795:
1027:) following from such a hypothetical trace formula. In 2011, Barnet-Lamb, Geraghty, Harris, and Taylor proved a generalized version of the Sato–Tate conjecture for an arbitrary non-CM holomorphic modular form of weight greater than or equal to two, by improving the potential modularity results of previous papers. The prior issues involved with the trace formula were solved by
549:
790:{\displaystyle \lim _{N\to \infty }{\frac {\#\{p\leq N:\alpha \leq \theta _{p}\leq \beta \}}{\#\{p\leq N\}}}={\frac {2}{\pi }}\int _{\alpha }^{\beta }\sin ^{2}\theta \,d\theta ={\frac {1}{\pi }}\left(\beta -\alpha +\sin(\alpha )\cos(\alpha )-\sin(\beta )\cos(\beta )\right)}
458:
902:
257:
542:
987:
1203:
1339:
is multiplicative, rather than additive. In practice this is the typical case, so the condition can be thought of as mild. In more classical terms, the result applies where the
286:
481:
501:
358:
815:
1796:
1791:
1682:
1654:
1568:
806:
179:
1459:
Harris, Michael; Shepherd-Barron, Nicholas; Taylor, Richard (2010), "A family of Calabi–Yau varieties and potential automorphy",
1039:
1016:
1734:
1631:
1559:
Harris, M. (2011). "An introduction to the stable trace formula". In Clozel, L.; Harris, M.; Labesse, J.-P.; Ngô, B. C. (eds.).
188:
1250:
1028:
312:
304:
81:
73:
1008:
In 2008, Clozel, Harris, Shepherd-Barron, and Taylor published a proof of the Sato–Tate conjecture for elliptic curves over
1739:
506:
1230:
1012:
satisfying a certain condition: of having multiplicative reduction at some prime, in a series of three joint papers.
320:
1269:
1081:, there is a conjectural correspondence between (unitarized) characteristic polynomials of Frobenius elements and
997:
949:
143:
42:
1820:
308:
128:
77:
1800:[d'après Clozel, Harris, Shepherd-Barron, Taylor], Bourbaki seminar June 2007 by Henri Carayol (PDF)
1164:
1825:
1765:
David, Chantal; Pappalardi, Francesco (1999-01-01). "Average
Frobenius distributions of elliptic curves".
1423:
1370:
1238:
1147:
336:
1805:
Video introducing
Elliptic curves and its relation to Sato-Tate conjecture, Imperial College London, 2014
1590:
1461:
1059:
323:, Harris, and Taylor in 2011. Several generalizations to other algebraic varieties and fields are open.
316:
1830:
1697:
Koblitz, Neal (1988), "Primality of the number of points on an elliptic curve over a finite field",
1428:
1375:
1063:
936:
28:
265:
1009:
296:
1563:. Vol. I: Stabilization of the trace formula. Boston: International Press. pp. 3–47.
1678:
1650:
1564:
1086:
1051:
1020:
920:
466:
1706:
1599:
1519:
1470:
1433:
1380:
486:
1720:
1533:
1484:
1445:
1392:
1786:
1716:
1529:
1480:
1441:
1388:
1273:
1082:
171:
elements, the conjecture gives an answer to the distribution of the second-order term for
124:
1109:) then gives the conjectured distribution, and the classical case is USp(2) =
1744:
1336:
924:
300:
289:
110:
65:
1632:"Richard Taylor, Institute for Advanced Study: 2015 Breakthrough Prize in Mathematics"
1814:
1328:
1234:
453:{\displaystyle p+1-N_{p}=2{\sqrt {p}}\cos \theta _{p}~~(0\leq \theta _{p}\leq \pi ).}
1213:
1130:
1102:
1078:
1055:
1032:
164:
132:
897:{\displaystyle {\frac {(p+1)-N_{p}}{2{\sqrt {p}}}}={\frac {a_{p}}{2{\sqrt {p}}}}}
1604:
1585:
1475:
1340:
1331:(and at least for elliptic curves over the rational numbers there are some such
1277:
916:
106:
98:
1506:
Barnet-Lamb, Thomas; Geraghty, David; Harris, Michael; Taylor, Richard (2011).
1406:
Clozel, Laurent; Harris, Michael; Taylor, Richard (2008). "Automorphy for some
1146:, the trace of Frobenius that appears in the formula. For the typical case (no
1804:
1437:
1384:
1126:
993:
139:
38:
1711:
1074:
1280:). The known analytic results on these answer even more precise questions.
1561:
The stable trace formula, Shimura varieties, and arithmetic applications
919:
accounting for the remainder and with the denominator as given they are
1524:
1507:
1066:. In particular there is a conjectural theory for curves of genus
1042:"for numerous breakthrough results in (...) the Sato–Tate conjecture."
1024:
1508:"A family of Calabi–Yau varieties and potential automorphy. II"
1586:"Galois representations arising from some compact Shimura varieties"
1216:(1988) provided detailed conjectures for the case of a prime number
1268:
In the case of an elliptic curve with complex multiplication, the
1110:
1094:
1619:
1547:
335:
be an elliptic curve defined over the rational numbers without
1735:"Concordia Mathematician Recognized for Research Excellence"
1000:(independently, and around 1960, published somewhat later).
1496:
See
Carayol's Bourbaki seminar of 17 June 2007 for details.
1241:
proved an averaged version of the Lang–Trotter conjecture.
907:
is between -1 and 1. Thus it can be expressed as cos
1050:
There are generalisations, involving the distribution of
1015:
Further results are conditional on improved forms of the
252:{\displaystyle N_{p}/p=1+\mathrm {O} (1/\!{\sqrt {p}})\ }
1023:
of a result for the product of two elliptic curves (not
288:, and the point of the conjecture is to predict how the
295:
The original conjecture and its generalization to all
1647:
1167:
1150:, trace ≠ 0) their formula states that the number of
952:
935:
doesn't have complex multiplication, states that the
818:
552:
509:
489:
469:
361:
268:
191:
156:denotes the number of points on the elliptic curve
87:
61:
48:
34:
24:
1197:
981:
896:
789:
536:
495:
475:
452:
280:
251:
1649:, Providence, RI: American Mathematical Society,
537:{\displaystyle 0\leq \alpha <\beta \leq \pi ,}
315:under mild assumptions in 2008, and completed by
235:
554:
146:independently posed the conjecture around 1960.
1645:Katz, Nicholas M. & Sarnak, Peter (1999),
1353:Taylor, Richard (2008). "Automorphy for some
16:Mathematical conjecture about elliptic curves
8:
1306:in the volume (O. F. G. Schilling, editor),
1304:Algebraic cycles and poles of zeta functions
1121:There are also more refined statements. The
632:
620:
612:
575:
19:
1073:Under the random matrix model developed by
982:{\displaystyle \sin ^{2}\theta \,d\theta .}
1792:Michael Harris notes, with statement (PDF)
1767:International Mathematics Research Notices
18:
1710:
1603:
1523:
1474:
1427:
1374:
1335:), the type in the singular fibre of the
1178:
1171:
1166:
969:
957:
951:
884:
874:
868:
855:
844:
819:
817:
692:
682:
670:
660:
655:
641:
600:
569:
557:
551:
508:
488:
468:
432:
407:
390:
378:
360:
267:
236:
230:
219:
202:
196:
190:
1038:In 2015, Richard Taylor was awarded the
1261:
1133:states the asymptotic number of primes
1669:Lang, Serge; Trotter, Hale F. (1976),
1198:{\displaystyle c{\sqrt {X}}/\log X\ }
7:
1787:Report on Barry Mazur giving context
1416:Publ. Math. Inst. Hautes Études Sci
1363:Publ. Math. Inst. Hautes Études Sci
915:; in geometric terms there are two
807:Hasse's theorem on elliptic curves
617:
572:
564:
275:
220:
180:Hasse's theorem on elliptic curves
14:
1040:Breakthrough Prize in Mathematics
463:Then, for every two real numbers
352:as the solution to the equation
119:obtained from an elliptic curve
1618:See p. 71 and Corollary 8.9 of
1410:-adic lifts of automorphic mod
1357:-adic lifts of automorphic mod
1308:Arithmetical Algebraic Geometry
1699:Pacific Journal of Mathematics
1251:Wigner semicircle distribution
834:
822:
779:
773:
764:
758:
746:
740:
731:
725:
561:
444:
419:
272:
243:
224:
109:statement about the family of
1:
1740:Canadian Mathematical Society
1671:Frobenius Distributions in GL
1361:Galois representations. II".
52:
1743:. 2013-04-15. Archived from
1302:It is mentioned in J. Tate,
1017:Arthur–Selberg trace formula
281:{\displaystyle p\to \infty }
1677:, Berlin: Springer-Verlag,
1605:10.4007/annals.2011.173.3.9
1476:10.4007/annals.2010.171.779
1272:is expressed in terms of a
1231:elliptic curve cryptography
1847:
1798:La Conjecture de Sato–Tate
1512:Publ. Res. Inst. Math. Sci
1208:with a specified constant
1438:10.1007/s10240-008-0016-1
1414:Galois representations".
1385:10.1007/s10240-008-0015-2
1712:10.2140/pjm.1988.131.157
309:Nicholas Shepherd-Barron
78:Nicholas Shepherd-Barron
1620:Barnet-Lamb et al. 2011
1548:Barnet-Lamb et al. 2011
1123:Lang–Trotter conjecture
476:{\displaystyle \alpha }
1584:Shin, Sug Woo (2011).
1310:, pages 93–110 (1965).
1199:
1148:complex multiplication
1137:with a given value of
1060:Galois representations
983:
898:
791:
538:
497:
496:{\displaystyle \beta }
477:
454:
339:. For a prime number
337:complex multiplication
282:
253:
1591:Annals of Mathematics
1462:Annals of Mathematics
1270:Hasse–Weil L-function
1200:
984:
899:
792:
539:
498:
478:
455:
283:
254:
1289:To normalise, put 2/
1239:Francesco Pappalardi
1165:
950:
929:Sato–Tate conjecture
816:
550:
507:
487:
467:
359:
266:
189:
103:Sato–Tate conjecture
20:Sato–Tate conjecture
1070: > 1.
1010:totally real fields
943:is proportional to
937:probability measure
665:
297:totally real fields
29:Arithmetic geometry
21:
1319:That is, for some
1195:
1158:is asymptotically
1052:Frobenius elements
979:
894:
787:
651:
568:
534:
493:
473:
450:
317:Thomas Barnet-Lamb
278:
249:
69:Thomas Barnet-Lamb
1807:(Last 15 minutes)
1684:978-0-387-07550-1
1656:978-0-8218-1017-0
1570:978-1-57146-227-5
1194:
1176:
1087:compact Lie group
1083:conjugacy classes
1021:conditional proof
921:complex conjugate
892:
889:
863:
860:
700:
649:
636:
553:
418:
415:
395:
248:
241:
163:defined over the
95:
94:
1838:
1775:
1774:
1762:
1756:
1755:
1753:
1752:
1731:
1725:
1723:
1714:
1694:
1688:
1687:
1666:
1660:
1659:
1642:
1636:
1635:
1628:
1622:
1616:
1610:
1609:
1607:
1598:(3): 1645–1741.
1581:
1575:
1574:
1556:
1550:
1544:
1538:
1537:
1527:
1525:10.2977/PRIMS/31
1503:
1497:
1494:
1488:
1487:
1478:
1456:
1450:
1449:
1431:
1403:
1397:
1396:
1378:
1350:
1344:
1343:is not integral.
1317:
1311:
1300:
1294:
1287:
1281:
1274:Hecke L-function
1266:
1204:
1202:
1201:
1196:
1192:
1182:
1177:
1172:
1064:étale cohomology
1058:involved in the
988:
986:
985:
980:
962:
961:
903:
901:
900:
895:
893:
891:
890:
885:
879:
878:
869:
864:
862:
861:
856:
850:
849:
848:
820:
796:
794:
793:
788:
786:
782:
701:
693:
675:
674:
664:
659:
650:
642:
637:
635:
615:
605:
604:
570:
567:
543:
541:
540:
535:
502:
500:
499:
494:
482:
480:
479:
474:
459:
457:
456:
451:
437:
436:
416:
413:
412:
411:
396:
391:
383:
382:
287:
285:
284:
279:
258:
256:
255:
250:
246:
242:
237:
234:
223:
206:
201:
200:
129:reduction modulo
125:rational numbers
57:
54:
22:
1846:
1845:
1841:
1840:
1839:
1837:
1836:
1835:
1821:Elliptic curves
1811:
1810:
1783:
1778:
1764:
1763:
1759:
1750:
1748:
1733:
1732:
1728:
1696:
1695:
1691:
1685:
1674:
1668:
1667:
1663:
1657:
1644:
1643:
1639:
1630:
1629:
1625:
1617:
1613:
1583:
1582:
1578:
1571:
1558:
1557:
1553:
1545:
1541:
1505:
1504:
1500:
1495:
1491:
1458:
1457:
1453:
1429:10.1.1.143.9755
1405:
1404:
1400:
1376:10.1.1.116.9791
1352:
1351:
1347:
1318:
1314:
1301:
1297:
1288:
1284:
1267:
1263:
1259:
1247:
1229:, motivated by
1228:
1163:
1162:
1145:
1119:
1048:
1046:Generalisations
1019:. Harris has a
1006:
992:This is due to
953:
948:
947:
880:
870:
851:
840:
821:
814:
813:
803:
706:
702:
666:
616:
596:
571:
548:
547:
505:
504:
485:
484:
465:
464:
428:
403:
374:
357:
356:
351:
329:
264:
263:
192:
187:
186:
176:
161:
154:
117:
111:elliptic curves
80:
76:
72:
70:
68:
55:
41:
17:
12:
11:
5:
1844:
1842:
1834:
1833:
1828:
1823:
1813:
1812:
1809:
1808:
1802:
1794:
1789:
1782:
1781:External links
1779:
1777:
1776:
1757:
1726:
1705:(1): 157–165,
1689:
1683:
1672:
1661:
1655:
1637:
1623:
1611:
1576:
1569:
1551:
1539:
1498:
1489:
1469:(2): 779–813,
1451:
1398:
1345:
1312:
1295:
1282:
1260:
1258:
1255:
1254:
1253:
1246:
1243:
1224:
1206:
1205:
1191:
1188:
1185:
1181:
1175:
1170:
1141:
1118:
1115:
1093:) =
1047:
1044:
1029:Michael Harris
1005:
1002:
990:
989:
978:
975:
972:
968:
965:
960:
956:
925:absolute value
905:
904:
888:
883:
877:
873:
867:
859:
854:
847:
843:
839:
836:
833:
830:
827:
824:
802:
799:
798:
797:
785:
781:
778:
775:
772:
769:
766:
763:
760:
757:
754:
751:
748:
745:
742:
739:
736:
733:
730:
727:
724:
721:
718:
715:
712:
709:
705:
699:
696:
691:
688:
685:
681:
678:
673:
669:
663:
658:
654:
648:
645:
640:
634:
631:
628:
625:
622:
619:
614:
611:
608:
603:
599:
595:
592:
589:
586:
583:
580:
577:
574:
566:
563:
560:
556:
533:
530:
527:
524:
521:
518:
515:
512:
492:
472:
461:
460:
449:
446:
443:
440:
435:
431:
427:
424:
421:
410:
406:
402:
399:
394:
389:
386:
381:
377:
373:
370:
367:
364:
347:
328:
325:
321:David Geraghty
313:Richard Taylor
305:Michael Harris
301:Laurent Clozel
299:was proved by
277:
274:
271:
260:
259:
245:
240:
233:
229:
226:
222:
218:
215:
212:
209:
205:
199:
195:
174:
159:
152:
115:
93:
92:
89:
88:First proof in
85:
84:
82:Richard Taylor
74:Michael Harris
71:David Geraghty
66:Laurent Clozel
63:
62:First proof by
59:
58:
50:
49:Conjectured in
46:
45:
36:
35:Conjectured by
32:
31:
26:
15:
13:
10:
9:
6:
4:
3:
2:
1843:
1832:
1829:
1827:
1826:Finite fields
1824:
1822:
1819:
1818:
1816:
1806:
1803:
1801:
1799:
1795:
1793:
1790:
1788:
1785:
1784:
1780:
1773:(4): 165–183.
1772:
1768:
1761:
1758:
1747:on 2017-02-01
1746:
1742:
1741:
1736:
1730:
1727:
1722:
1718:
1713:
1708:
1704:
1700:
1693:
1690:
1686:
1680:
1676:
1665:
1662:
1658:
1652:
1648:
1641:
1638:
1633:
1627:
1624:
1621:
1615:
1612:
1606:
1601:
1597:
1593:
1592:
1587:
1580:
1577:
1572:
1566:
1562:
1555:
1552:
1549:
1546:Theorem B of
1543:
1540:
1535:
1531:
1526:
1521:
1517:
1513:
1509:
1502:
1499:
1493:
1490:
1486:
1482:
1477:
1472:
1468:
1464:
1463:
1455:
1452:
1447:
1443:
1439:
1435:
1430:
1425:
1421:
1417:
1413:
1409:
1402:
1399:
1394:
1390:
1386:
1382:
1377:
1372:
1368:
1364:
1360:
1356:
1349:
1346:
1342:
1338:
1334:
1330:
1329:bad reduction
1326:
1322:
1316:
1313:
1309:
1305:
1299:
1296:
1292:
1286:
1283:
1279:
1276:(a result of
1275:
1271:
1265:
1262:
1256:
1252:
1249:
1248:
1244:
1242:
1240:
1236:
1235:Chantal David
1232:
1227:
1223:
1220:of points on
1219:
1215:
1211:
1189:
1186:
1183:
1179:
1173:
1168:
1161:
1160:
1159:
1157:
1153:
1149:
1144:
1140:
1136:
1132:
1128:
1124:
1116:
1114:
1112:
1108:
1104:
1100:
1098:
1092:
1088:
1084:
1080:
1076:
1071:
1069:
1065:
1061:
1057:
1056:Galois groups
1053:
1045:
1043:
1041:
1036:
1034:
1030:
1026:
1022:
1018:
1013:
1011:
1003:
1001:
999:
995:
976:
973:
970:
966:
963:
958:
954:
946:
945:
944:
942:
938:
934:
930:
927: 1. The
926:
922:
918:
914:
911:for an angle
910:
886:
881:
875:
871:
865:
857:
852:
845:
841:
837:
831:
828:
825:
812:
811:
810:
808:
800:
783:
776:
770:
767:
761:
755:
752:
749:
743:
737:
734:
728:
722:
719:
716:
713:
710:
707:
703:
697:
694:
689:
686:
683:
679:
676:
671:
667:
661:
656:
652:
646:
643:
638:
629:
626:
623:
609:
606:
601:
597:
593:
590:
587:
584:
581:
578:
558:
546:
545:
544:
531:
528:
525:
522:
519:
516:
513:
510:
490:
470:
447:
441:
438:
433:
429:
425:
422:
408:
404:
400:
397:
392:
387:
384:
379:
375:
371:
368:
365:
362:
355:
354:
353:
350:
346:
342:
338:
334:
326:
324:
322:
318:
314:
310:
306:
302:
298:
293:
291:
269:
238:
231:
227:
216:
213:
210:
207:
203:
197:
193:
185:
184:
183:
181:
177:
170:
166:
162:
155:
147:
145:
141:
137:
134:
133:prime numbers
130:
126:
122:
118:
112:
108:
104:
100:
90:
86:
83:
79:
75:
67:
64:
60:
51:
47:
44:
40:
37:
33:
30:
27:
23:
1797:
1770:
1766:
1760:
1749:. Retrieved
1745:the original
1738:
1729:
1702:
1698:
1692:
1670:
1664:
1646:
1640:
1626:
1614:
1595:
1589:
1579:
1560:
1554:
1542:
1518:(1): 29–98.
1515:
1511:
1501:
1492:
1466:
1460:
1454:
1419:
1415:
1411:
1407:
1401:
1366:
1362:
1358:
1354:
1348:
1332:
1324:
1320:
1315:
1307:
1303:
1298:
1290:
1285:
1264:
1225:
1221:
1217:
1214:Neal Koblitz
1209:
1207:
1155:
1151:
1142:
1138:
1134:
1131:Hale Trotter
1122:
1120:
1106:
1103:Haar measure
1096:
1090:
1079:Peter Sarnak
1072:
1067:
1049:
1037:
1033:Sug Woo Shin
1014:
1007:
991:
940:
932:
928:
912:
908:
906:
809:, the ratio
804:
462:
348:
344:
340:
332:
330:
294:
261:
172:
168:
165:finite field
157:
150:
148:
135:
120:
113:
102:
96:
1831:Conjectures
1369:: 183–239.
1341:j-invariant
1337:Néron model
1278:Max Deuring
1233:. In 1999,
1117:Refinements
917:eigenvalues
131:almost all
107:statistical
99:mathematics
56: 1960
1815:Categories
1751:2018-01-15
1675:extensions
1257:References
1127:Serge Lang
1125:(1976) of
994:Mikio Sato
503:for which
140:Mikio Sato
39:Mikio Sato
1424:CiteSeerX
1422:: 1–181.
1371:CiteSeerX
1293:in front.
1187:
1075:Nick Katz
1025:isogenous
998:John Tate
974:θ
967:θ
964:
838:−
777:β
771:
762:β
756:
750:−
744:α
738:
729:α
723:
714:α
711:−
708:β
698:π
687:θ
680:θ
677:
662:β
657:α
653:∫
647:π
627:≤
618:#
610:β
607:≤
598:θ
594:≤
591:α
582:≤
573:#
565:∞
562:→
529:π
526:≤
523:β
517:α
514:≤
491:β
471:α
442:π
439:≤
430:θ
426:≤
405:θ
401:
372:−
343:, define
327:Statement
276:∞
273:→
144:John Tate
123:over the
43:John Tate
1245:See also
1105:on USp(2
292:varies.
1721:0917870
1534:2827723
1485:2630056
1446:2470687
1393:2470688
1101:. The
1085:in the
931:, when
923:and of
801:Details
1719:
1681:
1653:
1567:
1532:
1483:
1444:
1426:
1391:
1373:
1323:where
1291:π
1193:
1154:up to
1031:, and
417:
414:
311:, and
290:O-term
247:
101:, the
1111:SU(2)
1089:USp(2
1004:Proof
178:. By
167:with
105:is a
25:Field
1679:ISBN
1651:ISBN
1565:ISBN
1327:has
1237:and
1129:and
1077:and
996:and
520:<
483:and
331:Let
142:and
91:2011
1771:199
1707:doi
1703:131
1600:doi
1596:173
1520:doi
1471:doi
1467:171
1434:doi
1420:108
1381:doi
1367:108
1184:log
1095:Sp(
1062:on
1054:in
955:sin
939:of
805:By
768:cos
753:sin
735:cos
720:sin
668:sin
555:lim
398:cos
262:as
149:If
127:by
97:In
1817::
1769:.
1737:.
1717:MR
1715:,
1701:,
1594:.
1588:.
1530:MR
1528:.
1516:47
1514:.
1510:.
1481:MR
1479:,
1465:,
1442:MR
1440:.
1432:.
1418:.
1389:MR
1387:.
1379:.
1365:.
1212:.
1113:.
1035:.
319:,
307:,
303:,
182:,
138:.
53:c.
1754:.
1724:.
1709::
1673:2
1634:.
1608:.
1602::
1573:.
1536:.
1522::
1473::
1448:.
1436::
1412:l
1408:l
1395:.
1383::
1359:l
1355:l
1333:p
1325:E
1321:p
1226:p
1222:E
1218:q
1210:c
1190:X
1180:/
1174:X
1169:c
1156:X
1152:p
1143:p
1139:a
1135:p
1107:n
1099:)
1097:n
1091:n
1068:n
977:.
971:d
959:2
941:θ
933:E
913:θ
909:θ
887:p
882:2
876:p
872:a
866:=
858:p
853:2
846:p
842:N
835:)
832:1
829:+
826:p
823:(
784:)
780:)
774:(
765:)
759:(
747:)
741:(
732:)
726:(
717:+
704:(
695:1
690:=
684:d
672:2
644:2
639:=
633:}
630:N
624:p
621:{
613:}
602:p
588::
585:N
579:p
576:{
559:N
532:,
511:0
448:.
445:)
434:p
423:0
420:(
409:p
393:p
388:2
385:=
380:p
376:N
369:1
366:+
363:p
349:p
345:θ
341:p
333:E
270:p
244:)
239:p
232:/
228:1
225:(
221:O
217:+
214:1
211:=
208:p
204:/
198:p
194:N
175:p
173:N
169:p
160:p
158:E
153:p
151:N
136:p
121:E
116:p
114:E
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.