990:
Ordinary representations. These are related to the representations of elliptic curves with ordinary (non-supersingular) reduction. More precisely, they are 2-dimensional representations that are reducible with a 1-dimensional subrepresentation, such that the inertia group acts in a certain way on the
921:
There are numerous conditions on representations given by some property of the representation restricted to a decomposition group of some prime. The terminology for these conditions is somewhat chaotic, with different authors inventing different names for the same condition and using the same name
952:
Finite flat representations. (This name is a little misleading, as they are really profinite rather than finite.) These can be constructed as a projective limit of representations of the Galois group on a finite flat
1147:
880:
836:. Other examples come from the Galois representations of modular forms and automorphic forms, and the Galois representations on â-adic cohomology groups of algebraic varieties.
1510:
991:
submodule and the quotient. The exact condition depends on the author; for example it might act trivially on the quotient and by the character Δ on the submodule.
625:
There are also Galois representations that arise from auxiliary objects and can be used to study Galois groups. An important family of examples are the
1566:
1085:
243:
913:
These are representations over a finite field of characteristic â. They often arise as the reduction mod â of an â-adic representation.
482:
1005:
998:
representations. This means that the representations restricted to an open subgroup of finite index has some specified property.
1761:
1722:
1697:
642:
531:
module. It leaves the question of the gap between free and projective, for which a large theory has now been built up.
1823:
1790:
851:
1849:
1046:
1004:
Semistable representations. These are two dimensional representations related to the representations coming from
598:
is a local field, the multiplicative group of its separable closure is a module for the absolute Galois group of
1571:
690:
1814:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 1, Berlin-Heidelberg-New York-Tokyo:
1782:
984:
908:
882:. â-adic representations with finite image are often called Artin representations. Via an isomorphism of
829:
543:
455:
839:
Unlike Artin representations, â-adic representations can have infinite image. For example, the image of
970:
478:
1747:
788:
607:
603:
221:
926:
Abelian representations. This means that the image of the Galois group in the representations is
694:
274:
144:
1692:, Proc. Sympos. Pure Math., vol. 55, Providence, R.I.: Amer. Math. Soc., pp. 365â392,
973:. These are irreducible in the sense that the only subrepresentation is the whole space or zero.
1854:
1491:
554:
Many objects that arise in number theory are naturally Galois representations. For example, if
315:-module, and one can ask what its structure is. This is an arithmetic question, in that by the
1756:, Proc. Sympos. Pure Math., vol. 33, Providence, R.I.: Amer. Math. Soc., pp. 3â26,
686:
124:
71:
1732:
682:
539:
400:), is there a normal integral basis? The answer is yes, as one sees by identifying it with
316:
161:
1833:
1807:
1800:
1740:
1688:
Kudla, Stephen S. (1994), "The local
Langlands correspondence: the non-archimedean case",
1334: â â, then it is simpler to study the so-called WeilâDeligne representations of
8:
1024:
Wildly ramified representations. These are non-trivial on the (first) ramification group.
720:
671:
615:
47:
1301:
1012:
747:
709:
512:. Then Noether's theorem states that tame ramification is necessary and sufficient for
67:
1706:
1819:
1786:
1757:
1718:
1693:
1421:
1077:
934:
611:
520:
459:
345:
299:
169:
116:
87:
328:
1829:
1796:
1736:
809:
698:
566:
427:
213:
210:
128:
331:
1. If the same is true for the integers, that is equivalent to the existence of a
1815:
1728:
1038:
961:
630:
451:
375:
173:
158:
155:
43:
940:
BarsottiâTate representations. These are similar to finite flat representations.
1202:
435:
1843:
927:
778:
535:
91:
1751:
1506:
1431:
1001:
Reducible representations. These have a proper non-zero sub-representation.
980:
954:
732:
559:
467:
194:
136:
83:
59:
39:
655:
introduced a class of Galois representations of the absolute Galois group
1073:
933:
Absolutely irreducible representations. These remain irreducible over an
626:
79:
63:
20:
652:
132:
108:
1428:
1142:{\displaystyle r_{K}:C_{K}{\tilde {\rightarrow }}W_{K}^{\text{ab}}}
1021:
Unramified representations. These are trivial on the inertia group.
28:
1011:
Tamely ramified representations. These are trivial on the (first)
960:
Good representations. These are related to the representations of
685:. Artin's study of these representations led him to formulate the
979:
Modular representations. These are representations coming from a
719:
and the usual (Euclidean) topology on complex vector spaces, the
370:. This is an interesting question even (perhaps especially) when
268:
1486:
These representations are the same as the representations over
643:
Artin conductor § Artin representation and Artin character
1549:). These latter have the nice feature that the continuity of
1272:
whose image is infinite and therefore is not a character of
542:
has a normal integral basis. This may be seen by using the
1753:
Automorphic forms, representations, and L-functions, Part 2
922:
with different meanings. Some of these conditions include:
549:
1705:
1308:, then the â-adic cohomology of the geometric fibre of
78:-module. The study of Galois modules for extensions of
1557:, thus making the situation more algebraic in flavor.
1513:
sets up a bijection between â-adic representations of
985:
representations over fields of positive characteristic
916:
832:
and the â-adic Tate modules of abelian varieties over
1088:
854:
527:. It is certainly therefore necessary for it to be a
1028:
546:
to embed the abelian field into a cyclotomic field.
176:
are Galois modules for the absolute Galois group of
1319:which, via Ï, induces an â-adic representation of
1141:
874:
1553:is only with respect to the discrete topology on
768:is either a finite-dimensional vector space over
289:be the corresponding Galois group. Then the ring
1841:
1781:, Fields Institute monographs, Providence, RI:
1344:
875:{\displaystyle \mathbf {Z} _{\ell }^{\times }}
1812:Galois module structure of algebraic integers
1715:Grundlehren der Mathematischen Wissenschaften
723:of an Artin representation is always finite.
674:finite-dimensional linear representations of
269:Galois module structure of algebraic integers
1709:; Schmidt, Alexander; Wingberg, Kay (2000),
1230:can have strictly more representations than
1667:)th power of the (arithmetic) Frobenius of
1567:Compatible system of â-adic representations
1330:is a local field of residue characteristic
1717:, vol. 323, Berlin: Springer-Verlag,
1416:is a finite-dimensional vector space over
1296:. These arise naturally from geometry: if
1037:is a local or global field, the theory of
1216:can be considered as a representation of
848:under the â-adic cyclotomic character is
726:
534:A classical result, based on a result of
1806:
1590:
1258:. Thus, the absolute value character on
902:
828:). The first examples to arise were the
636:
450:), as can be deduced from the theory of
74:, but can also be used as a synonym for
1750:(1979), "Number theoretic background",
550:Galois representations in number theory
1842:
1776:
1529:) and WeilâDeligne representations of
708:Because of the incompatibility of the
689:and conjecture what is now called the
184:
147:, its first cohomology group is zero).
1687:
1357:be a field of characteristic zero. A
1244:the continuous complex characters of
1746:
1643:is the size of the residue field of
976:Minimally ramified representations.
917:Local conditions on representations
13:
1770:
1289:is defined in the same way as for
1055:, a continuous group homomorphism
594:(see HilbertâSpeiser theorem). If
462:does not. This is an example of a
14:
1866:
1584:
1501:If the residue characteristic of
1279:(as all such have finite image).
1029:Representations of the Weil group
446:have normal integral bases (over
426:In fact all the subfields of the
281:be a Galois extension of a field
1388:a continuous group homomorphism
857:
246:. A Galois module ρ :
1312:is an â-adic representation of
1251:are in bijection with those of
1209:. Via Ï, any representation of
1596:
1572:Arboreal Galois representation
1118:
1115:
777:(the algebraic closure of the
1:
1783:American Mathematical Society
1681:
943:Crystabelline representations
909:Modular representation theory
1659:is equivalent to the −
1345:WeilâDeligne representations
1282:An â-adic representation of
1018:Trianguline representations.
946:Crystalline representations.
895:they can be identified with
538:, is that a tamely ramified
54:is frequently used when the
7:
1711:Cohomology of Number Fields
1560:
1359:WeilâDeligne representation
971:Irreducible representations
967:HodgeâTate representations.
830:â-adic cyclotomic character
558:is a Galois extension of a
123:is a Galois module for the
97:
10:
1871:
1777:Snaith, Victor P. (1994),
1006:semistable elliptic curves
906:
640:
458:). On the other hand, the
1163:or the idele class group
608:global class field theory
1577:
1511:â-adic monodromy theorem
1178:is local or global) and
983:, but can also refer to
949:de Rham representations.
604:local class field theory
586:for the Galois group of
579:is a Galois module over
474:). What matters here is
306:can be considered as an
197:(with valuation denoted
90:is an important tool in
16:Mathematical terminology
1850:Algebraic number theory
1779:Galois module structure
899:Artin representations.
602:and its study leads to
544:KroneckerâWeber theorem
456:HilbertâSpeiser theorem
275:algebraic number theory
1545:(or equivalently over
1353:be a local field. Let
1265:yields a character of
1174:(depending on whether
1143:
876:
727:â-adic representations
472:perhaps known earlier?
352:give a free basis for
1505:is different from â,
1205:of the Weil group of
1144:
907:Further information:
903:Mod â representations
877:
737:â-adic representation
687:Artin reciprocity law
683:complex vector spaces
668:Artin representations
641:Further information:
637:Artin representations
333:normal integral basis
125:absolute Galois group
72:representation theory
52:Galois representation
1086:
964:with good reduction.
852:
616:separable extensions
540:abelian number field
317:normal basis theorem
145:Hilbert's theorem 90
109:multiplicative group
1237:. For example, via
1138:
871:
651:be a number field.
627:â-adic Tate modules
610:, the union of the
492:, and taking still
466:condition found by
185:Ramification theory
1808:Fröhlich, Albrecht
1492:WeilâDeligne group
1420:equipped with the
1302:projective variety
1139:
1124:
1078:topological groups
1013:ramification group
872:
855:
789:finitely generated
748:group homomorphism
710:profinite topology
612:idele class groups
481:. In terms of the
346:conjugate elements
300:algebraic integers
216:with Galois group
62:over a field or a
1763:978-0-8218-1437-6
1724:978-3-540-66671-4
1699:978-0-8218-1635-6
1422:discrete topology
1136:
1121:
935:algebraic closure
631:abelian varieties
622:is used instead.
521:projective module
428:cyclotomic fields
170:â-adic cohomology
117:separable closure
1862:
1836:
1803:
1766:
1743:
1707:Neukirch, JĂŒrgen
1702:
1675:
1635:
1634:
1633:
1630:
1615:
1612:
1600:
1594:
1588:
1541:
1525:
1446:
1411:
1384:) consisting of
1200:
1199:
1198:
1195:
1187:
1184:
1148:
1146:
1145:
1140:
1137:
1134:
1132:
1123:
1122:
1114:
1111:
1110:
1098:
1097:
1071:
1039:class formations
887:
881:
879:
878:
873:
870:
865:
860:
824:
810:integral closure
804:
795:
773:
763:
746:is a continuous
691:Artin conjecture
670:. These are the
567:ring of integers
452:Gaussian periods
418:
399:
398:
385:For example, if
254:) is said to be
214:Galois extension
129:cohomology group
88:group cohomology
1870:
1869:
1865:
1864:
1863:
1861:
1860:
1859:
1840:
1839:
1826:
1816:Springer-Verlag
1793:
1773:
1771:Further reading
1764:
1725:
1700:
1690:Motives, Part 2
1684:
1679:
1678:
1672:
1655:) is such that
1641:
1631:
1626:
1617:
1616:
1613:
1608:
1607:
1606:|| is given by
1601:
1597:
1589:
1585:
1580:
1563:
1544:
1537:
1534:
1528:
1521:
1518:
1480:
1434:
1405:
1398:
1389:
1370:
1347:
1339:
1324:
1317:
1294:
1287:
1277:
1270:
1263:
1256:
1249:
1242:
1235:
1228:
1221:
1214:
1196:
1191:
1189:
1188:
1185:
1180:
1179:
1168:
1157:
1133:
1128:
1113:
1112:
1106:
1102:
1093:
1089:
1087:
1084:
1083:
1069:
1062:
1056:
1053:
1031:
962:elliptic curves
919:
911:
905:
890:
883:
866:
861:
856:
853:
850:
849:
847:
827:
820:
818:
807:
800:
799:-module (where
798:
791:
786:
776:
769:
756:
750:
744:
729:
717:
693:concerning the
679:
660:
645:
639:
584:
573:
552:
517:
416:
396:
394:
376:rational number
369:
360:
343:
335:, i.e. of α in
319:one knows that
314:
297:
271:
263:
240:
187:
174:geometric fibre
100:
17:
12:
11:
5:
1868:
1858:
1857:
1852:
1838:
1837:
1824:
1804:
1791:
1772:
1769:
1768:
1767:
1762:
1744:
1723:
1703:
1698:
1683:
1680:
1677:
1676:
1670:
1639:
1595:
1582:
1581:
1579:
1576:
1575:
1574:
1569:
1562:
1559:
1542:
1532:
1526:
1516:
1484:
1483:
1478:
1425:
1401:
1396:
1372:(or simply of
1368:
1346:
1343:
1337:
1322:
1315:
1292:
1285:
1275:
1268:
1261:
1254:
1247:
1240:
1233:
1226:
1219:
1212:
1203:abelianization
1166:
1155:
1150:
1149:
1131:
1127:
1120:
1117:
1109:
1105:
1101:
1096:
1092:
1067:
1060:
1051:
1030:
1027:
1026:
1025:
1022:
1019:
1016:
1009:
1002:
999:
992:
988:
977:
974:
968:
965:
958:
950:
947:
944:
941:
938:
931:
918:
915:
904:
901:
888:
869:
864:
859:
843:
825:
816:
805:
796:
784:
779:â-adic numbers
774:
754:
742:
728:
725:
715:
677:
658:
638:
635:
614:of all finite
582:
571:
551:
548:
515:
460:Gaussian field
436:roots of unity
424:
423:
365:
356:
344:such that its
339:
310:
293:
270:
267:
261:
238:
186:
183:
182:
181:
172:groups of its
148:
103:Given a field
99:
96:
15:
9:
6:
4:
3:
2:
1867:
1856:
1855:Galois theory
1853:
1851:
1848:
1847:
1845:
1835:
1831:
1827:
1825:3-540-11920-5
1821:
1817:
1813:
1809:
1805:
1802:
1798:
1794:
1792:0-8218-0264-X
1788:
1784:
1780:
1775:
1774:
1765:
1759:
1755:
1754:
1749:
1745:
1742:
1738:
1734:
1730:
1726:
1720:
1716:
1712:
1708:
1704:
1701:
1695:
1691:
1686:
1685:
1673:
1666:
1662:
1658:
1654:
1650:
1646:
1642:
1629:
1624:
1620:
1611:
1605:
1599:
1592:
1591:Fröhlich 1983
1587:
1583:
1573:
1570:
1568:
1565:
1564:
1558:
1556:
1552:
1548:
1540:
1535:
1524:
1519:
1512:
1508:
1504:
1499:
1497:
1493:
1489:
1481:
1475: â
1474:
1470:
1466:
1462:
1458:
1454:
1450:
1445:
1441:
1437:
1433:
1430:
1426:
1423:
1419:
1415:
1409:
1404:
1399:
1392:
1387:
1386:
1385:
1383:
1379:
1376:) is a pair (
1375:
1371:
1364:
1360:
1356:
1352:
1342:
1340:
1333:
1329:
1325:
1318:
1311:
1307:
1303:
1299:
1295:
1288:
1280:
1278:
1271:
1264:
1257:
1250:
1243:
1236:
1229:
1222:
1215:
1208:
1204:
1194:
1183:
1177:
1173:
1169:
1162:
1158:
1129:
1125:
1107:
1103:
1099:
1094:
1090:
1082:
1081:
1080:
1079:
1075:
1070:
1063:
1054:
1048:
1044:
1040:
1036:
1023:
1020:
1017:
1014:
1010:
1007:
1003:
1000:
997:
993:
989:
986:
982:
978:
975:
972:
969:
966:
963:
959:
956:
951:
948:
945:
942:
939:
937:of the field.
936:
932:
929:
925:
924:
923:
914:
910:
900:
898:
894:
886:
867:
862:
846:
842:
837:
835:
831:
823:
815:
811:
803:
794:
790:
783:
780:
772:
767:
761:
757:
749:
745:
738:
734:
724:
722:
718:
711:
706:
704:
702:
696:
692:
688:
684:
680:
673:
669:
666:, now called
665:
661:
654:
650:
644:
634:
632:
628:
623:
621:
617:
613:
609:
605:
601:
597:
593:
589:
585:
578:
574:
568:
564:
561:
557:
547:
545:
541:
537:
536:David Hilbert
532:
530:
526:
522:
518:
511:
508:to the power
507:
503:
499:
496: =
495:
491:
487:
484:
480:
477:
473:
469:
465:
461:
457:
453:
449:
445:
441:
437:
433:
429:
421:
414:
411:
410:
409:
407:
403:
392:
389: =
388:
383:
381:
377:
373:
368:
364:
359:
355:
351:
347:
342:
338:
334:
330:
326:
322:
318:
313:
309:
305:
301:
296:
292:
288:
284:
280:
276:
273:In classical
266:
264:
257:
253:
249:
245:
244:inertia group
241:
234:
230:
226:
223:
219:
215:
212:
208:
204:
200:
196:
192:
179:
175:
171:
167:
164:over a field
163:
160:
157:
153:
149:
146:
142:
138:
134:
130:
127:. Its second
126:
122:
118:
114:
110:
106:
102:
101:
95:
93:
92:number theory
89:
85:
84:global fields
81:
77:
73:
69:
65:
61:
58:-module is a
57:
53:
49:
45:
41:
37:
33:
31:
26:
25:Galois module
22:
1811:
1778:
1752:
1714:
1710:
1689:
1668:
1664:
1660:
1656:
1652:
1648:
1644:
1637:
1627:
1622:
1618:
1609:
1603:
1598:
1586:
1554:
1550:
1546:
1538:
1530:
1522:
1514:
1507:Grothendieck
1502:
1500:
1495:
1487:
1485:
1476:
1472:
1468:
1464:
1460:
1456:
1452:
1448:
1443:
1439:
1435:
1432:endomorphism
1417:
1413:
1407:
1402:
1394:
1390:
1381:
1377:
1373:
1366:
1362:
1358:
1354:
1350:
1348:
1335:
1331:
1327:
1320:
1313:
1309:
1305:
1300:is a smooth
1297:
1290:
1283:
1281:
1273:
1266:
1259:
1252:
1245:
1238:
1231:
1224:
1217:
1210:
1206:
1192:
1181:
1175:
1171:
1164:
1160:
1153:
1151:
1065:
1058:
1049:
1042:
1041:attaches to
1034:
1032:
995:
994:Potentially
981:modular form
955:group scheme
920:
912:
896:
892:
884:
844:
840:
838:
833:
821:
813:
801:
792:
781:
770:
765:
759:
752:
740:
736:
733:prime number
730:
713:
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576:
569:
562:
560:number field
555:
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533:
528:
524:
513:
509:
505:
504:must divide
501:
497:
493:
489:
485:
483:discriminant
479:ramification
475:
471:
468:Emmy Noether
463:
447:
444:prime number
443:
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425:
419:
412:
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232:
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217:
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198:
195:valued field
190:
188:
177:
165:
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140:
137:Brauer group
120:
112:
104:
75:
60:vector space
55:
51:
40:Galois group
35:
29:
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1593:, p. 8
1223:. However,
1074:isomorphism
731:Let â be a
500:, no prime
327:-module of
242:denote its
64:free module
50:. The term
21:mathematics
1844:Categories
1834:0501.12012
1801:0830.11042
1748:Tate, John
1741:0948.11001
1682:References
1463:)= ||
1447:such that
1047:Weil group
703:-functions
695:holomorphy
672:continuous
653:Emil Artin
323:is a free
285:, and let
258:if ρ(
256:unramified
201:) and let
133:isomorphic
86:and their
38:being the
1429:nilpotent
1119:~
1116:→
1072:, and an
1057:Ï :
996:something
897:bona fide
868:×
863:ℓ
751:Ï :
464:necessary
265:) = {1}.
222:extension
220:. For an
168:then the
44:extension
1810:(1983),
1561:See also
1471:for all
1438: :
1412:, where
1393: :
519:to be a
408:) where
98:Examples
42:of some
1733:1737196
1602:Here ||
1490:of the
1380:,
1201:is the
928:abelian
808:is the
787:) or a
415:= exp(2
395:√
374:is the
135:to the
115:) of a
66:over a
34:, with
32:-module
1832:
1822:
1799:
1789:
1760:
1739:
1731:
1721:
1696:
1636:where
1632:
1614:
1520:(over
1197:
1186:
1152:where
764:where
758:â Aut(
699:Artin
606:. For
565:, the
378:field
348:under
277:, let
250:â Aut(
235:, let
211:finite
162:scheme
159:proper
156:smooth
107:, the
48:fields
1578:Notes
1536:over
1400:â Aut
1361:over
1326:. If
1304:over
891:with
735:. An
721:image
523:over
454:(the
361:over
209:be a
193:be a
154:is a
80:local
27:is a
1820:ISBN
1787:ISBN
1758:ISBN
1719:ISBN
1694:ISBN
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1349:Let
1045:its
647:Let
529:free
476:tame
438:for
434:-th
430:for
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143:(by
68:ring
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