691:
737:
are equal to products of automorphic L-functions of general linear groups. A proof of
Langlands functoriality would also lead towards a thorough understanding of the analytic properties of automorphic L-functions.
541:
343:
722:
products of representations of GL(m) and GL(n). The resulting Rankin-Selberg L-functions satisfy a number of analytic properties, their functional equation being first proved via the
384:
584:
189:
411:
434:
249:
229:
209:
145:
1123:
591:
1320:
1057:
1029:
951:
910:
869:
837:
790:
1305:
441:
1167:
1284:
1350:
1248:
1116:
147:-functions should have the following properties (which have been proved in some cases but are still conjectural in other cases).
817:
Automorphic forms, representations and L-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2
256:
1289:
1274:
1386:
1310:
861:
821:
1214:
723:
1109:
1391:
1381:
774:
348:
48:
719:
747:
714:) and verified analytic continuation and the functional equation, by using a generalization of the method in
730:
353:
1279:
1200:
548:
1315:
1153:
1243:
1158:
416:
The L-function is expected to have an analytic continuation as a meromorphic function of all complex
67:
711:
82:
765:
1269:
1223:
1091:
985:
153:
78:
815:
1335:
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1053:
1025:
947:
906:
865:
833:
786:
389:
968:
Jacquet, H.; Piatetski-Shapiro, I. I.; Shalika, J. A. (1983), "Rankin-Selberg
Convolutions",
1253:
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1017:
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825:
778:
715:
94:
86:
1067:
1039:
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847:
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1063:
1035:
1013:
957:
935:
916:
894:
886:
875:
843:
796:
761:
734:
52:
927:
419:
234:
214:
194:
130:
764:(1991), "Lectures on automorphic L-functions", in Coates, John; Taylor, M. J. (eds.),
1375:
1184:
1139:
1360:
1355:
829:
811:
807:
686:{\displaystyle \epsilon (s,\pi ,r)=\prod _{v}\epsilon (s,\pi _{v},r_{v},\psi _{v})}
90:
59:
855:
782:
20:
1101:
1132:
1007:
997:
1047:
1006:
Langlands, R. P. (1970), "Problems in the theory of automorphic forms",
1095:
1021:
989:
943:
902:
706:
constructed the automorphic L-functions for general linear groups with
1087:
981:
536:{\displaystyle L(s,\pi ,r)=\epsilon (s,\pi ,r)L(1-s,\pi ,r^{\lor })}
893:, Lecture Notes in Mathematics, vol. 1254, Berlin, New York:
934:, Lecture Notes in Mathematics, vol. 260, Berlin, New York:
1105:
860:, Fields Institute Monographs, vol. 20, Providence, R.I.:
733:
conjectures imply that automorphic L-functions of a connected
854:
Cogdell, James W.; Kim, Henry H.; Murty, Maruti Ram (2004),
1012:, Lecture Notes in Math, vol. 170, Berlin, New York:
338:{\displaystyle L(s,\pi ,r)=\prod _{v}L(s,\pi _{v},r_{v})}
773:, London Math. Soc. Lecture Note Ser., vol. 153,
594:
551:
444:
422:
392:
356:
259:
237:
217:
197:
156:
133:
1334:
1298:
1262:
1236:
1193:
1146:
889:; Piatetski-Shapiro, Ilya; Rallis, Stephen (1987),
685:
578:
535:
428:
405:
378:
337:
243:
223:
203:
183:
139:
1009:Lectures in modern analysis and applications, III
891:Explicit Constructions of Automorphic L-Functions
1074:Shahidi, F. (1981), "On certain "L"-functions",
62:and a finite-dimensional complex representation
703:
1117:
116:
8:
386:is a tensor product of the representations
1124:
1110:
1102:
718:. Ubiquitous in the Langlands Program are
767:L-functions and arithmetic (Durham, 1989)
674:
661:
648:
626:
593:
550:
524:
443:
421:
397:
391:
370:
355:
326:
313:
291:
258:
236:
216:
196:
155:
132:
119:gave surveys of automorphic L-functions.
106:
102:
98:
710:the standard representation (so-called
820:, vol. XXXIII, Providence, R.I.:
810:(1979), "Automorphic L-functions", in
379:{\displaystyle \pi =\otimes \pi _{v}}
112:
7:
1306:Birch and Swinnerton-Dyer conjecture
436:, and satisfy a functional equation
191:should be a product over the places
857:Lectures on automorphic L-functions
579:{\displaystyle \epsilon (s,\pi ,r)}
586:is a product of "local constants"
14:
1351:Main conjecture of Iwasawa theory
932:Zeta Functions of Simple Algebras
1285:Ramanujan–Petersson conjecture
1275:Generalized Riemann hypothesis
1171:-functions of Hecke characters
1046:Langlands, Robert P. (1971) ,
680:
635:
616:
598:
573:
555:
530:
499:
493:
475:
466:
448:
332:
300:
281:
263:
178:
160:
1:
1244:Analytic class number formula
862:American Mathematical Society
822:American Mathematical Society
704:Godement & Jacquet (1972)
1249:Riemann–von Mangoldt formula
783:10.1017/CBO9780511526053.003
695:almost all of which are 1.
184:{\displaystyle L(s,\pi ,r)}
117:Arthur & Gelbart (1991)
1408:
996:Langlands, Robert (1967),
830:10.1090/pspum/033.2/546608
775:Cambridge University Press
349:automorphic representation
93:. They were introduced by
49:automorphic representation
1052:, Yale University Press,
930:; Jacquet, Hervé (1972),
814:; Casselman, W. (eds.),
748:Grand Riemann hypothesis
724:Langlands–Shahidi method
406:{\displaystyle \pi _{v}}
43:) of a complex variable
1201:Dedekind zeta functions
731:Langlands functoriality
687:
580:
537:
430:
407:
380:
339:
245:
225:
205:
185:
141:
1321:Bloch–Kato conjecture
1316:Beilinson conjectures
1299:Algebraic conjectures
1154:Riemann zeta function
699:General linear groups
688:
581:
538:
431:
408:
381:
340:
246:
226:
206:
186:
142:
1387:Zeta and L-functions
1326:Langlands conjecture
1311:Deligne's conjecture
1263:Analytic conjectures
999:Letter to Prof. Weil
712:standard L-functions
592:
549:
442:
420:
390:
354:
257:
235:
215:
195:
154:
131:
68:Langlands dual group
16:Mathematical concept
1280:Lindelöf hypothesis
83:Dirichlet character
77:, generalizing the
47:, associated to an
1270:Riemann hypothesis
1194:Algebraic examples
1022:10.1007/BFb0079065
1016:, pp. 18–61,
944:10.1007/BFb0070263
903:10.1007/BFb0078125
824:, pp. 27–61,
683:
631:
576:
533:
426:
403:
376:
335:
296:
241:
221:
201:
181:
137:
79:Dirichlet L-series
1392:Langlands program
1382:Automorphic forms
1369:
1368:
1147:Analytic examples
1059:978-0-300-01395-5
1031:978-3-540-05284-5
953:978-3-540-05797-0
912:978-3-540-17848-4
871:978-0-8218-3516-6
839:978-0-8218-1437-6
792:978-0-521-38619-7
777:, pp. 1–59,
622:
545:where the factor
429:{\displaystyle s}
413:of local groups.
287:
244:{\displaystyle L}
224:{\displaystyle F}
204:{\displaystyle v}
140:{\displaystyle L}
1399:
1290:Artin conjecture
1254:Weil conjectures
1126:
1119:
1112:
1103:
1098:
1070:
1042:
1002:
992:
964:
923:
887:Gelbart, Stephen
882:
850:
803:
772:
762:Gelbart, Stephen
729:In general, the
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87:Mellin transform
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1088:10.2307/2374219
1073:
1060:
1045:
1032:
1014:Springer-Verlag
1005:
995:
982:10.2307/2374264
967:
954:
936:Springer-Verlag
928:Godement, Roger
926:
913:
895:Springer-Verlag
885:
872:
853:
840:
806:
793:
770:
760:Arthur, James;
759:
756:
744:
735:reductive group
701:
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589:
547:
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520:
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393:
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366:
352:
351:
322:
309:
255:
254:
233:
232:
213:
212:
193:
192:
152:
151:
150:The L-function
129:
128:
125:
53:reductive group
17:
12:
11:
5:
1405:
1403:
1395:
1394:
1389:
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1234:
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1203:
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1195:
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1190:
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1187:
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1144:
1143:
1131:
1129:
1128:
1121:
1114:
1106:
1100:
1099:
1082:(2): 297–355,
1076:Amer. J. Math.
1071:
1058:
1049:Euler products
1043:
1030:
1003:
993:
976:(2): 367–464,
970:Amer. J. Math.
965:
952:
924:
911:
883:
870:
851:
838:
804:
791:
755:
752:
751:
750:
743:
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720:Rankin-Selberg
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31:is a function
15:
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10:
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4:
3:
2:
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1199:
1198:
1196:
1192:
1186:
1185:Selberg class
1183:
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1174:
1172:
1170:
1166:
1164:
1162:
1157:
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1149:
1145:
1141:
1140:number theory
1137:
1135:
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1120:
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945:
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881:
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867:
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859:
858:
852:
849:
845:
841:
835:
831:
827:
823:
819:
818:
813:
812:Borel, Armand
809:
808:Borel, Armand
805:
802:
798:
794:
788:
784:
780:
776:
769:
768:
763:
758:
757:
753:
749:
746:
745:
741:
739:
736:
732:
727:
725:
721:
717:
716:Tate's thesis
713:
709:
705:
698:
696:
693:
675:
671:
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658:
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641:
638:
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627:
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619:
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561:
558:
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490:
487:
484:
481:
478:
472:
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96:
92:
88:
84:
80:
76:
72:
69:
65:
61:
57:
54:
50:
46:
42:
38:
34:
30:
28:
22:
1361:Euler system
1356:Selmer group
1340:
1336:
1225:
1216:
1207:
1177:
1176:Automorphic
1175:
1168:
1160:
1133:
1079:
1075:
1048:
1008:
998:
973:
969:
931:
890:
856:
816:
766:
728:
707:
702:
694:
588:
544:
438:
415:
346:
253:
149:
127:Automorphic
126:
113:Borel (1979)
111:
91:modular form
74:
70:
63:
60:global field
55:
44:
40:
36:
32:
26:
25:automorphic
24:
18:
1215:Hasse–Weil
251:functions.
21:mathematics
1376:Categories
1343:-functions
1228:-functions
1219:-functions
1210:-functions
1180:-functions
1163:-functions
1159:Dirichlet
1136:-functions
754:References
123:Properties
672:ψ
646:π
633:ϵ
624:∏
608:π
596:ϵ
565:π
553:ϵ
526:∨
515:π
506:−
485:π
473:ϵ
458:π
395:π
368:π
364:⊗
358:π
347:Here the
311:π
289:∏
273:π
231:of local
170:π
95:Langlands
29:-function
1237:Theorems
1224:Motivic
742:See also
85:and the
1096:2374219
1068:0419366
1040:0302614
990:2374264
962:0342495
921:0892097
880:2071722
848:0546608
801:1110389
97: (
66:of the
58:over a
51:Ď€ of a
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1206:Artin
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1038:
1028:
988:
960:
950:
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909:
878:
868:
846:
836:
799:
789:
1092:JSTOR
986:JSTOR
771:(PDF)
89:of a
81:of a
23:, an
1054:ISBN
1026:ISBN
948:ISBN
907:ISBN
866:ISBN
834:ISBN
787:ISBN
115:and
107:1971
103:1970
99:1967
1138:in
1084:doi
1080:103
1018:doi
978:doi
974:105
940:doi
899:doi
826:doi
779:doi
211:of
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73:of
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1090:,
1078:,
1064:MR
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972:,
958:MR
956:,
946:,
938:,
917:MR
915:,
905:,
897:,
876:MR
874:,
864:,
844:MR
842:,
832:,
797:MR
795:,
785:,
726:.
105:,
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1226:L
1217:L
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650:v
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639:s
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611:,
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602:s
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571:r
568:,
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559:s
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491:r
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482:,
479:s
476:(
470:=
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452:s
449:(
446:L
424:s
399:v
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328:v
324:r
320:,
315:v
307:,
304:s
301:(
298:L
293:v
285:=
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279:r
276:,
270:,
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