1143:
1494:
967:
1344:
257:
386:
831:
1359:
201:
939:
678:
631:
580:
538:
479:
436:
329:
110:
1138:{\displaystyle \#\operatorname {Bun} _{G}(X)(\mathbf {F} _{q})=q^{\dim \operatorname {Bun} _{G}(X)}\operatorname {tr} (\phi ^{-1}|H^{*}(\operatorname {Bun} _{G}(X);\mathbb {Z} _{l}))}
1249:
1190:
290:
143:
53:
732:
1216:
867:
1524:
1238:
761:
1537:
neither left nor right side in the formula converges. Thus, the formula states that the two sides converge to finite numbers and that those numbers coincide.
1555:
1778:
1754:
1716:
768:
889:
767:. It is not of finite type but locally of finite type; one thus usually uses a stratification by open substacks of finite type (cf. the
214:
334:
1732:
J. Heinloth, A.H.W. Schmitt, The
Cohomology Ring of Moduli Stacks of Principal Bundles over Curves, 2010 preprint, available at
1790:
1800:
781:
1489:{\displaystyle \operatorname {tr} (\phi ^{-1}|V_{*})=\sum _{i=0}^{\infty }(-1)^{i}\operatorname {tr} (\phi ^{-1}|V_{i})}
1815:
1347:
155:
1149:
905:
644:
597:
546:
504:
490:
445:
402:
295:
76:
878:
1339:{\displaystyle \#\operatorname {Bun} _{G}(X)(\mathbf {F} _{q})=\sum _{P}{1 \over \#\operatorname {Aut} (P)}}
899:
1564:
771:), also for parahoric G over curve X see and for G only a flat group scheme of finite type over X see.
591:
1741:
1166:
266:
119:
29:
1527:
893:
687:
1497:
1241:
56:
1199:
1604:
836:
1750:
1712:
494:
1693:
1704:
1614:
20:
1764:
1749:, Annals of Mathematics Studies, vol. 199, Princeton, NJ: Princeton University Press,
1726:
1626:
1502:
1223:
1760:
1722:
1622:
1599:
Arasteh Rad, E.; Hartl, Urs (2021), "Uniformizing the moduli stacks of global G-shtukas",
1193:
737:
583:
113:
482:
1809:
958:
950:
498:
954:
681:
442:
is over the field of complex numbers. Roughly, in the complex case, one can define
59:
17:
1708:
587:
1653:
1795:
1618:
543:
In the finite field case, it is not common to define the homotopy type of
1703:, Trends in Mathematics, Basel: Birkhäuser/Springer, pp. 123–153,
493:. Replacing the quotient stack (which is not a topological space) by a
1655:
The
Lefschetz Trace Formula for the Moduli Stack of Principal Bundles
945:
is over a finite field, introduced by
Behrend in 1993. It states: if
204:
1733:
1609:
778:
is a split reductive group, then the set of connected components
252:{\displaystyle X\times _{\mathbf {F} _{q}}\operatorname {Spec} R}
1779:
Lectures on moduli of principal G-bundles over algebraic curves
381:{\displaystyle \operatorname {Bun} _{G}(X)(\mathbf {F} _{q})}
1694:"Lectures on the moduli stack of vector bundles on a curve"
1557:
Tamagawa
Numbers in the Function Field Case (Lecture 2)
1505:
1362:
1252:
1226:
1202:
1169:
970:
908:
839:
833:
is in a natural bijection with the fundamental group
826:{\displaystyle \pi _{0}(\operatorname {Bun} _{G}(X))}
784:
740:
690:
647:
600:
549:
507:
448:
405:
337:
298:
269:
217:
158:
122:
79:
32:
1518:
1488:
1346:, the sum running over all isomorphism classes of
1338:
1232:
1210:
1184:
1137:
933:
861:
825:
755:
726:
672:
625:
574:
532:
473:
430:
380:
323:
284:
251:
195:
137:
104:
47:
1661:(PhD thesis), University of California, Berkeley
196:{\displaystyle \operatorname {Bun} _{G}(X)(R)=}
1671:
1743:Weil's conjecture for function fields, Vol. 1
8:
1701:Affine flag manifolds and principal bundles
934:{\displaystyle \operatorname {Bun} _{G}(X)}
673:{\displaystyle \operatorname {Bun} _{G}(X)}
626:{\displaystyle \operatorname {Bun} _{G}(X)}
575:{\displaystyle \operatorname {Bun} _{G}(X)}
533:{\displaystyle \operatorname {Bun} _{G}(X)}
485:of the space of holomorphic connections on
474:{\displaystyle \operatorname {Bun} _{G}(X)}
431:{\displaystyle \operatorname {Bun} _{G}(X)}
324:{\displaystyle \operatorname {Bun} _{G}(X)}
105:{\displaystyle \operatorname {Bun} _{G}(X)}
1601:International Mathematics Research Notices
1608:
1510:
1504:
1477:
1468:
1459:
1440:
1421:
1410:
1394:
1385:
1376:
1361:
1309:
1303:
1287:
1282:
1260:
1251:
1225:
1204:
1203:
1201:
1176:
1172:
1171:
1168:
1123:
1119:
1118:
1096:
1083:
1074:
1065:
1032:
1021:
1005:
1000:
978:
969:
913:
907:
844:
838:
802:
789:
783:
739:
689:
652:
646:
605:
599:
554:
548:
512:
506:
497:(which is a topological space) gives the
453:
447:
410:
404:
369:
364:
342:
336:
303:
297:
276:
271:
268:
232:
227:
225:
216:
163:
157:
129:
124:
121:
84:
78:
39:
34:
31:
1740:Gaitsgory, Dennis; Lurie, Jacob (2019),
1639:
1586:
1546:
898:This is a (conjectural) version of the
1674:, Chapter 5: The Trace Formula for Bun
7:
890:Weil conjecture on Tamagawa numbers
438:can also be defined when the curve
1530:on the right absolutely converges.
1422:
1315:
1253:
971:
14:
67:moduli stack of principal bundles
1734:http://www.uni-essen.de/~hm0002/
1283:
1185:{\displaystyle \mathbb {Z} _{l}}
1001:
769:Harder–Narasimhan stratification
365:
285:{\displaystyle \mathbf {F} _{q}}
272:
228:
138:{\displaystyle \mathbf {F} _{q}}
125:
48:{\displaystyle \mathbf {F} _{q}}
35:
1791:Geometric Langlands conjectures
1699:, in Schmitt, Alexander (ed.),
263:In particular, the category of
16:In algebraic geometry, given a
1801:Moduli stack of vector bundles
1554:Lurie, Jacob (April 3, 2013),
1483:
1469:
1452:
1437:
1427:
1400:
1386:
1369:
1330:
1324:
1293:
1278:
1275:
1269:
1159:is a prime number that is not
1132:
1129:
1111:
1105:
1089:
1075:
1058:
1047:
1041:
1011:
996:
993:
987:
928:
922:
856:
850:
820:
817:
811:
795:
750:
744:
727:{\displaystyle (g(X)-1)\dim G}
712:
703:
697:
691:
667:
661:
620:
614:
582:. But one can still define a (
569:
563:
527:
521:
468:
462:
425:
419:
375:
360:
357:
351:
318:
312:
187:
181:
178:
172:
99:
93:
1:
883:
1211:{\displaystyle \mathbb {C} }
1709:10.1007/978-3-0346-0288-4_4
862:{\displaystyle \pi _{1}(G)}
1832:
1672:Gaitsgory & Lurie 2019
1196:is viewed as a subring of
957:with semisimple connected
887:
876:
1692:Heinloth, Jochen (2010),
1652:Behrend, Kai A. (1991),
211:over the relative curve
1150:Behrend's trace formula
900:Lefschetz trace formula
884:Behrend's trace formula
873:The Atiyah–Bott formula
1520:
1490:
1426:
1340:
1234:
1212:
1186:
1139:
935:
863:
827:
757:
728:
674:
627:
576:
534:
475:
432:
382:
325:
286:
253:
197:
139:
106:
49:
1521:
1519:{\displaystyle V_{*}}
1491:
1406:
1341:
1235:
1233:{\displaystyle \phi }
1213:
1187:
1140:
936:
864:
828:
758:
729:
675:
628:
577:
535:
476:
433:
388:, is the category of
383:
326:
287:
254:
198:
140:
107:
50:
1503:
1360:
1250:
1224:
1200:
1167:
968:
906:
837:
782:
756:{\displaystyle g(X)}
738:
688:
645:
598:
547:
505:
446:
403:
335:
296:
267:
215:
156:
120:
77:
30:
26:over a finite field
1642:, Proposition 2.1.2
1619:10.1093/imrn/rnz223
1603:(21): 16121–16192,
1589:, Proposition 2.1.2
1498:graded vector space
1242:geometric Frobenius
879:Atiyah–Bott formula
1816:Algebraic geometry
1516:
1486:
1336:
1308:
1230:
1208:
1182:
1135:
931:
859:
823:
753:
724:
670:
623:
572:
530:
471:
428:
378:
321:
282:
249:
193:
135:
116:given by: for any
102:
45:
1756:978-0-691-18214-8
1718:978-3-0346-0287-7
1630:; see Theorem 2.5
1334:
1299:
1152:for the details)
894:Behrend's formula
641:It is known that
495:homotopy quotient
1823:
1767:
1748:
1729:
1698:
1679:
1669:
1663:
1662:
1660:
1649:
1643:
1637:
1631:
1629:
1612:
1596:
1590:
1584:
1578:
1577:
1576:
1575:
1569:
1563:, archived from
1562:
1551:
1525:
1523:
1522:
1517:
1515:
1514:
1495:
1493:
1492:
1487:
1482:
1481:
1472:
1467:
1466:
1445:
1444:
1425:
1420:
1399:
1398:
1389:
1384:
1383:
1345:
1343:
1342:
1337:
1335:
1333:
1310:
1307:
1292:
1291:
1286:
1265:
1264:
1239:
1237:
1236:
1231:
1217:
1215:
1214:
1209:
1207:
1191:
1189:
1188:
1183:
1181:
1180:
1175:
1148:where (see also
1144:
1142:
1141:
1136:
1128:
1127:
1122:
1101:
1100:
1088:
1087:
1078:
1073:
1072:
1051:
1050:
1037:
1036:
1010:
1009:
1004:
983:
982:
940:
938:
937:
932:
918:
917:
868:
866:
865:
860:
849:
848:
832:
830:
829:
824:
807:
806:
794:
793:
763:is the genus of
762:
760:
759:
754:
733:
731:
730:
725:
679:
677:
676:
671:
657:
656:
637:Basic properties
632:
630:
629:
624:
610:
609:
581:
579:
578:
573:
559:
558:
539:
537:
536:
531:
517:
516:
480:
478:
477:
472:
458:
457:
437:
435:
434:
429:
415:
414:
387:
385:
384:
379:
374:
373:
368:
347:
346:
330:
328:
327:
322:
308:
307:
291:
289:
288:
283:
281:
280:
275:
258:
256:
255:
250:
239:
238:
237:
236:
231:
203:the category of
202:
200:
199:
194:
168:
167:
144:
142:
141:
136:
134:
133:
128:
111:
109:
108:
103:
89:
88:
54:
52:
51:
46:
44:
43:
38:
21:projective curve
1831:
1830:
1826:
1825:
1824:
1822:
1821:
1820:
1806:
1805:
1787:
1774:
1772:Further reading
1757:
1746:
1739:
1719:
1696:
1691:
1688:
1683:
1682:
1677:
1670:
1666:
1658:
1651:
1650:
1646:
1638:
1634:
1598:
1597:
1593:
1585:
1581:
1573:
1571:
1567:
1560:
1553:
1552:
1548:
1543:
1526:, provided the
1506:
1501:
1500:
1473:
1455:
1436:
1390:
1372:
1358:
1357:
1354:and convergent.
1314:
1281:
1256:
1248:
1247:
1222:
1221:
1198:
1197:
1194:l-adic integers
1170:
1165:
1164:
1117:
1092:
1079:
1061:
1028:
1017:
999:
974:
966:
965:
909:
904:
903:
896:
886:
881:
875:
840:
835:
834:
798:
785:
780:
779:
736:
735:
686:
685:
648:
643:
642:
639:
601:
596:
595:
550:
545:
544:
508:
503:
502:
449:
444:
443:
406:
401:
400:
363:
338:
333:
332:
299:
294:
293:
270:
265:
264:
226:
221:
213:
212:
159:
154:
153:
123:
118:
117:
114:algebraic stack
80:
75:
74:
33:
28:
27:
12:
11:
5:
1829:
1827:
1819:
1818:
1808:
1807:
1804:
1803:
1798:
1793:
1786:
1783:
1782:
1781:
1773:
1770:
1769:
1768:
1755:
1737:
1730:
1717:
1687:
1684:
1681:
1680:
1675:
1664:
1644:
1632:
1591:
1579:
1545:
1544:
1542:
1539:
1532:
1531:
1513:
1509:
1485:
1480:
1476:
1471:
1465:
1462:
1458:
1454:
1451:
1448:
1443:
1439:
1435:
1432:
1429:
1424:
1419:
1416:
1413:
1409:
1405:
1402:
1397:
1393:
1388:
1382:
1379:
1375:
1371:
1368:
1365:
1355:
1332:
1329:
1326:
1323:
1320:
1317:
1313:
1306:
1302:
1298:
1295:
1290:
1285:
1280:
1277:
1274:
1271:
1268:
1263:
1259:
1255:
1245:
1229:
1219:
1206:
1179:
1174:
1146:
1145:
1134:
1131:
1126:
1121:
1116:
1113:
1110:
1107:
1104:
1099:
1095:
1091:
1086:
1082:
1077:
1071:
1068:
1064:
1060:
1057:
1054:
1049:
1046:
1043:
1040:
1035:
1031:
1027:
1024:
1020:
1016:
1013:
1008:
1003:
998:
995:
992:
989:
986:
981:
977:
973:
930:
927:
924:
921:
916:
912:
885:
882:
877:Main article:
874:
871:
858:
855:
852:
847:
843:
822:
819:
816:
813:
810:
805:
801:
797:
792:
788:
752:
749:
746:
743:
723:
720:
717:
714:
711:
708:
705:
702:
699:
696:
693:
669:
666:
663:
660:
655:
651:
638:
635:
622:
619:
616:
613:
608:
604:
571:
568:
565:
562:
557:
553:
529:
526:
523:
520:
515:
511:
483:quotient stack
470:
467:
464:
461:
456:
452:
427:
424:
421:
418:
413:
409:
392:-bundles over
377:
372:
367:
362:
359:
356:
353:
350:
345:
341:
320:
317:
314:
311:
306:
302:
279:
274:
261:
260:
248:
245:
242:
235:
230:
224:
220:
192:
189:
186:
183:
180:
177:
174:
171:
166:
162:
132:
127:
101:
98:
95:
92:
87:
83:
42:
37:
13:
10:
9:
6:
4:
3:
2:
1828:
1817:
1814:
1813:
1811:
1802:
1799:
1797:
1794:
1792:
1789:
1788:
1784:
1780:
1776:
1775:
1771:
1766:
1762:
1758:
1752:
1745:
1744:
1738:
1735:
1731:
1728:
1724:
1720:
1714:
1710:
1706:
1702:
1695:
1690:
1689:
1685:
1673:
1668:
1665:
1657:
1656:
1648:
1645:
1641:
1640:Heinloth 2010
1636:
1633:
1628:
1624:
1620:
1616:
1611:
1606:
1602:
1595:
1592:
1588:
1587:Heinloth 2010
1583:
1580:
1570:on 2013-04-11
1566:
1559:
1558:
1550:
1547:
1540:
1538:
1536:
1529:
1511:
1507:
1499:
1478:
1474:
1463:
1460:
1456:
1449:
1446:
1441:
1433:
1430:
1417:
1414:
1411:
1407:
1403:
1395:
1391:
1380:
1377:
1373:
1366:
1363:
1356:
1353:
1349:
1327:
1321:
1318:
1311:
1304:
1300:
1296:
1288:
1272:
1266:
1261:
1257:
1246:
1243:
1227:
1220:
1195:
1177:
1163:and the ring
1162:
1158:
1155:
1154:
1153:
1151:
1124:
1114:
1108:
1102:
1097:
1093:
1084:
1080:
1069:
1066:
1062:
1055:
1052:
1044:
1038:
1033:
1029:
1025:
1022:
1018:
1014:
1006:
990:
984:
979:
975:
964:
963:
962:
960:
959:generic fiber
956:
952:
948:
944:
925:
919:
914:
910:
901:
895:
891:
880:
872:
870:
853:
845:
841:
814:
808:
803:
799:
790:
786:
777:
772:
770:
766:
747:
741:
721:
718:
715:
709:
706:
700:
694:
684:of dimension
683:
664:
658:
653:
649:
636:
634:
617:
611:
606:
602:
593:
589:
585:
566:
560:
555:
551:
541:
524:
518:
513:
509:
500:
499:homotopy type
496:
492:
488:
484:
465:
459:
454:
450:
441:
422:
416:
411:
407:
397:
395:
391:
370:
354:
348:
343:
339:
315:
309:
304:
300:
277:
246:
243:
240:
233:
222:
218:
210:
208:
190:
184:
175:
169:
164:
160:
152:
151:
150:
148:
130:
115:
96:
90:
85:
81:
73:, denoted by
72:
68:
65:over it, the
64:
61:
58:
55:and a smooth
40:
25:
22:
19:
1742:
1700:
1667:
1654:
1647:
1635:
1600:
1594:
1582:
1572:, retrieved
1565:the original
1556:
1549:
1534:
1533:
1351:
1160:
1156:
1147:
955:group scheme
946:
942:
897:
775:
773:
764:
682:smooth stack
640:
542:
486:
439:
398:
393:
389:
262:
206:
146:
70:
66:
62:
60:group scheme
23:
15:
1777:C. Sorger,
1678:(X), p. 260
491:gauge group
399:Similarly,
331:, that is,
292:-points of
1686:References
1574:2014-01-30
888:See also:
588:cohomology
205:principal
1796:Ran space
1610:1302.6351
1535:A priori,
1512:∗
1461:−
1457:ϕ
1450:
1431:−
1423:∞
1408:∑
1396:∗
1378:−
1374:ϕ
1367:
1348:G-bundles
1322:
1316:#
1301:∑
1267:
1254:#
1228:ϕ
1103:
1085:∗
1067:−
1063:ϕ
1056:
1039:
1026:
985:
972:#
920:
842:π
809:
787:π
719:
707:−
659:
612:
561:
519:
460:
417:
349:
310:
244:
223:×
170:
145:-algebra
91:
1810:Category
1785:See also
592:homology
209:-bundles
112:, is an
1765:3887650
1727:3013029
1627:4338216
1240:is the
961:, then
953:affine
489:by the
481:as the
1763:
1753:
1725:
1715:
1625:
1528:series
1496:for a
951:smooth
734:where
584:smooth
57:affine
18:smooth
1747:(PDF)
1697:(PDF)
1659:(PDF)
1605:arXiv
1568:(PDF)
1561:(PDF)
1541:Notes
949:is a
941:when
680:is a
69:over
1751:ISBN
1713:ISBN
902:for
892:and
590:and
241:Spec
1705:doi
1615:doi
1350:on
1319:Aut
1258:Bun
1192:of
1094:Bun
1030:Bun
1023:dim
976:Bun
911:Bun
800:Bun
774:If
716:dim
650:Bun
603:Bun
594:of
552:Bun
510:Bun
501:of
451:Bun
408:Bun
340:Bun
301:Bun
161:Bun
82:Bun
1812::
1761:MR
1759:,
1723:MR
1721:,
1711:,
1623:MR
1621:,
1613:,
1447:tr
1364:tr
1053:tr
869:.
633:.
586:)
540:.
396:.
149:,
1736:.
1707::
1676:G
1617::
1607::
1508:V
1484:)
1479:i
1475:V
1470:|
1464:1
1453:(
1442:i
1438:)
1434:1
1428:(
1418:0
1415:=
1412:i
1404:=
1401:)
1392:V
1387:|
1381:1
1370:(
1352:X
1331:)
1328:P
1325:(
1312:1
1305:P
1297:=
1294:)
1289:q
1284:F
1279:(
1276:)
1273:X
1270:(
1262:G
1244:.
1218:.
1205:C
1178:l
1173:Z
1161:p
1157:l
1133:)
1130:)
1125:l
1120:Z
1115:;
1112:)
1109:X
1106:(
1098:G
1090:(
1081:H
1076:|
1070:1
1059:(
1048:)
1045:X
1042:(
1034:G
1019:q
1015:=
1012:)
1007:q
1002:F
997:(
994:)
991:X
988:(
980:G
947:G
943:X
929:)
926:X
923:(
915:G
857:)
854:G
851:(
846:1
821:)
818:)
815:X
812:(
804:G
796:(
791:0
776:G
765:X
751:)
748:X
745:(
742:g
722:G
713:)
710:1
704:)
701:X
698:(
695:g
692:(
668:)
665:X
662:(
654:G
621:)
618:X
615:(
607:G
570:)
567:X
564:(
556:G
528:)
525:X
522:(
514:G
487:X
469:)
466:X
463:(
455:G
440:X
426:)
423:X
420:(
412:G
394:X
390:G
376:)
371:q
366:F
361:(
358:)
355:X
352:(
344:G
319:)
316:X
313:(
305:G
278:q
273:F
259:.
247:R
234:q
229:F
219:X
207:G
191:=
188:)
185:R
182:(
179:)
176:X
173:(
165:G
147:R
131:q
126:F
100:)
97:X
94:(
86:G
71:X
63:G
41:q
36:F
24:X
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.