1187:
To show that all L-functions associated to
Shimura varieties – thus to any motive defined by a Shimura variety – can be expressed in terms of the automorphic L-functions of is weaker, even very much weaker, than to show that all motivic L-functions are equal to such L-functions. Moreover, although
372:
724:
1156:
of weight 2. Indeed, it was in the process of generalization of this theorem that Goro
Shimura introduced his varieties and proved his reciprocity law. Zeta functions of Shimura varieties associated with the group
858:
298:
771:
1164:
over other number fields and its inner forms (i.e. multiplicative groups of quaternion algebras) were studied by
Eichler, Shimura, Kuga, Sato, and Ihara. On the basis of their results,
1113:
and, therefore, arithmetical significance. It forms the starting point in his formulation of the reciprocity law, where an important role is played by certain arithmetically defined
1188:
the stronger statement is expected to be valid, there is, so far as I know, no very compelling reason to expect that all motivic L-functions will be attached to
Shimura varieties.
938:
during the 1960s. Shimura's approach, later presented in his monograph, was largely phenomenological, pursuing the widest generalizations of the reciprocity law formulation of
946:, who proceeded to isolate the abstract features that played a role in Shimura's theory. In Deligne's formulation, Shimura varieties are parameter spaces of certain types of
468:
431:
629:
1175:
defined over a number field would be a product of positive and negative powers of automorphic L-functions, i.e. it should arise from a collection of
1145:
1458:
1379:
1256:
1388:
SĂ©minaire
Bourbaki, 23ème année (1970/71), Exp. No. 389, pp. 123–165. Lecture Notes in Math., Vol. 244, Springer, Berlin, 1971.
1431:
1352:
931:
75:
theory. Shimura showed that while initially defined analytically, they are arithmetic objects, in the sense that they admit models
1287:
Qualification: many examples are known, and the sense in which they all "come from" Shimura varieties is a somewhat abstract one.
962:
with level structure. In many cases, the moduli problems to which
Shimura varieties are solutions have been likewise identified.
119:
of a
Shimura variety are more amenable to study than general automorphic forms; in particular, there is a construction attaching
1409:, Proc. Sympos. Pure Math., XXXIII (Corvallis, OR, 1977), Part 2, pp. 247–289, Amer. Math. Soc., Providence, R.I., 1979.
1312:
1224:
1179:. However philosophically natural it may be to expect such a description, statements of this type have only been proved when
1129:
774:
1562:
1499:
1344:
1149:
1125:
805:
367:{\displaystyle {\mathfrak {g}}\otimes \mathbb {C} ={\mathfrak {k}}\oplus {\mathfrak {p}}^{+}\oplus {\mathfrak {p}}^{-},}
1557:
1516:
1494:
1567:
1450:
1176:
574:
736:
730:
215:
33:
1301:
1213:
1083:
57:
1079:
939:
100:
72:
61:
1064:
120:
1489:
719:{\displaystyle \operatorname {Sh} _{K}(G,X)=G(\mathbb {Q} )\backslash X\times G(\mathbb {A} _{f})/K}
1403:
Variétés de
Shimura: interprétation modulaire, et techniques de construction de modèles canoniques,
1110:
1030:
909:
905:
76:
37:
436:
399:
979:
245:
96:
1037:, one gets a Shimura curve, and curves arising from this construction are already compact (i.e.
95:
remarked that
Shimura varieties form a natural realm of examples for which equivalence between
1454:
1427:
1375:
1348:
1169:
1141:
480:
108:
49:
29:
1472:
1358:
1297:
1271:
1209:
1165:
1058:
1038:
982:
529:
226:
159:
112:
92:
1468:
1437:
1413:
1392:
1476:
1464:
1434:
1410:
1389:
1362:
1121:
947:
566:
238:
171:
163:
41:
1369:
1396:
959:
943:
792:
88:
1426:
Lecture Notes in
Mathematics, 900. Springer-Verlag, Berlin-New York, 1982. ii+414 pp.
1551:
1444:
1052:
1045:
951:
25:
17:
1308:
1220:
1153:
1152:
of a modular curve is a product of L-functions associated to explicitly determined
1095:
1072:
955:
935:
620:
167:
80:
68:
1317:
Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics
1229:
Automorphic Forms, Representations, and L-Functions: Symposium in Pure Mathematics
1520:
1480:, paperback edition by Cambridge University Press, 2001, ISBN 978-0-521-00419-0.
1044:
Some examples of Shimura curves with explicitly known equations are given by the
1419:
1336:
1275:
595:
116:
1481:
1105:. This important result due to Shimura shows that Shimura varieties, which
1302:"Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen"
1214:"Automorphic Representations, Shimura Varieties, and Motives. Ein Märchen"
1513:, Proc. Symp. Pure Math, 55:2, Amer. Math. Soc. (1994), pp. 447–523
1231:. Vol. XXXIII Part 1. Chelsea Publishing Company. pp. 205–246.
1128:. Conditional results have been obtained on this conjecture, assuming a
1449:, Mathematical Sciences Research Institute Publications, vol. 35,
67:
Special instances of Shimura varieties were originally introduced by
942:
theory. In retrospect, the name "Shimura variety" was introduced by
1319:. Vol. XXXIII Part 1. Chelsea Publishing Company. p. 208.
1124:
of sets of special points on a Shimura variety is described by the
91:
created an axiomatic framework for the work of Shimura. In 1979,
950:. Thus they form a natural higher-dimensional generalization of
1528:
The semi-simple zeta function of quaternionic Shimura varieties
276:, only weights (0,0), (1,−1), (−1,1) may occur in
750:
678:
532:(possibly, disconnected) such that for every representation
1168:
made a prediction that the Hasse-Weil zeta function of any
1542:
Introduction to Arithmetic Theory of Automorphic Functions
1371:
Harmonic Analysis, the Trace Formula and Shimura Varieties
993:
gives rise to a canonical Shimura variety. Its dimension
569:; moreover, it forms a variation of Hodge structure, and
64:
are among the best known classes of Shimura varieties.
1482:
Read This: The Eightfold Way, reviewed by Ruth Michler
1368:
James Arthur, David Ellwood, and Robert Kottwitz (ed)
1343:, CRM Monograph Series, vol. 22, Providence, RI:
1341:
Quaternion orders, quadratic forms, and Shimura curves
1094:
Each Shimura variety can be defined over a canonical
904:
For special types of hermitian symmetric domains and
808:
739:
632:
602:. For every sufficiently small compact open subgroup
439:
402:
301:
1530:, Lecture Notes in Mathematics, 1657, Springer, 1997
1204:
1202:
795:over all sufficiently small compact open subgroups
791:) are complex algebraic varieties and they form an
1407:Automorphic forms, representations and L-functions
1140:Shimura varieties play an outstanding role in the
853:{\displaystyle (\operatorname {Sh} _{K}(G,X))_{K}}
852:
765:
718:
462:
425:
366:
1374:, Clay Mathematics Proceedings, vol 4, AMS, 2005
1183:is a Shimura variety. In the words of Langlands:
1509:, in U. Jannsen, S. Kleiman. J.-P. Serre (ed.),
1185:
1109:are only complex manifolds, have an algebraic
1086:, also known as Hilbert–Blumenthal varieties.
470:) on the second (respectively, third) summand.
396:) acts trivially on the first summand and via
1424:Hodge cycles, motives, and Shimura varieties.
8:
1078:Other examples of Shimura varieties include
997:is the number of infinite places over which
766:{\displaystyle \Gamma _{i}\backslash X^{+}}
71:in the course of his generalization of the
56:are the one-dimensional Shimura varieties.
1523:, in Arthur, Ellwood, and Kottwitz (2005)
934:were introduced in a series of papers of
844:
816:
807:
773:, where the plus superscript indicates a
757:
744:
738:
708:
699:
695:
694:
671:
670:
637:
631:
452:
441:
440:
438:
412:
411:
406:
401:
355:
349:
348:
338:
332:
331:
321:
320:
313:
312:
303:
302:
300:
52:but are families of algebraic varieties.
1255:Klingler, Bruno; Yafaev, Andrei (2014),
1198:
285:, i.e. the complexified Lie algebra of
87:of the Shimura variety. In the 1970s,
24:is a higher-dimensional analogue of a
7:
1242:
1241:Elkies, section 4.4 (pp. 94–97) in (
1535:The Collected Works of Goro Shimura
1146:Eichler–Shimura congruence relation
1090:Canonical models and special points
974:be a totally real number field and
880:associated with the Shimura datum (
350:
333:
322:
304:
741:
524:It follows from these axioms that
14:
1521:Introduction to Shimura varieties
863:admits a natural right action of
264:satisfying the following axioms:
162:of the multiplicative group from
1418:Pierre Deligne, James S. Milne,
1144:. The prototypical theorem, the
1029:)), fixing a sufficiently small
1005: = 1 (for example, if
912:of the form Γ \
1130:generalized Riemann hypothesis
1120:The qualitative nature of the
841:
837:
825:
809:
729:is a finite disjoint union of
705:
690:
675:
667:
658:
646:
573:is a finite disjoint union of
446:
417:
214:) consisting of a (connected)
1:
1507:Shimura varieties and motives
1345:American Mathematical Society
1136:Role in the Langlands program
565:) is a holomorphic family of
528:has a unique structure of a
512:such that the projection of
463:{\displaystyle {\bar {z}}/z}
426:{\displaystyle z/{\bar {z}}}
289:decomposes into a direct sum
48:. Shimura varieties are not
1495:Encyclopedia of Mathematics
1276:10.4007/annals.2014.180.3.2
1257:"The André-Oort conjecture"
1177:automorphic representations
989:. The multiplicative group
731:locally symmetric varieties
575:hermitian symmetric domains
1584:
1451:Cambridge University Press
1443:Levy, Silvio, ed. (1999),
1001:splits. In particular, if
28:that arises as a quotient
216:reductive algebraic group
42:reductive algebraic group
34:Hermitian symmetric space
1150:Hasse–Weil zeta function
1084:Hilbert modular surfaces
504:does not admit a factor
483:on the adjoint group of
475:The adjoint action of h(
62:Siegel modular varieties
58:Hilbert modular surfaces
1080:Picard modular surfaces
221:defined over the field
1190:
940:complex multiplication
888:) and denoted Sh(
854:
799:. This inverse system
767:
720:
464:
427:
368:
121:Galois representations
73:complex multiplication
1488:Milne, J.S. (2001) ,
1264:Annals of Mathematics
1126:André–Oort conjecture
1065:First Hurwitz triplet
855:
768:
721:
596:ring of finite adeles
495:The adjoint group of
465:
428:
369:
1563:Zeta and L-functions
1335:Alsina, Montserrat;
906:congruence subgroups
876:). It is called the
806:
737:
630:
437:
400:
299:
1386:Travaux de Shimura.
1148:, implies that the
1111:field of definition
1031:arithmetic subgroup
1017: ⊗
910:algebraic varieties
775:connected component
561: ⋅
50:algebraic varieties
38:congruence subgroup
1558:Algebraic geometry
1313:Casselman, William
1225:Casselman, William
850:
777:. The varieties Sh
763:
716:
460:
423:
364:
107:postulated in the
1568:Automorphic forms
1490:"Shimura variety"
1460:978-0-521-66066-2
1446:The Eightfold Way
1422:, Kuang-yen Shi,
1380:978-0-8218-3844-0
1298:Langlands, Robert
1210:Langlands, Robert
1170:algebraic variety
1142:Langlands program
932:compactifications
481:Cartan involution
449:
420:
174:, whose group of
113:Automorphic forms
109:Langlands program
1575:
1502:
1479:
1401:Pierre Deligne,
1384:Pierre Deligne,
1365:
1321:
1320:
1306:
1294:
1288:
1285:
1279:
1278:
1261:
1252:
1246:
1239:
1233:
1232:
1218:
1206:
1166:Robert Langlands
1059:Macbeath surface
983:division algebra
948:Hodge structures
859:
857:
856:
851:
849:
848:
821:
820:
772:
770:
769:
764:
762:
761:
749:
748:
725:
723:
722:
717:
712:
704:
703:
698:
674:
642:
641:
567:Hodge structures
530:complex manifold
469:
467:
466:
461:
456:
451:
450:
442:
432:
430:
429:
424:
422:
421:
413:
410:
373:
371:
370:
365:
360:
359:
354:
353:
343:
342:
337:
336:
326:
325:
316:
308:
307:
227:rational numbers
160:Weil restriction
115:realized in the
93:Robert Langlands
1583:
1582:
1578:
1577:
1576:
1574:
1573:
1572:
1548:
1547:
1546:
1537:(2003), vol 1–5
1526:Harry Reimann,
1487:
1461:
1442:
1355:
1334:
1330:
1325:
1324:
1304:
1296:
1295:
1291:
1286:
1282:
1259:
1254:
1253:
1249:
1240:
1236:
1216:
1208:
1207:
1200:
1195:
1163:
1138:
1122:Zariski closure
1092:
1024:
968:
960:elliptic curves
921:
902:
878:Shimura variety
875:
840:
812:
804:
803:
782:
753:
740:
735:
734:
693:
633:
628:
627:
618:
593:
583:
581:Shimura variety
553:), the family (
544:
503:
491:
435:
434:
433:(respectively,
398:
397:
347:
330:
297:
296:
284:
263:
239:conjugacy class
172:algebraic group
170:. It is a real
164:complex numbers
157:
149:
134:
129:
111:can be tested.
22:Shimura variety
12:
11:
5:
1581:
1579:
1571:
1570:
1565:
1560:
1550:
1549:
1545:
1544:
1538:
1533:Goro Shimura,
1531:
1524:
1514:
1503:
1485:
1459:
1440:
1416:
1399:
1382:
1366:
1353:
1331:
1329:
1326:
1323:
1322:
1289:
1280:
1270:(3): 867–925,
1266:, 2nd Series,
1247:
1234:
1197:
1196:
1194:
1191:
1161:
1137:
1134:
1115:special points
1091:
1088:
1069:
1068:
1062:
1056:
1048:of low genus:
1046:Hurwitz curves
1022:
967:
964:
952:modular curves
917:
901:
898:
871:
861:
860:
847:
843:
839:
836:
833:
830:
827:
824:
819:
815:
811:
793:inverse system
778:
760:
756:
752:
747:
743:
727:
726:
715:
711:
707:
702:
697:
692:
689:
686:
683:
680:
677:
673:
669:
666:
663:
660:
657:
654:
651:
648:
645:
640:
636:
614:
589:
582:
579:
540:
522:
521:
499:
493:
487:
472:
471:
459:
455:
448:
445:
419:
416:
409:
405:
380:where for any
377:
376:
375:
374:
363:
358:
352:
346:
341:
335:
329:
324:
319:
315:
311:
306:
291:
290:
280:
259:
153:
141:
133:
130:
128:
125:
89:Pierre Deligne
54:Shimura curves
13:
10:
9:
6:
4:
3:
2:
1580:
1569:
1566:
1564:
1561:
1559:
1556:
1555:
1553:
1543:
1540:Goro Shimura
1539:
1536:
1532:
1529:
1525:
1522:
1518:
1515:
1512:
1508:
1504:
1501:
1497:
1496:
1491:
1486:
1483:
1478:
1474:
1470:
1466:
1462:
1456:
1452:
1448:
1447:
1441:
1439:
1436:
1433:
1432:3-540-11174-3
1429:
1425:
1421:
1417:
1415:
1412:
1408:
1404:
1400:
1398:
1394:
1391:
1387:
1383:
1381:
1377:
1373:
1372:
1367:
1364:
1360:
1356:
1354:0-8218-3359-6
1350:
1346:
1342:
1338:
1333:
1332:
1327:
1318:
1314:
1310:
1309:Borel, Armand
1303:
1299:
1293:
1290:
1284:
1281:
1277:
1273:
1269:
1265:
1258:
1251:
1248:
1244:
1238:
1235:
1230:
1226:
1222:
1221:Borel, Armand
1215:
1211:
1205:
1203:
1199:
1192:
1189:
1184:
1182:
1178:
1174:
1171:
1167:
1160:
1155:
1154:modular forms
1151:
1147:
1143:
1135:
1133:
1131:
1127:
1123:
1118:
1116:
1112:
1108:
1104:
1100:
1097:
1089:
1087:
1085:
1081:
1076:
1075:of degree 7.
1074:
1066:
1063:
1060:
1057:
1054:
1053:Klein quartic
1051:
1050:
1049:
1047:
1042:
1040:
1036:
1032:
1028:
1020:
1016:
1012:
1008:
1004:
1000:
996:
992:
988:
984:
981:
977:
973:
965:
963:
961:
957:
956:moduli spaces
953:
949:
945:
941:
937:
933:
929:
925:
920:
915:
911:
907:
899:
897:
895:
891:
887:
883:
879:
874:
870:
866:
845:
834:
831:
828:
822:
817:
813:
802:
801:
800:
798:
794:
790:
786:
781:
776:
758:
754:
745:
732:
713:
709:
700:
687:
684:
681:
664:
661:
655:
652:
649:
643:
638:
634:
626:
625:
624:
622:
617:
613:
609:
605:
601:
597:
592:
588:
580:
578:
576:
572:
568:
564:
560:
556:
552:
548:
543:
539:
535:
531:
527:
519:
515:
511:
508:defined over
507:
502:
498:
494:
490:
486:
482:
478:
474:
473:
457:
453:
443:
414:
407:
403:
395:
391:
387:
383:
379:
378:
361:
356:
344:
339:
327:
317:
309:
295:
294:
293:
292:
288:
283:
279:
275:
271:
267:
266:
265:
262:
258:
254:
250:
247:
246:homomorphisms
243:
240:
236:
232:
228:
224:
220:
217:
213:
209:
205:
204:Shimura datum
201:
197:
193:
190:and group of
189:
185:
181:
177:
173:
169:
165:
161:
156:
152:
148:
144:
139:
132:Shimura datum
131:
126:
124:
122:
118:
114:
110:
106:
104:
98:
94:
90:
86:
82:
78:
74:
70:
65:
63:
59:
55:
51:
47:
44:defined over
43:
39:
35:
31:
27:
26:modular curve
23:
19:
18:number theory
1541:
1534:
1527:
1510:
1506:
1493:
1445:
1423:
1406:
1402:
1385:
1370:
1340:
1337:Bayer, Pilar
1316:
1292:
1283:
1267:
1263:
1250:
1237:
1228:
1186:
1180:
1172:
1158:
1139:
1119:
1114:
1106:
1103:reflex field
1102:
1098:
1096:number field
1093:
1077:
1073:Fermat curve
1070:
1043:
1034:
1026:
1018:
1014:
1010:
1006:
1002:
998:
994:
990:
986:
975:
971:
969:
936:Goro Shimura
930:) and their
927:
923:
918:
913:
903:
893:
889:
885:
881:
877:
872:
868:
864:
862:
796:
788:
784:
779:
733:of the form
728:
621:double coset
615:
611:
607:
603:
599:
590:
586:
584:
570:
562:
558:
554:
550:
546:
541:
537:
533:
525:
523:
517:
513:
509:
505:
500:
496:
488:
484:
479:) induces a
476:
393:
389:
385:
381:
286:
281:
277:
273:
269:
260:
256:
252:
248:
241:
234:
230:
222:
218:
211:
207:
203:
199:
195:
191:
187:
183:
179:
175:
168:real numbers
154:
150:
146:
142:
137:
135:
102:
101:automorphic
85:reflex field
84:
81:number field
69:Goro Shimura
66:
53:
45:
21:
15:
1517:J. S. Milne
1420:Arthur Ogus
1101:called the
1071:and by the
520:is trivial.
206:is a pair (
194:-points is
1552:Categories
1505:J. Milne,
1477:0941.00006
1363:1073.11040
1328:References
1067:(genus 14)
1039:projective
980:quaternion
954:viewed as
127:Definition
117:cohomology
105:-functions
1500:EMS Press
1243:Levy 1999
1061:(genus 7)
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310:⊗
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123:to them.
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1227:(eds.).
1212:(1979).
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545:→
384:∈
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255:→
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