787:
264:
982:
596:
1465:
1557:. In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler product and establishes the multiplicativity of
360:
1160:
96:
814:
782:{\displaystyle T_{m}f(z)=m^{k-1}\sum _{\left({\begin{smallmatrix}a&b\\c&d\end{smallmatrix}}\right)\in \Gamma \backslash M_{m}}(cz+d)^{-k}f\left({\frac {az+b}{cz+d}}\right),}
444:
1290:
493:, as well as moderate growth at infinity; these conditions are preserved by the summation, and so Hecke operators preserve the space of modular forms of a given weight.
1574:
are also called "Hecke algebras", although sometimes the link to Hecke operators is not entirely obvious. These algebras include certain quotients of the
1758:
279:
1610:
808:
assures that the image of a form with integer
Fourier coefficients has integer Fourier coefficients. This can be rewritten in the form
1046:
385:
Hecke operators can be realized in a number of contexts. The simplest meaning is combinatorial, namely as taking for a given integer
259:{\displaystyle \Delta (z)=q\left(\prod _{n=1}^{\infty }(1-q^{n})\right)^{24}=\sum _{n=1}^{\infty }\tau (n)q^{n},\quad q=e^{2\pi iz},}
1470:
Thus for normalized cuspidal Hecke eigenforms of integer weight, their
Fourier coefficients coincide with their Hecke eigenvalues.
1595:
1803:
1643:
977:{\displaystyle T_{m}f(z)=m^{k-1}\sum _{a,d>0,ad=m}{\frac {1}{d^{k}}}\sum _{b{\pmod {d}}}f\left({\frac {az+b}{d}}\right),}
17:
1711:
Hecke, E. (1937b), "Über
Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. II.",
1673:
Hecke, E. (1937a), "Über
Modulfunktionen und die Dirichletschen Reihen mit Eulerscher Produktentwicklung. I.",
1571:
1665:
1660:
1165:
One can see from this explicit formula that Hecke operators with different indices commute and that if
412:
52:
1753:
1460:{\displaystyle T_{m}f=a_{m}f,\quad a_{m}a_{n}=\sum _{r>0,r|(m,n)}r^{k-1}a_{mn/r^{2}},\ m,n\geq 1.}
490:
1655:
370:
1514:
270:
1818:
87:
47:), is a certain kind of "averaging" operator that plays a significant role in the structure of
1713:
1675:
1510:
1489:
486:
1771:
1742:
1704:
8:
1628:
1799:
1777:
1730:
1692:
1639:
1583:
482:
1582:. The presence of this commutative operator algebra plays a significant role in the
1767:
1738:
1722:
1700:
1684:
1605:
1534:
1530:
1518:
1485:
799:
508:
approach, this translates to double cosets with respect to some compact subgroups.
63:
481:
are particular kinds of functions of a lattice, subject to conditions making them
1635:
403:
652:
1623:
1812:
1749:
1734:
1696:
1600:
1538:
1526:
1522:
1509:
coprime to the level acting on the space of cusp forms of a given weight are
1479:
554:
501:
374:
366:
497:
478:
48:
32:
1579:
532:
79:
40:
28:
1787:
1726:
1688:
1575:
355:{\displaystyle \tau (mn)=\tau (m)\tau (n)\quad {\text{ for }}(m,n)=1.}
1209:
75:
71:
454:
1525:
for these Hecke operators. Each of these basic forms possesses an
70:) used Hecke operators on modular forms in a paper on the special
1484:
Algebras of Hecke operators are called "Hecke algebras", and are
1754:"On Mr. Ramanujan's empirical expansions of modular functions."
1155:{\displaystyle b_{n}=\sum _{r>0,r|(m,n)}r^{k-1}a_{mn/r^{2}}.}
1202:
is preserved by the Hecke operators. If a (non-zero) cusp form
505:
987:
which leads to the formula for the
Fourier coefficients of
1521:
implies that there is a basis of modular forms that are
1630:
Modular functions and
Dirichlet series in number theory
690:
496:
Another way to express Hecke operators is by means of
1293:
1049:
817:
599:
415:
282:
99:
90:, expressing the coefficients of the Ramanujan form,
1759:Proceedings of the Cambridge Philosophical Society
1627:
1459:
1154:
976:
781:
438:
354:
258:
1810:
377:which realise some individual Hecke operators.
1792:Elliptic Modular Forms and Their Applications
83:
44:
380:
473:and two dimensions, there are three such
1018:in terms of the Fourier coefficients of
1748:
1622:
365:The idea goes back to earlier work of
78:, ahead of the general theory given by
67:
14:
1811:
1710:
1672:
1611:Wiles's proof of Fermat's Last Theorem
1586:of modular forms and generalisations.
590:th Hecke operator acts by the formula
1541:with the local factor for each prime
1596:Eichler–Shimura congruence relation
929:
922:
511:
24:
1634:(2nd ed.), Berlin, New York:
687:
426:
197:
140:
100:
51:of modular forms and more general
25:
1830:
1473:
651:
439:{\displaystyle \sum f(\Lambda ')}
31:, in particular in the theory of
1572:Other related mathematical rings
449:with the sum taken over all the
1798:, Universitext, Springer, 2008
1326:
802:and the normalization constant
325:
227:
1387:
1375:
1371:
1100:
1088:
1084:
933:
923:
837:
831:
721:
705:
619:
613:
433:
422:
343:
331:
322:
316:
310:
304:
295:
286:
211:
205:
164:
145:
109:
103:
13:
1:
1616:
1492:theory, the Hecke operators
1551:, a quadratic polynomial in
7:
1661:Encyclopedia of Mathematics
1589:
86:). Mordell proved that the
53:automorphic representations
10:
1835:
1796:The 1-2-3 of Modular Forms
1477:
58:
371:algebraic correspondences
1196:of cusp forms of weight
381:Mathematical description
1515:Petersson inner product
1270:. Hecke eigenforms are
1212:of all Hecke operators
567:. Given a modular form
531:integral matrices with
271:multiplicative function
1782:A course in arithmetic
1547:is the inverse of the
1529:. More precisely, its
1461:
1210:simultaneous eigenform
1156:
978:
783:
504:. In the contemporary
440:
356:
260:
201:
144:
88:Ramanujan tau function
1714:Mathematische Annalen
1676:Mathematische Annalen
1490:elliptic modular form
1462:
1157:
979:
784:
441:
357:
261:
181:
124:
1513:with respect to the
1291:
1047:
815:
597:
467:. For example, with
413:
280:
97:
1488:. In the classical
1727:10.1007/BF01594180
1689:10.1007/BF01594160
1457:
1391:
1185:, so the subspace
1152:
1104:
974:
938:
895:
779:
704:
678:
677:
483:analytic functions
436:
352:
256:
1804:978-3-540-74117-6
1778:Jean-Pierre Serre
1750:Mordell, Louis J.
1645:978-0-387-97127-8
1584:harmonic analysis
1517:. Therefore, the
1486:commutative rings
1441:
1350:
1223:with eigenvalues
1063:
965:
913:
911:
859:
770:
641:
406:of fixed rank to
329:
18:Modular eigenform
16:(Redirected from
1826:
1774:
1745:
1707:
1669:
1656:"Hecke operator"
1650:(See chapter 8.)
1648:
1633:
1606:Abstract algebra
1567:
1556:
1549:Hecke polynomial
1546:
1535:Dirichlet series
1531:Mellin transform
1519:spectral theorem
1508:
1502:
1466:
1464:
1463:
1458:
1439:
1435:
1434:
1433:
1432:
1423:
1407:
1406:
1390:
1374:
1346:
1345:
1336:
1335:
1319:
1318:
1303:
1302:
1283:
1269:
1259:
1233:
1222:
1207:
1201:
1195:
1184:
1174:
1161:
1159:
1158:
1153:
1148:
1147:
1146:
1145:
1136:
1120:
1119:
1103:
1087:
1059:
1058:
1039:
1017:
983:
981:
980:
975:
970:
966:
961:
947:
937:
936:
912:
910:
909:
897:
894:
858:
857:
827:
826:
807:
800:upper half-plane
797:
788:
786:
785:
780:
775:
771:
769:
755:
741:
732:
731:
703:
702:
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683:
679:
640:
639:
609:
608:
589:
583:
577:
566:
552:
539:
530:
526:
512:Explicit formula
489:with respect to
476:
472:
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460:
452:
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442:
437:
432:
401:
390:
361:
359:
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21:
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1828:
1827:
1825:
1824:
1823:
1809:
1808:
1654:
1646:
1636:Springer-Verlag
1624:Apostol, Tom M.
1619:
1592:
1558:
1552:
1542:
1504:
1501:
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1482:
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1428:
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1419:
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1408:
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1392:
1370:
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1310:
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1044:
1035:
1019:
1013:
996:
988:
948:
946:
942:
921:
917:
905:
901:
896:
863:
847:
843:
822:
818:
816:
813:
812:
803:
793:
756:
742:
740:
736:
724:
720:
697:
693:
676:
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670:
664:
663:
658:
650:
646:
645:
629:
625:
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585:
579:
568:
557:
551:
541:
535:
528:
525:
517:
514:
474:
468:
462:
458:
450:
425:
414:
411:
410:
402:defined on the
392:
386:
383:
328: for
326:
281:
278:
277:
238:
234:
218:
214:
196:
185:
172:
158:
154:
139:
128:
123:
119:
118:
98:
95:
94:
61:
41:Erich Hecke
23:
22:
15:
12:
11:
5:
1832:
1822:
1821:
1807:
1806:
1785:
1775:
1746:
1708:
1670:
1652:
1644:
1618:
1615:
1614:
1613:
1608:
1603:
1598:
1591:
1588:
1576:group algebras
1539:Euler products
1523:eigenfunctions
1497:
1478:Main article:
1475:
1474:Hecke algebras
1472:
1468:
1467:
1456:
1453:
1450:
1447:
1444:
1438:
1431:
1427:
1422:
1418:
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1380:
1377:
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1366:
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1360:
1357:
1353:
1349:
1344:
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1334:
1330:
1325:
1322:
1317:
1313:
1309:
1306:
1301:
1297:
1279:
1265:
1256:
1248:
1239:
1228:
1217:
1190:
1180:
1170:
1163:
1162:
1151:
1144:
1140:
1135:
1131:
1128:
1124:
1118:
1115:
1112:
1108:
1102:
1099:
1096:
1093:
1090:
1086:
1082:
1079:
1076:
1073:
1070:
1066:
1062:
1057:
1053:
1031:
1009:
992:
985:
984:
973:
969:
964:
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954:
951:
945:
941:
935:
932:
928:
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920:
916:
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904:
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890:
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881:
878:
875:
872:
869:
866:
862:
856:
853:
850:
846:
842:
839:
836:
833:
830:
825:
821:
790:
789:
778:
774:
768:
765:
762:
759:
754:
751:
748:
745:
739:
735:
730:
727:
723:
719:
716:
713:
710:
707:
700:
696:
692:
689:
686:
682:
674:
671:
669:
666:
665:
662:
659:
657:
654:
653:
649:
644:
638:
635:
632:
628:
624:
621:
618:
615:
612:
607:
603:
549:
527:be the set of
521:
513:
510:
447:
446:
435:
431:
428:
424:
421:
418:
391:some function
382:
379:
375:modular curves
369:, who treated
363:
362:
351:
348:
345:
342:
339:
336:
333:
324:
321:
318:
315:
312:
309:
306:
303:
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127:
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117:
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108:
105:
102:
60:
57:
37:Hecke operator
9:
6:
4:
3:
2:
1831:
1820:
1819:Modular forms
1817:
1816:
1814:
1805:
1801:
1797:
1793:
1789:
1786:
1783:
1779:
1776:
1773:
1769:
1765:
1761:
1760:
1755:
1751:
1747:
1744:
1740:
1736:
1732:
1728:
1724:
1720:
1717:(in German),
1716:
1715:
1709:
1706:
1702:
1698:
1694:
1690:
1686:
1682:
1679:(in German),
1678:
1677:
1671:
1667:
1663:
1662:
1657:
1653:
1651:
1647:
1641:
1637:
1632:
1631:
1625:
1621:
1620:
1612:
1609:
1607:
1604:
1602:
1601:Hecke algebra
1599:
1597:
1594:
1593:
1587:
1585:
1581:
1577:
1573:
1569:
1565:
1561:
1555:
1550:
1545:
1540:
1536:
1532:
1528:
1527:Euler product
1524:
1520:
1516:
1512:
1507:
1500:
1496:
1491:
1487:
1481:
1480:Hecke algebra
1471:
1454:
1451:
1448:
1445:
1442:
1436:
1429:
1425:
1420:
1416:
1413:
1409:
1403:
1400:
1397:
1393:
1384:
1381:
1378:
1367:
1364:
1361:
1358:
1355:
1351:
1347:
1342:
1338:
1332:
1328:
1323:
1320:
1315:
1311:
1307:
1304:
1299:
1295:
1287:
1286:
1285:
1278:
1273:
1264:
1255:
1251:
1247:
1242:
1238:
1231:
1227:
1220:
1216:
1211:
1206:
1200:
1193:
1189:
1179:
1169:
1149:
1142:
1138:
1133:
1129:
1126:
1122:
1116:
1113:
1110:
1106:
1097:
1094:
1091:
1080:
1077:
1074:
1071:
1068:
1064:
1060:
1055:
1051:
1043:
1042:
1041:
1038:
1034:
1030:
1026:
1022:
1016:
1012:
1008:
1004:
1000:
995:
991:
971:
967:
962:
958:
955:
952:
949:
943:
939:
930:
926:
918:
914:
906:
902:
898:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
860:
854:
851:
848:
844:
840:
834:
828:
823:
819:
811:
810:
809:
806:
801:
796:
776:
772:
766:
763:
760:
757:
752:
749:
746:
743:
737:
733:
728:
725:
717:
714:
711:
708:
698:
694:
684:
680:
672:
667:
660:
655:
647:
642:
636:
633:
630:
626:
622:
616:
610:
605:
601:
593:
592:
591:
588:
582:
575:
571:
564:
560:
556:
555:modular group
548:
544:
538:
534:
524:
520:
509:
507:
503:
502:modular group
499:
498:double cosets
494:
492:
488:
484:
480:
479:Modular forms
471:
465:
456:
429:
419:
416:
409:
408:
407:
405:
399:
395:
389:
378:
376:
372:
368:
367:Adolf Hurwitz
349:
346:
340:
337:
334:
319:
313:
307:
301:
298:
292:
289:
283:
276:
275:
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219:
215:
208:
202:
192:
189:
186:
182:
178:
173:
168:
159:
155:
151:
148:
135:
132:
129:
125:
120:
115:
112:
106:
93:
92:
91:
89:
85:
81:
77:
73:
69:
65:
56:
54:
50:
49:vector spaces
46:
42:
39:, studied by
38:
34:
33:modular forms
30:
19:
1795:
1791:
1781:
1763:
1757:
1718:
1712:
1680:
1674:
1659:
1649:
1629:
1580:braid groups
1570:
1563:
1559:
1553:
1548:
1543:
1511:self-adjoint
1505:
1498:
1494:
1483:
1469:
1276:
1271:
1262:
1253:
1249:
1245:
1240:
1236:
1229:
1225:
1218:
1214:
1204:
1198:
1191:
1187:
1177:
1167:
1164:
1036:
1032:
1028:
1024:
1020:
1014:
1010:
1006:
1002:
998:
993:
989:
986:
804:
794:
791:
586:
580:
573:
569:
562:
558:
553:be the full
546:
542:
536:
522:
518:
515:
495:
469:
463:
448:
397:
393:
387:
384:
364:
268:
62:
36:
26:
1766:: 117–124,
1721:: 316–351,
1027:) = Σ
533:determinant
491:homotheties
487:homogeneous
84:1937a,1937b
45:1937a,1937b
29:mathematics
1788:Don Zagier
1772:46.0605.01
1743:0016.35503
1705:0015.40202
1617:References
1272:normalized
798:is in the
578:of weight
1735:0025-5831
1697:0025-5831
1666:EMS Press
1537:that has
1452:≥
1401:−
1352:∑
1114:−
1065:∑
915:∑
861:∑
852:−
726:−
691:∖
688:Γ
685:∈
643:∑
634:−
461:of index
455:subgroups
453:that are
427:Λ
417:∑
314:τ
302:τ
284:τ
243:π
203:τ
198:∞
183:∑
152:−
141:∞
126:∏
101:Δ
76:Ramanujan
72:cusp form
1813:Category
1752:(1917),
1683:: 1–28,
1626:(1990),
1590:See also
1274:so that
475:Λ′
451:Λ′
430:′
404:lattices
373:between
1668:, 2001
1533:is the
1284:, then
1005:)) = Σ
500:in the
82: (
66: (
64:Mordell
59:History
43: (
1802:
1770:
1741:
1733:
1703:
1695:
1642:
1560:τ
1440:
1246:λ
1226:λ
792:where
584:, the
543:Γ
506:adelic
398:Λ
1794:, in
1503:with
1234:then
1208:is a
1175:then
269:is a
80:Hecke
1800:ISBN
1731:ISSN
1693:ISSN
1640:ISBN
1359:>
1260:and
1072:>
874:>
561:(2,
540:and
516:Let
485:and
68:1917
35:, a
1768:JFM
1739:Zbl
1723:doi
1719:114
1701:Zbl
1685:doi
1681:114
1578:of
1282:= 1
1268:≠ 0
1183:= 0
1173:= 0
927:mod
545:=
529:2×2
470:n=2
457:of
74:of
27:In
1815::
1790:,
1780:,
1764:19
1762:,
1756:,
1737:,
1729:,
1699:,
1691:,
1664:,
1658:,
1638:,
1568:.
1455:1.
1244:=
1040::
559:SL
477:.
350:1.
273::
174:24
55:.
1784:.
1725::
1687::
1566:)
1564:n
1562:(
1554:p
1544:p
1506:n
1499:n
1495:T
1449:n
1446:,
1443:m
1437:,
1430:2
1426:r
1421:/
1417:n
1414:m
1410:a
1404:1
1398:k
1394:r
1388:)
1385:n
1382:,
1379:m
1376:(
1372:|
1368:r
1365:,
1362:0
1356:r
1348:=
1343:n
1339:a
1333:m
1329:a
1324:,
1321:f
1316:m
1312:a
1308:=
1305:f
1300:m
1296:T
1280:1
1277:a
1266:1
1263:a
1257:1
1254:a
1250:m
1241:m
1237:a
1230:m
1219:m
1215:T
1205:f
1199:k
1192:k
1188:S
1181:0
1178:b
1171:0
1168:a
1150:.
1143:2
1139:r
1134:/
1130:n
1127:m
1123:a
1117:1
1111:k
1107:r
1101:)
1098:n
1095:,
1092:m
1089:(
1085:|
1081:r
1078:,
1075:0
1069:r
1061:=
1056:n
1052:b
1037:q
1033:n
1029:a
1025:z
1023:(
1021:f
1015:q
1011:n
1007:b
1003:z
1001:(
999:f
997:(
994:m
990:T
972:,
968:)
963:d
959:b
956:+
953:z
950:a
944:(
940:f
934:)
931:d
924:(
919:b
907:k
903:d
899:1
892:m
889:=
886:d
883:a
880:,
877:0
871:d
868:,
865:a
855:1
849:k
845:m
841:=
838:)
835:z
832:(
829:f
824:m
820:T
805:m
795:z
777:,
773:)
767:d
764:+
761:z
758:c
753:b
750:+
747:z
744:a
738:(
734:f
729:k
722:)
718:d
715:+
712:z
709:c
706:(
699:m
695:M
681:)
673:d
668:c
661:b
656:a
648:(
637:1
631:k
627:m
623:=
620:)
617:z
614:(
611:f
606:m
602:T
587:m
581:k
576:)
574:z
572:(
570:f
565:)
563:Z
550:1
547:M
537:m
523:m
519:M
464:n
459:Λ
434:)
423:(
420:f
400:)
396:(
394:f
388:n
347:=
344:)
341:n
338:,
335:m
332:(
323:)
320:n
317:(
311:)
308:m
305:(
299:=
296:)
293:n
290:m
287:(
254:,
249:z
246:i
240:2
236:e
232:=
229:q
225:,
220:n
216:q
212:)
209:n
206:(
193:1
190:=
187:n
179:=
169:)
165:)
160:n
156:q
149:1
146:(
136:1
133:=
130:n
121:(
116:q
113:=
110:)
107:z
104:(
20:)
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