711:
885:
359:
977:
608:
1173:
469:
291:
1058:
1265:
In 2005, Chandrashekhar Khare obtained a proof of the level 1 case of Serre conjecture, and in 2008 a proof of the full conjecture in collaboration with Jean-Pierre
Wintenberger.
767:
1087:
249:
168:
1004:
738:
193:
548:
510:
122:
arises from a modular form. A stronger version of this conjecture specifies the weight and level of the modular form. The conjecture in the level 1 case was proved by
814:
1110:
1024:
640:
382:
216:
1240:
1197:
1130:
787:
632:
822:
1259:
299:
1395:
Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "On Serre's reciprocity conjecture for 2-dimensional mod p representations of Gal(Q/Q)",
896:
559:
1688:
1712:
1718:
17:
1743:
1583:(1975), "Valeurs propres des opérateurs de Hecke modulo l", Journées Arithmétiques de Bordeaux (Conf., Univ. Bordeaux, 1974),
1665:
1199:
are explicitly conjectured in Serre's article. In addition, he derives a number of results from this conjecture, among them
1138:
1738:
396:
1652:
Stein, William A.; Ribet, Kenneth A. (2001), "Lectures on Serre's conjectures", in Conrad, Brian; Rubin, Karl (eds.),
254:
1657:
1029:
743:
1700:
1063:
225:
144:
1616:
1284:
1553:
1516:
1372:
1333:
985:
719:
1350:
1255:
1200:
513:
127:
78:
176:
475:
46:
518:
480:
1733:
1548:
1531:
Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)",
1511:
1367:
1348:
Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (II)",
1328:
219:
1494:
Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)",
1311:
Khare, Chandrashekhar; Wintenberger, Jean-Pierre (2009), "Serre's modularity conjecture (I)",
706:{\displaystyle \rho _{f}\colon G_{\mathbb {Q} }\rightarrow \mathrm {GL} _{2}({\mathcal {O}}),}
1397:
1207:(although this implies Fermat's Last Theorem, Serre proves it directly from his conjecture).
171:
139:
115:
796:
1675:
1645:
1600:
1540:
1503:
1359:
1320:
1251:
1095:
1009:
367:
201:
123:
74:
1215:
The strong form of Serre's conjecture describes the level and weight of the modular form.
8:
551:
1544:
1507:
1363:
1324:
1250:
A proof of the level 1 and small weight cases of the conjecture was obtained in 2004 by
1439:
1225:
1204:
1182:
1115:
772:
617:
31:
1661:
1633:
1607:
1588:
1580:
388:
103:
56:
1203:
and the now-proven
Taniyama–Weil (or Taniyama–Shimura) conjecture, now known as the
1625:
1558:
1521:
1476:
1467:
Khare, Chandrashekhar (2006), "Serre's modularity conjecture: The level one case",
1449:
1406:
1377:
1338:
1293:
1282:
Khare, Chandrashekhar (2006), "Serre's modularity conjecture: The level one case",
769:. This representation is characterized by the condition that for all prime numbers
1629:
1480:
1297:
880:{\displaystyle \operatorname {Trace} (\rho _{f}(\operatorname {Frob} _{p}))=a_{p}}
1671:
1641:
1596:
1427:
1219:
1411:
126:
in 2005, and a proof of the full conjecture was completed jointly by Khare and
1562:
1525:
1381:
1342:
1727:
1637:
1592:
354:{\displaystyle \rho \colon G_{\mathbb {Q} }\rightarrow \mathrm {GL} _{2}(F).}
972:{\displaystyle \det(\rho _{f}(\operatorname {Frob} _{p}))=p^{k-1}\chi (p).}
119:
1708:
95:
603:{\displaystyle \chi \colon \mathbb {Z} /N\mathbb {Z} \rightarrow F^{*}}
1453:
1704:
1444:
384:
is odd, meaning the image of complex conjugation has determinant -1.
790:
614:
a theorem due to
Shimura, Deligne, and Serre-Deligne attaches to
1610:(1987), "Sur les représentations modulaires de degré 2 de Gal(
1656:, IAS/Park City Math. Ser., vol. 9, Providence, R.I.:
982:
Reducing this representation modulo the maximal ideal of
1092:
Serre's conjecture asserts that for any representation
1228:
1185:
1141:
1118:
1098:
1066:
1032:
1012:
988:
899:
825:
799:
775:
746:
722:
643:
620:
562:
521:
483:
399:
370:
302:
257:
228:
204:
179:
147:
114:), states that an odd, irreducible, two-dimensional
1654:
Arithmetic algebraic geometry (Park City, UT, 1999)
1234:
1191:
1168:{\displaystyle {\overline {\rho _{f}}}\cong \rho }
1167:
1124:
1104:
1081:
1052:
1018:
998:
971:
879:
808:
781:
761:
732:
705:
626:
602:
542:
504:
463:
376:
353:
285:
243:
210:
187:
162:
1530:
1493:
1428:"The level 1 weight 2 case of Serre's conjecture"
1394:
1347:
1310:
740:is the ring of integers in a finite extension of
464:{\displaystyle f=q+a_{2}q^{2}+a_{3}q^{3}+\cdots }
1725:
900:
222:, continuous, two-dimensional representation of
1179:The level and weight of the conjectural form
1210:
1651:
1425:
286:{\displaystyle F=\mathbb {F} _{\ell ^{r}}}
1552:
1515:
1443:
1410:
1371:
1332:
1222:of the representation, with the power of
1073:
749:
663:
583:
570:
315:
266:
235:
181:
154:
1112:as above, there is a modular eigenform
1053:{\displaystyle {\overline {\rho _{f}}}}
14:
1726:
1689:Wiles's proof of Fermat's Last Theorem
1606:
1579:
1466:
1281:
111:
107:
762:{\displaystyle \mathbb {Q} _{\ell }}
24:
991:
725:
692:
677:
674:
329:
326:
25:
1755:
1744:Conjectures that have been proved
1694:
1432:Revista Matemática Iberoamericana
1082:{\displaystyle G_{\mathbb {Q} }}
244:{\displaystyle G_{\mathbb {Q} }}
163:{\displaystyle G_{\mathbb {Q} }}
1719:Lectures on Serre's conjectures
1487:
1460:
1419:
1388:
1304:
1275:
999:{\displaystyle {\mathcal {O}}}
963:
957:
932:
929:
916:
903:
861:
858:
845:
832:
733:{\displaystyle {\mathcal {O}}}
697:
687:
669:
587:
537:
531:
499:
493:
345:
339:
321:
133:
13:
1:
1701:Serre's Modularity Conjecture
1658:American Mathematical Society
1630:10.1215/S0012-7094-87-05413-5
1573:
1481:10.1215/S0012-7094-06-13434-8
1298:10.1215/S0012-7094-06-13434-8
100:Serre's modularity conjecture
38:Serre's modularity conjecture
1707:given on October 25, 2007 (
1154:
1045:
188:{\displaystyle \mathbb {Q} }
138:The conjecture concerns the
7:
1682:
1412:10.4007/annals.2009.169.229
27:Conjecture in number theory
18:Serre modularity conjecture
10:
1760:
543:{\displaystyle k=k(\rho )}
505:{\displaystyle N=N(\rho )}
29:
1739:Theorems in number theory
1617:Duke Mathematical Journal
1563:10.1007/s00222-009-0206-6
1526:10.1007/s00222-009-0205-7
1469:Duke Mathematical Journal
1426:Dieulefait, Luis (2007),
1382:10.1007/s00222-009-0206-6
1343:10.1007/s00222-009-0205-7
1285:Duke Mathematical Journal
1218:The optimal level is the
84:
70:
62:
52:
42:
1533:Inventiones Mathematicae
1496:Inventiones Mathematicae
1351:Inventiones Mathematicae
1313:Inventiones Mathematicae
1268:
1256:Jean-Pierre Wintenberger
1245:
1211:Optimal level and weight
128:Jean-Pierre Wintenberger
79:Jean-Pierre Wintenberger
30:Not to be confused with
1713:other version of slides
47:Algebraic number theory
1236:
1193:
1169:
1126:
1106:
1083:
1054:
1020:
1000:
973:
881:
810:
809:{\displaystyle N\ell }
783:
763:
734:
707:
628:
604:
544:
506:
465:
378:
355:
287:
245:
220:absolutely irreducible
212:
189:
164:
1703:50 minute lecture by
1398:Annals of Mathematics
1237:
1201:Fermat's Last Theorem
1194:
1170:
1127:
1107:
1105:{\displaystyle \rho }
1084:
1055:
1021:
1019:{\displaystyle \ell }
1001:
974:
882:
811:
784:
764:
735:
708:
629:
605:
545:
507:
466:
379:
377:{\displaystyle \rho }
364:Additionally, assume
356:
288:
246:
213:
211:{\displaystyle \rho }
190:
172:rational number field
165:
140:absolute Galois group
116:Galois representation
104:Jean-Pierre Serre
1660:, pp. 143–232,
1252:Chandrashekhar Khare
1226:
1183:
1139:
1116:
1096:
1064:
1030:
1010:
986:
897:
823:
797:
773:
744:
720:
641:
618:
560:
519:
481:
397:
368:
300:
255:
251:over a finite field
226:
202:
177:
145:
124:Chandrashekhar Khare
75:Chandrashekhar Khare
1545:2009InMat.178..505K
1508:2009InMat.178..485K
1364:2009InMat.178..505K
1325:2009InMat.178..485K
552:Nebentype character
39:
1608:Serre, Jean-Pierre
1587:, 24–25: 109–117,
1581:Serre, Jean-Pierre
1232:
1205:modularity theorem
1189:
1165:
1122:
1102:
1079:
1050:
1016:
996:
969:
877:
806:
779:
759:
730:
703:
624:
600:
540:
502:
461:
387:To any normalized
374:
351:
283:
241:
208:
185:
160:
37:
32:Modularity theorem
1667:978-0-8218-2173-2
1262:, independently.
1235:{\displaystyle l}
1192:{\displaystyle f}
1157:
1125:{\displaystyle f}
1048:
782:{\displaystyle p}
634:a representation
627:{\displaystyle f}
389:modular eigenform
92:
91:
57:Jean-Pierre Serre
16:(Redirected from
1751:
1678:
1648:
1613:
1603:
1567:
1565:
1556:
1528:
1519:
1491:
1485:
1483:
1464:
1458:
1456:
1447:
1438:(3): 1115–1124,
1423:
1417:
1415:
1414:
1392:
1386:
1384:
1375:
1345:
1336:
1308:
1302:
1300:
1279:
1241:
1239:
1238:
1233:
1198:
1196:
1195:
1190:
1174:
1172:
1171:
1166:
1158:
1153:
1152:
1143:
1131:
1129:
1128:
1123:
1111:
1109:
1108:
1103:
1088:
1086:
1085:
1080:
1078:
1077:
1076:
1059:
1057:
1056:
1051:
1049:
1044:
1043:
1034:
1025:
1023:
1022:
1017:
1005:
1003:
1002:
997:
995:
994:
978:
976:
975:
970:
953:
952:
928:
927:
915:
914:
886:
884:
883:
878:
876:
875:
857:
856:
844:
843:
815:
813:
812:
807:
788:
786:
785:
780:
768:
766:
765:
760:
758:
757:
752:
739:
737:
736:
731:
729:
728:
712:
710:
709:
704:
696:
695:
686:
685:
680:
668:
667:
666:
653:
652:
633:
631:
630:
625:
609:
607:
606:
601:
599:
598:
586:
578:
573:
549:
547:
546:
541:
511:
509:
508:
503:
470:
468:
467:
462:
454:
453:
444:
443:
431:
430:
421:
420:
383:
381:
380:
375:
360:
358:
357:
352:
338:
337:
332:
320:
319:
318:
292:
290:
289:
284:
282:
281:
280:
279:
269:
250:
248:
247:
242:
240:
239:
238:
217:
215:
214:
209:
194:
192:
191:
186:
184:
169:
167:
166:
161:
159:
158:
157:
102:, introduced by
40:
36:
21:
1759:
1758:
1754:
1753:
1752:
1750:
1749:
1748:
1724:
1723:
1697:
1685:
1668:
1611:
1576:
1571:
1570:
1554:10.1.1.228.8022
1517:10.1.1.518.4611
1492:
1488:
1465:
1461:
1454:10.4171/rmi/525
1424:
1420:
1393:
1389:
1373:10.1.1.228.8022
1334:10.1.1.518.4611
1309:
1305:
1280:
1276:
1271:
1260:Luis Dieulefait
1248:
1227:
1224:
1223:
1220:Artin conductor
1213:
1184:
1181:
1180:
1148:
1144:
1142:
1140:
1137:
1136:
1117:
1114:
1113:
1097:
1094:
1093:
1072:
1071:
1067:
1065:
1062:
1061:
1039:
1035:
1033:
1031:
1028:
1027:
1026:representation
1011:
1008:
1007:
990:
989:
987:
984:
983:
942:
938:
923:
919:
910:
906:
898:
895:
894:
871:
867:
852:
848:
839:
835:
824:
821:
820:
798:
795:
794:
774:
771:
770:
753:
748:
747:
745:
742:
741:
724:
723:
721:
718:
717:
691:
690:
681:
673:
672:
662:
661:
657:
648:
644:
642:
639:
638:
619:
616:
615:
594:
590:
582:
574:
569:
561:
558:
557:
520:
517:
516:
482:
479:
478:
449:
445:
439:
435:
426:
422:
416:
412:
398:
395:
394:
369:
366:
365:
333:
325:
324:
314:
313:
309:
301:
298:
297:
275:
271:
270:
265:
264:
256:
253:
252:
234:
233:
229:
227:
224:
223:
203:
200:
199:
180:
178:
175:
174:
153:
152:
148:
146:
143:
142:
136:
77:
35:
28:
23:
22:
15:
12:
11:
5:
1757:
1747:
1746:
1741:
1736:
1722:
1721:
1716:
1696:
1695:External links
1693:
1692:
1691:
1684:
1681:
1680:
1679:
1666:
1649:
1624:(1): 179–230,
1604:
1575:
1572:
1569:
1568:
1539:(3): 505–586,
1502:(3): 485–504,
1486:
1475:(3): 557–589,
1459:
1418:
1405:(1): 229–253,
1387:
1358:(3): 505–586,
1319:(3): 485–504,
1303:
1292:(3): 557–589,
1273:
1272:
1270:
1267:
1247:
1244:
1231:
1212:
1209:
1188:
1177:
1176:
1164:
1161:
1156:
1151:
1147:
1121:
1101:
1075:
1070:
1047:
1042:
1038:
1015:
993:
980:
979:
968:
965:
962:
959:
956:
951:
948:
945:
941:
937:
934:
931:
926:
922:
918:
913:
909:
905:
902:
888:
887:
874:
870:
866:
863:
860:
855:
851:
847:
842:
838:
834:
831:
828:
805:
802:
778:
756:
751:
727:
714:
713:
702:
699:
694:
689:
684:
679:
676:
671:
665:
660:
656:
651:
647:
623:
612:
611:
597:
593:
589:
585:
581:
577:
572:
568:
565:
539:
536:
533:
530:
527:
524:
501:
498:
495:
492:
489:
486:
472:
471:
460:
457:
452:
448:
442:
438:
434:
429:
425:
419:
415:
411:
408:
405:
402:
373:
362:
361:
350:
347:
344:
341:
336:
331:
328:
323:
317:
312:
308:
305:
278:
274:
268:
263:
260:
237:
232:
207:
183:
156:
151:
135:
132:
90:
89:
86:
85:First proof in
82:
81:
72:
71:First proof by
68:
67:
64:
63:Conjectured in
60:
59:
54:
53:Conjectured by
50:
49:
44:
26:
9:
6:
4:
3:
2:
1756:
1745:
1742:
1740:
1737:
1735:
1734:Modular forms
1732:
1731:
1729:
1720:
1717:
1714:
1710:
1706:
1702:
1699:
1698:
1690:
1687:
1686:
1677:
1673:
1669:
1663:
1659:
1655:
1650:
1647:
1643:
1639:
1635:
1631:
1627:
1623:
1619:
1618:
1609:
1605:
1602:
1598:
1594:
1590:
1586:
1582:
1578:
1577:
1564:
1560:
1555:
1550:
1546:
1542:
1538:
1534:
1527:
1523:
1518:
1513:
1509:
1505:
1501:
1497:
1490:
1482:
1478:
1474:
1470:
1463:
1455:
1451:
1446:
1441:
1437:
1433:
1429:
1422:
1413:
1408:
1404:
1400:
1399:
1391:
1383:
1379:
1374:
1369:
1365:
1361:
1357:
1353:
1352:
1344:
1340:
1335:
1330:
1326:
1322:
1318:
1314:
1307:
1299:
1295:
1291:
1287:
1286:
1278:
1274:
1266:
1263:
1261:
1257:
1253:
1243:
1229:
1221:
1216:
1208:
1206:
1202:
1186:
1162:
1159:
1149:
1145:
1135:
1134:
1133:
1119:
1099:
1090:
1068:
1040:
1036:
1013:
966:
960:
954:
949:
946:
943:
939:
935:
924:
920:
911:
907:
893:
892:
891:
872:
868:
864:
853:
849:
840:
836:
829:
826:
819:
818:
817:
803:
800:
792:
776:
754:
700:
682:
658:
654:
649:
645:
637:
636:
635:
621:
595:
591:
579:
575:
566:
563:
556:
555:
554:
553:
534:
528:
525:
522:
515:
496:
490:
487:
484:
477:
458:
455:
450:
446:
440:
436:
432:
427:
423:
417:
413:
409:
406:
403:
400:
393:
392:
391:
390:
385:
371:
348:
342:
334:
310:
306:
303:
296:
295:
294:
276:
272:
261:
258:
230:
221:
205:
196:
173:
149:
141:
131:
129:
125:
121:
117:
113:
109:
105:
101:
97:
87:
83:
80:
76:
73:
69:
65:
61:
58:
55:
51:
48:
45:
41:
33:
19:
1653:
1621:
1615:
1584:
1536:
1532:
1499:
1495:
1489:
1472:
1468:
1462:
1445:math/0412099
1435:
1431:
1421:
1402:
1396:
1390:
1355:
1349:
1316:
1312:
1306:
1289:
1283:
1277:
1264:
1249:
1217:
1214:
1178:
1091:
1006:gives a mod
981:
889:
715:
613:
473:
386:
363:
197:
137:
120:finite field
99:
93:
550:, and some
134:Formulation
96:mathematics
1728:Categories
1585:Astérisque
1574:References
1132:such that
1705:Ken Ribet
1638:0012-7094
1593:0303-1179
1549:CiteSeerX
1512:CiteSeerX
1368:CiteSeerX
1329:CiteSeerX
1258:, and by
1242:removed.
1163:ρ
1160:≅
1155:¯
1146:ρ
1100:ρ
1046:¯
1037:ρ
1014:ℓ
955:χ
947:−
908:ρ
837:ρ
830:
804:ℓ
755:ℓ
670:→
655::
646:ρ
596:∗
588:→
567::
564:χ
535:ρ
497:ρ
459:⋯
372:ρ
322:→
307::
304:ρ
273:ℓ
206:ρ
130:in 2008.
1683:See also
816:we have
1676:1860042
1646:0885783
1601:0382173
1541:Bibcode
1504:Bibcode
1360:Bibcode
1321:Bibcode
791:coprime
170:of the
118:over a
106: (
1709:slides
1674:
1664:
1644:
1636:
1614:/Q)",
1599:
1591:
1551:
1514:
1370:
1331:
716:where
514:weight
218:be an
1711:PDF,
1440:arXiv
1269:Notes
1246:Proof
827:Trace
476:level
43:Field
1715:PDF)
1662:ISBN
1634:ISSN
1589:ISSN
1529:and
1346:and
1254:and
921:Frob
890:and
850:Frob
198:Let
112:1987
108:1975
88:2008
66:1975
1626:doi
1559:doi
1537:178
1522:doi
1500:178
1477:doi
1473:134
1450:doi
1407:doi
1403:169
1378:doi
1356:178
1339:doi
1317:178
1294:doi
1290:134
1060:of
901:det
793:to
474:of
94:In
1730::
1672:MR
1670:,
1642:MR
1640:,
1632:,
1622:54
1620:,
1597:MR
1595:,
1557:,
1547:,
1535:,
1520:,
1510:,
1498:,
1471:,
1448:,
1436:23
1434:,
1430:,
1401:,
1376:,
1366:,
1354:,
1337:,
1327:,
1315:,
1288:,
1089:.
789:,
512:,
293:.
195:.
110:,
98:,
1628::
1612:Q
1566:.
1561::
1543::
1524::
1506::
1484:.
1479::
1457:.
1452::
1442::
1416:.
1409::
1385:.
1380::
1362::
1341::
1323::
1301:.
1296::
1230:l
1187:f
1175:.
1150:f
1120:f
1074:Q
1069:G
1041:f
992:O
967:.
964:)
961:p
958:(
950:1
944:k
940:p
936:=
933:)
930:)
925:p
917:(
912:f
904:(
873:p
869:a
865:=
862:)
859:)
854:p
846:(
841:f
833:(
801:N
777:p
750:Q
726:O
701:,
698:)
693:O
688:(
683:2
678:L
675:G
664:Q
659:G
650:f
622:f
610:,
592:F
584:Z
580:N
576:/
571:Z
538:)
532:(
529:k
526:=
523:k
500:)
494:(
491:N
488:=
485:N
456:+
451:3
447:q
441:3
437:a
433:+
428:2
424:q
418:2
414:a
410:+
407:q
404:=
401:f
349:.
346:)
343:F
340:(
335:2
330:L
327:G
316:Q
311:G
277:r
267:F
262:=
259:F
236:Q
231:G
182:Q
155:Q
150:G
34:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.