933:
929:
924:
922:
918:
914:
907:
903:
895:
893:
891:
887:
883:
879:
875:
871:
867:
863:
859:
855:
851:
847:
843:
837:
836:Tate's thesis
830:Tate's thesis
824:
820:
816:
812:
808:
804:
803:
799:
795:
791:
787:
783:
782:
778:
775:
757:
743:
739:
738:
734:
728:
727:
722:
719:
715:
714:
709:
708:
705:Special cases
704:
702:
700:
696:
692:
690:
684:
680:
676:
672:
670:
664:
660:
654:
652:
648:
644:
640:
636:
631:
629:
625:
621:
617:
613:
609:
605:
601:
596:
594:
590:
586:
582:
578:
573:
569:
565:
561:
557:
535:
532:
529:
523:
520:
515:
512:
504:
498:
492:
486:
481:
478:
472:
469:
466:
459:
451:
450:
449:
448:
444:
440:
436:
434:
428:
420:
418:
413:
407:
404:
400:
395:
391:
387:
383:
379:
375:
371:
367:
363:
358:
354:
350:
345:
341:
337:
330:
326:
321:
317:
313:
309:
304:
298:
294:
290:
286:
282:
278:
273:
269:
264:
260:
256:
251:
247:
243:
239:
232:
228:
223:
219:
215:
211:
204:
200:
196:
192:
188:
185:
181:
177:
169:
167:
161:
157:
153:
149:
145:
141:
137:
133:
129:
124:
122:
121:
115:
110:
108:
104:
100:
96:
92:
88:
84:
76:
74:
72:
68:
64:
60:
58:
52:
50:
45:
41:
37:
33:
32:number theory
19:
1355:
1334:
1300:
1296:
1252:
1208:
1163:
1132:
1123:
1095:
1069:
1060:
1037:
1028:
1019:
1014:, chapter 16
1006:
991:
975:
967:
963:
955:
951:
947:
943:
939:
935:
927:
925:
909:
901:
899:
889:
845:
841:
839:
822:
818:
814:
810:
806:
797:
793:
789:
785:
773:
724:
717:
711:
688:
668:
662:
655:
650:
646:
642:
638:
634:
632:
627:
623:
619:
615:
611:
607:
603:
599:
597:
592:
588:
584:
580:
576:
571:
567:
559:
555:
553:
442:
438:
432:
430:
426:
424:
411:
408:
402:
398:
393:
389:
385:
377:
373:
369:
365:
361:
356:
352:
348:
343:
339:
335:
328:
324:
319:
315:
311:
307:
302:
296:
292:
288:
284:
280:
276:
271:
267:
262:
258:
254:
249:
245:
241:
230:
226:
221:
217:
209:
202:
198:
194:
190:
186:
184:number field
175:
173:
159:
155:
151:
147:
143:
139:
135:
131:
127:
125:
119:
111:
95:number field
82:
80:
56:
53:larger than
48:
35:
29:
1351:Weil, André
1309:A. Fröhlich
1221:. pp.
926:Indeed let
886:adele group
687:Hasse–Weil
103:idele group
44:Erich Hecke
1371:Categories
1343:1179.11041
1323:Tate, J.T.
1317:1179.11041
1289:0956.11021
1241:0717.11045
1198:1040.11043
1155:0605.14032
1104:0153.07403
1081:References
860:, applied
858:Emil Artin
691:-functions
671:-functions
635:admissible
564:ideal norm
562:means the
435:-functions
138:such that
77:Definition
59:-functions
55:Dirichlet
51:-functions
1295:J. Tate,
906:algebraic
884:) on the
874:Weil 1966
854:John Tate
697:(or even
536:χ
513:−
487:χ
460:∑
182:. For a
128:conductor
87:character
1353:(1966),
1333:(eds.).
1251:(1999).
1122:(eds.).
735:Examples
661:: their
279:) where
266:and let
1281:1697859
1223:334–343
982:of the
699:motives
614:not in
384:modulo
244:. Let
216:, with
214:modulus
89:of the
1341:
1315:
1287:
1279:
1269:
1239:
1229:
1196:
1186:
1153:
1143:
1102:
1010:As in
930:be an
910:type A
788:)) = |
730:field.
667:Artin
431:Hecke
323:, and
238:places
189:, let
1360:(PDF)
998:Notes
673:, as
351:) in
300:, ord
208:be a
93:of a
85:is a
1303:edd
1267:ISBN
1227:ISBN
1184:ISBN
1141:ISBN
990:for
805:for
560:N(I)
229:and
126:The
34:, a
1339:Zbl
1313:Zbl
1285:Zbl
1237:Zbl
1194:Zbl
1176:doi
1151:Zbl
1100:Zbl
970:of
962:of
900:An
888:of
872:by
784:χ((
701:).
314:in
291:in
240:of
154:if
97:or
30:In
1373::
1329:;
1307:,
1283:.
1277:MR
1275:.
1265:.
1255:.
1235:.
1225:.
1192:.
1182:.
1174:.
1166:.
1149:.
1139:.
1118:;
1090:;
1046:^
923:.
852:.
800:|)
796:/|
792:|(
723:A
710:A
653:.
406:.
193:=
174:A
166:.
81:A
73:.
1345:.
1291:.
1243:.
1217:/
1200:.
1178::
1157:.
1106:.
992:E
976:p
968:p
964:E
956:S
952:F
948:F
944:K
940:K
936:F
928:E
912:0
890:K
846:s
844:(
842:L
823:i
819:a
815:a
811:n
807:s
798:a
794:a
790:a
786:a
774:p
758:+
753:R
720:.
718:m
689:L
669:L
663:L
651:m
647:S
643:m
639:I
628:p
624:p
620:I
616:S
612:p
608:S
604:I
600:S
593:L
589:L
585:L
581:s
577:L
572:m
568:P
556:m
539:)
533:,
530:s
527:(
524:L
521:=
516:s
509:)
505:I
502:(
499:N
496:)
493:I
490:(
482:1
479:=
476:)
473:m
470:,
467:I
464:(
443:L
439:L
433:L
427:L
415:∞
412:m
403:m
399:P
397:/
394:m
390:I
386:m
378:K
374:K
370:a
366:K
362:a
357:m
353:P
349:a
344:m
340:I
336:m
332:∞
329:m
325:a
320:f
316:m
312:v
308:a
306:(
303:v
297:f
293:m
289:v
285:m
281:a
277:a
272:m
268:P
263:f
259:m
255:K
250:m
246:I
242:K
234:∞
231:m
227:K
222:f
218:m
212:-
210:K
206:∞
203:m
199:f
195:m
191:m
187:K
164:v
162:O
160:m
156:χ
152:m
148:χ
144:m
140:χ
136:m
132:χ
120:C
57:L
49:L
20:)
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