1981:
2309:
1526:
1220:
124:
1871:
1412:
1086:
998:
1784:
2555:
190:
2424:
1673:
1283:
2111:
300:
450:
53:, and every elliptic curve is isomorphic to its dual, but this fails for higher-dimensional abelian varieties, so the concept of dual becomes more interesting in higher dimensions.
1588:
2190:
2073:
is often a line bundle so one may usually leave the left derived functors underived in the above expression. Note also that one can analogously define a
Fourier-Mukai transform
261:
2071:
2341:
1876:
2195:
1135:
1027:
1709:
915:
376:
2155:
2047:
1310:
222:
2020:
1546:
1350:
1330:
1106:
939:
2370:
1417:
1593:
Hence, we obtain a contravariant endofunctor on the category of abelian varieties which squares to the identity. This kind of functor is often called a
1143:
72:
2637:
2538:
1792:
1358:
1032:
944:
1737:
133:
2375:
1611:
1225:
2685:
2076:
874:
2690:
266:
1720:
766:
724:
416:
731:
1551:
2160:
587:
528:
227:
2669:
1976:{\displaystyle \Phi _{\mathcal {K}}^{X\to Y}(\cdot )=Rq_{*}({\mathcal {K}}\otimes _{L}Lp^{*}(\cdot ))}
2650:
2586:
2510:
1723:
and shows that the bounded derived category cannot necessarily distinguish non-isomorphic varieties.
2304:{\displaystyle \Phi _{P_{A}}^{{\hat {A}}\to A}\circ \Phi _{P_{A}}^{A\to {\hat {A}}}\cong \iota ^{*}}
2052:
743:
2314:
827:
575:
43:
1606:
456:
in the sense of the above definition. Moreover, this family is universal, that is, to any family
1111:
1003:
531:
one can state the above result as follows. The contravariant functor, which associates to each
2633:
2534:
1678:
888:
2567:
1712:
755:
717:
671:
667:
355:
2133:
2025:
1288:
195:
706:
623:
36:
452:, called the Poincaré bundle, which is a family of degree 0 line bundles parametrized by
1528:. One can show using this description that this map is an isogeny of the same degree as
17:
2005:
1531:
1335:
1315:
1091:
924:
678:
663:
325:
50:
2346:
1521:{\displaystyle ({\hat {f}}\times 1_{A})^{*}P_{A}\cong (f\times 1_{\hat {B}})^{*}P_{B}}
2679:
2625:
858:
710:
686:
870:
682:
647:
643:
639:
2610:
2464:
2444:
2113:
using the same kernel, by just interchanging the projection maps in the formula.
563:, is representable. The universal element representing this functor is the pair (
524:
given by tensor product of line bundles, which makes it into an abelian variety.
28:
2665:
2571:
1215:{\displaystyle f\times 1_{\hat {B}}:A\times {\hat {B}}\to B\times {\hat {B}}}
2556:"Duality between D(X) and D(\hat{X}) with its application to Picard sheaves"
689:
had given a general definition of
Albanese variety. For an abelian variety
119:{\displaystyle \operatorname {Pic} ^{0}(A)\subset \operatorname {Pic} (A)}
1355:
By the aforementioned functorial description, there is then a morphism
918:
754:
class as its dual. An explicit isogeny can be constructed by use of an
751:
742:
with its own Picard variety. This in a sense is just a consequence of
1605:
A celebrated theorem of Mukai states that there is an isomorphism of
792:
into an isomorphic copy is itself finite. In that case, the quotient
709:
of the identity element of what in contemporary terminology is the
630:
is coprime to the characteristic of the base. In general - for all
1866:{\displaystyle \Phi _{\mathcal {K}}^{X\to Y}:D^{b}(X)\to D^{b}(Y)}
1332:. This is then the required family of degree zero line bundles on
877:, and the quotient taken is now a quotient by a subgroup scheme.
746:. For general abelian varieties, still over the complex numbers,
2049:
is exact on the level of coherent sheaves, and in applications
941:
is finite-to-one and surjective.) We will construct an isogeny
1088:
is the same as giving a family of degree zero line bundles on
666:. In that case there is a general form of duality between the
324:(over the same field), which is the solution to the following
539:
the set of families of degree 0 line bundles parametrised by
2086:
2058:
1933:
1886:
1802:
1743:
574:
This association is a duality in the sense that there is a
2531:
Commutative
Algebra with a View Toward Algebraic Goemetry
2664:
This article incorporates material from Dual isogeny on
328:. A family of degree 0 line bundles parametrized by a
2378:
2349:
2317:
2198:
2163:
2136:
2116:
The statement of Mukai's theorem is then as follows.
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2055:
2028:
2008:
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1006:
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136:
75:
401:
is a trivial line bundle (here 0 is the identity of
1407:{\displaystyle {\hat {f}}:{\hat {B}}\to {\hat {A}}}
1081:{\displaystyle {\hat {f}}:{\hat {B}}\to {\hat {A}}}
993:{\displaystyle {\hat {f}}:{\hat {B}}\to {\hat {A}}}
2418:
2364:
2335:
2303:
2184:
2149:
2105:
2065:
2041:
2014:
1975:
1865:
1779:{\displaystyle {\mathcal {K}}\in D^{b}(X\times Y)}
1778:
1703:
1667:
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1520:
1406:
1344:
1324:
1304:
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1100:
1080:
1021:
992:
933:
909:
444:
370:
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255:
216:
184:
118:
681:; this was realised, for definitions in terms of
586:(defined via the Poincaré bundle) and that it is
2670:Creative Commons Attribution/Share-Alike License
1786:is a complex of coherent sheaves, we define the
185:{\displaystyle m^{*}L\cong p^{*}L\otimes q^{*}L}
2419:{\displaystyle \Phi _{P_{A}}^{A\to {\hat {A}}}}
1668:{\displaystyle D^{b}(A)\cong D^{b}({\hat {A}})}
1278:{\displaystyle (f\times 1_{\hat {B}})^{*}P_{B}}
658:The theory was first put into a good form when
1719:. Historically, this was the first use of the
126:to be the subgroup consisting of line bundles
2106:{\displaystyle \Phi _{\mathcal {K}}^{Y\to X}}
8:
834:, a universal line bundle can be defined on
520:, so there is a natural group operation on
295:{\displaystyle \operatorname {Pic} ^{0}(A)}
224:are the multiplication and projection maps
811:is isomorphic to the dual abelian variety
516:correspond to line bundles of degree 0 on
2632:(2nd ed.). Oxford University Press.
2403:
2402:
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559:the mapping induced by the pullback with
436:
418:
357:
274:
268:
238:
229:
197:
173:
157:
141:
135:
80:
74:
49:. A 1-dimensional abelian variety is an
1711:denotes the bounded derived category of
2435:
618:-torsion of an abelian variety and the
590:, i.e. it associates to all morphisms
512:is a point, we see that the points of
445:{\displaystyle P\to A\times A^{\vee }}
2372:is the shift functor. In particular,
7:
1029:, which says that the data of a map
1000:using the functorial description of
646:of each other. This generalizes the
2126:be an abelian variety of dimension
1583:{\displaystyle {\hat {\hat {f}}}=f}
830:. In terms of this definition, the
738:gives rise to an identification of
65:be an abelian variety over a field
2380:
2240:
2200:
2185:{\displaystyle A\times {\hat {A}}}
2081:
1881:
1797:
1140:To this end, consider the isogeny
256:{\displaystyle A\times _{k}A\to A}
25:
508:. Applying this to the case when
480:is isomorphic to the pullback of
2533:. Springer-Verlag. p. 521.
1312:is the Poincare line bundle for
921:of abelian varieties. (That is,
464:is associated a unique morphism
336:is defined to be a line bundle
2668:, which is licensed under the
2408:
2399:
2359:
2350:
2327:
2298:
2289:
2268:
2259:
2228:
2222:
2176:
2095:
2066:{\displaystyle {\mathcal {K}}}
1970:
1967:
1961:
1928:
1909:
1903:
1895:
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1854:
1841:
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1832:
1811:
1773:
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1698:
1692:
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1568:
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1389:
1383:
1368:
1256:
1248:
1229:
1206:
1191:
1185:
1163:
1121:
1072:
1063:
1057:
1042:
1013:
984:
975:
969:
954:
901:
697:itself, so the dual should be
642:of dual abelian varieties are
423:
289:
283:
247:
113:
107:
95:
89:
1:
2336:{\displaystyle \iota :A\to A}
2157:the Poincare line bundle on
390:} is a degree 0 line bundle,
263:respectively. An element of
873:that is a scheme-theoretic
2707:
2343:is the inversion map, and
1130:{\displaystyle {\hat {B}}}
1022:{\displaystyle {\hat {A}}}
869:) has to be in terms of a
693:, the Albanese variety is
2612:Abelian Varieties (v2.00)
2572:10.1017/S002776300001922X
1991:are the projections onto
622:-torsion of its dual are
614:in a compatible way. The
2609:Milne, James S. (2008).
2529:Eisenbud, David (1995).
1704:{\displaystyle D^{b}(X)}
910:{\displaystyle f:A\to B}
588:contravariant functorial
578:between the double dual
409:Then there is a variety
18:Autoduality of Jacobians
2649:Bhatt, Bhargav (2017).
2585:Bhatt, Bhargav (2017).
2554:Mukai, Shigeru (1981).
2509:Bhatt, Bhargav (2017).
1788:Fourier-Mukai transform
1721:Fourier-Mukai transform
1600:
767:holomorphic line bundle
725:compact Riemann surface
35:can be defined from an
2420:
2366:
2337:
2305:
2186:
2151:
2107:
2067:
2043:
2016:
1977:
1873:to be the composition
1867:
1780:
1705:
1669:
1584:
1542:
1522:
1408:
1346:
1326:
1306:
1279:
1216:
1131:
1102:
1082:
1023:
994:
935:
911:
849:The construction when
732:principal polarization
529:representable functors
446:
372:
371:{\displaystyle t\in T}
317:one then associates a
296:
257:
218:
186:
120:
2421:
2367:
2338:
2306:
2187:
2152:
2150:{\displaystyle P_{A}}
2108:
2068:
2044:
2042:{\displaystyle p^{*}}
2017:
1978:
1868:
1781:
1706:
1670:
1585:
1543:
1523:
1409:
1347:
1327:
1307:
1305:{\displaystyle P_{B}}
1280:
1217:
1132:
1103:
1083:
1024:
995:
936:
912:
822:extends to any field
818:This construction of
769:), when the subgroup
765:(i.e. in this case a
650:for elliptic curves.
447:
378:, the restriction of
373:
297:
258:
219:
217:{\displaystyle m,p,q}
187:
121:
2376:
2347:
2315:
2196:
2161:
2134:
2077:
2053:
2026:
2006:
1877:
1793:
1738:
1679:
1612:
1552:
1532:
1418:
1359:
1336:
1316:
1289:
1226:
1144:
1112:
1092:
1033:
1004:
945:
925:
889:
861:. The definition of
716:For the case of the
484:along the morphism 1
417:
356:
319:dual abelian variety
304:degree 0 line bundle
267:
228:
196:
134:
73:
33:dual abelian variety
2426:is an isomorphism.
2415:
2275:
2235:
2102:
1902:
1818:
1734:are varieties, and
853:has characteristic
828:characteristic zero
784:of translations on
707:connected component
626:to each other when
576:natural isomorphism
527:In the language of
393:the restriction of
2416:
2379:
2362:
2333:
2301:
2239:
2199:
2182:
2147:
2103:
2080:
2063:
2039:
2022:is flat and hence
2012:
1973:
1880:
1863:
1796:
1776:
1701:
1665:
1607:derived categories
1580:
1538:
1518:
1404:
1342:
1322:
1302:
1275:
1212:
1127:
1108:, parametrized by
1098:
1078:
1019:
990:
931:
907:
730:, the choice of a
442:
413:and a line bundle
368:
292:
253:
214:
182:
116:
2686:Abelian varieties
2652:Abelian Varieties
2639:978-0-19-560528-0
2630:Abelian Varieties
2588:Abelian Varieties
2540:978-3-540-78122-6
2512:Abelian Varieties
2497:Abelian Varieties
2484:Abelian Varieties
2466:Abelian Varieties
2452:. pp. 35–36.
2446:Abelian Varieties
2411:
2271:
2225:
2179:
2015:{\displaystyle p}
1659:
1595:dualizing functor
1571:
1566:
1541:{\displaystyle f}
1494:
1433:
1401:
1386:
1371:
1345:{\displaystyle A}
1325:{\displaystyle B}
1251:
1209:
1188:
1166:
1124:
1101:{\displaystyle A}
1075:
1060:
1045:
1016:
987:
972:
957:
934:{\displaystyle f}
662:was the field of
42:, defined over a
16:(Redirected from
2698:
2691:Duality theories
2659:
2657:
2643:
2619:
2617:
2596:
2595:
2593:
2582:
2576:
2575:
2551:
2545:
2544:
2526:
2520:
2519:
2517:
2506:
2500:
2493:
2487:
2480:
2474:
2473:
2471:
2463:Milne, James S.
2460:
2454:
2453:
2451:
2443:Milne, James S.
2440:
2425:
2423:
2422:
2417:
2414:
2413:
2412:
2404:
2394:
2393:
2392:
2371:
2369:
2368:
2365:{\displaystyle }
2363:
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2334:
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2019:
2018:
2013:
1982:
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1959:
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1937:
1936:
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1901:
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1853:
1852:
1831:
1830:
1817:
1806:
1805:
1785:
1783:
1782:
1777:
1760:
1759:
1747:
1746:
1713:coherent sheaves
1710:
1708:
1707:
1702:
1691:
1690:
1674:
1672:
1671:
1666:
1661:
1660:
1652:
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1300:
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1274:
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1264:
1263:
1254:
1253:
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1244:
1221:
1219:
1218:
1213:
1211:
1210:
1202:
1190:
1189:
1181:
1169:
1168:
1167:
1159:
1136:
1134:
1133:
1128:
1126:
1125:
1117:
1107:
1105:
1104:
1099:
1087:
1085:
1084:
1079:
1077:
1076:
1068:
1062:
1061:
1053:
1047:
1046:
1038:
1028:
1026:
1025:
1020:
1018:
1017:
1009:
999:
997:
996:
991:
989:
988:
980:
974:
973:
965:
959:
958:
950:
940:
938:
937:
932:
916:
914:
913:
908:
881:The Dual Isogeny
756:invertible sheaf
718:Jacobian variety
672:complete variety
668:Albanese variety
460:parametrized by
451:
449:
448:
443:
441:
440:
377:
375:
374:
369:
301:
299:
298:
293:
279:
278:
262:
260:
259:
254:
243:
242:
223:
221:
220:
215:
191:
189:
188:
183:
178:
177:
162:
161:
146:
145:
125:
123:
122:
117:
85:
84:
21:
2706:
2705:
2701:
2700:
2699:
2697:
2696:
2695:
2676:
2675:
2655:
2648:
2640:
2624:
2615:
2608:
2605:
2600:
2599:
2591:
2584:
2583:
2579:
2553:
2552:
2548:
2541:
2528:
2527:
2523:
2515:
2508:
2507:
2503:
2499:, p.123 onwards
2494:
2490:
2481:
2477:
2469:
2462:
2461:
2457:
2449:
2442:
2441:
2437:
2432:
2384:
2374:
2373:
2345:
2344:
2313:
2312:
2279:
2244:
2204:
2194:
2193:
2159:
2158:
2137:
2132:
2131:
2075:
2074:
2051:
2050:
2029:
2024:
2023:
2004:
2003:
1951:
1938:
1918:
1875:
1874:
1844:
1822:
1791:
1790:
1751:
1736:
1735:
1726:Recall that if
1682:
1677:
1676:
1637:
1615:
1610:
1609:
1603:
1601:Mukai's Theorem
1550:
1549:
1530:
1529:
1508:
1498:
1481:
1459:
1449:
1439:
1416:
1415:
1357:
1356:
1334:
1333:
1314:
1313:
1292:
1287:
1286:
1265:
1255:
1238:
1224:
1223:
1153:
1142:
1141:
1110:
1109:
1090:
1089:
1031:
1030:
1002:
1001:
943:
942:
923:
922:
887:
886:
883:
832:Poincaré bundle
750:is in the same
664:complex numbers
656:
602:dual morphisms
487:
432:
415:
414:
354:
353:
270:
265:
264:
234:
226:
225:
194:
193:
169:
153:
137:
132:
131:
76:
71:
70:
59:
37:abelian variety
23:
22:
15:
12:
11:
5:
2704:
2702:
2694:
2693:
2688:
2678:
2677:
2661:
2660:
2645:
2644:
2638:
2626:Mumford, David
2621:
2620:
2604:
2601:
2598:
2597:
2577:
2546:
2539:
2521:
2501:
2488:
2475:
2455:
2434:
2433:
2431:
2428:
2410:
2407:
2401:
2398:
2391:
2387:
2382:
2361:
2358:
2355:
2352:
2332:
2329:
2326:
2323:
2320:
2300:
2297:
2294:
2291:
2286:
2282:
2278:
2270:
2267:
2261:
2258:
2251:
2247:
2242:
2238:
2233:
2230:
2224:
2221:
2211:
2207:
2202:
2178:
2175:
2169:
2166:
2144:
2140:
2100:
2097:
2094:
2088:
2083:
2060:
2036:
2032:
2011:
1999:respectively.
1972:
1969:
1966:
1963:
1958:
1954:
1950:
1945:
1941:
1935:
1930:
1925:
1921:
1917:
1914:
1911:
1908:
1905:
1900:
1897:
1894:
1888:
1883:
1862:
1859:
1856:
1851:
1847:
1843:
1840:
1837:
1834:
1829:
1825:
1821:
1816:
1813:
1810:
1804:
1799:
1775:
1772:
1769:
1766:
1763:
1758:
1754:
1750:
1745:
1700:
1697:
1694:
1689:
1685:
1664:
1658:
1655:
1649:
1644:
1640:
1636:
1633:
1630:
1627:
1622:
1618:
1602:
1599:
1579:
1576:
1570:
1565:
1562:
1537:
1515:
1511:
1505:
1501:
1493:
1490:
1484:
1480:
1477:
1474:
1471:
1466:
1462:
1456:
1452:
1446:
1442:
1438:
1432:
1429:
1423:
1400:
1397:
1391:
1385:
1382:
1376:
1370:
1367:
1341:
1321:
1299:
1295:
1272:
1268:
1262:
1258:
1250:
1247:
1241:
1237:
1234:
1231:
1208:
1205:
1199:
1196:
1193:
1187:
1184:
1178:
1175:
1172:
1165:
1162:
1156:
1152:
1149:
1123:
1120:
1097:
1074:
1071:
1065:
1059:
1056:
1050:
1044:
1041:
1015:
1012:
986:
983:
977:
971:
968:
962:
956:
953:
930:
906:
903:
900:
897:
894:
882:
879:
847:
846:
809:
808:
782:
781:
744:Abel's theorem
679:Picard variety
655:
652:
485:
439:
435:
431:
428:
425:
422:
407:
406:
391:
367:
364:
361:
326:moduli problem
291:
288:
285:
282:
277:
273:
252:
249:
246:
241:
237:
233:
213:
210:
207:
204:
201:
181:
176:
172:
168:
165:
160:
156:
152:
149:
144:
140:
115:
112:
109:
106:
103:
100:
97:
94:
91:
88:
83:
79:
58:
55:
51:elliptic curve
24:
14:
13:
10:
9:
6:
4:
3:
2:
2703:
2692:
2689:
2687:
2684:
2683:
2681:
2674:
2673:
2671:
2667:
2654:
2653:
2647:
2646:
2641:
2635:
2631:
2627:
2623:
2622:
2614:
2613:
2607:
2606:
2602:
2594:. p. 43.
2590:
2589:
2581:
2578:
2573:
2569:
2565:
2561:
2557:
2550:
2547:
2542:
2536:
2532:
2525:
2522:
2518:. p. 38.
2514:
2513:
2505:
2502:
2498:
2492:
2489:
2485:
2479:
2476:
2472:. p. 36.
2468:
2467:
2459:
2456:
2448:
2447:
2439:
2436:
2429:
2427:
2405:
2396:
2389:
2385:
2356:
2353:
2330:
2324:
2321:
2318:
2295:
2292:
2284:
2280:
2276:
2265:
2256:
2249:
2245:
2236:
2231:
2219:
2209:
2205:
2173:
2167:
2164:
2142:
2138:
2129:
2125:
2121:
2117:
2114:
2098:
2092:
2034:
2030:
2009:
2000:
1998:
1994:
1990:
1986:
1964:
1956:
1952:
1948:
1943:
1939:
1923:
1919:
1915:
1912:
1906:
1898:
1892:
1857:
1849:
1845:
1835:
1827:
1823:
1819:
1814:
1808:
1789:
1770:
1767:
1764:
1756:
1752:
1748:
1733:
1729:
1724:
1722:
1718:
1714:
1695:
1687:
1683:
1653:
1642:
1638:
1634:
1628:
1620:
1616:
1608:
1598:
1596:
1591:
1577:
1574:
1560:
1535:
1513:
1509:
1503:
1488:
1482:
1478:
1475:
1469:
1464:
1460:
1454:
1444:
1440:
1436:
1427:
1395:
1380:
1374:
1365:
1353:
1339:
1319:
1297:
1293:
1270:
1266:
1260:
1245:
1239:
1235:
1232:
1203:
1197:
1194:
1182:
1176:
1173:
1170:
1160:
1154:
1150:
1147:
1138:
1118:
1095:
1069:
1054:
1048:
1039:
1010:
981:
966:
960:
951:
928:
920:
904:
898:
895:
892:
880:
878:
876:
872:
868:
864:
860:
859:scheme theory
856:
852:
844:
840:
837:
836:
835:
833:
829:
825:
821:
816:
814:
806:
802:
798:
795:
794:
793:
791:
787:
779:
775:
772:
771:
770:
768:
764:
760:
757:
753:
749:
745:
741:
737:
733:
729:
726:
722:
719:
714:
712:
711:Picard scheme
708:
704:
700:
696:
692:
688:
685:, as soon as
684:
680:
676:
673:
669:
665:
661:
653:
651:
649:
645:
644:Cartier duals
641:
640:group schemes
637:
633:
629:
625:
621:
617:
613:
609:
605:
601:
597:
593:
589:
585:
581:
577:
572:
570:
566:
562:
558:
554:
550:
546:
542:
538:
534:
530:
525:
523:
519:
515:
511:
507:
503:
499:
495:
491:
483:
479:
475:
471:
467:
463:
459:
455:
437:
433:
429:
426:
420:
412:
404:
400:
396:
392:
389:
385:
381:
365:
362:
359:
351:
350:
349:
347:
343:
339:
335:
331:
327:
323:
320:
316:
311:
309:
305:
286:
280:
275:
271:
250:
244:
239:
235:
231:
211:
208:
205:
202:
199:
179:
174:
170:
166:
163:
158:
154:
150:
147:
142:
138:
129:
110:
104:
101:
98:
92:
86:
81:
77:
68:
64:
56:
54:
52:
48:
45:
41:
38:
34:
30:
19:
2663:
2662:
2651:
2629:
2611:
2587:
2580:
2563:
2559:
2549:
2530:
2524:
2511:
2504:
2496:
2491:
2483:
2478:
2465:
2458:
2445:
2438:
2127:
2123:
2119:
2118:
2115:
2001:
1996:
1992:
1988:
1984:
1787:
1731:
1727:
1725:
1716:
1604:
1594:
1592:
1354:
1139:
884:
871:group scheme
866:
862:
854:
850:
848:
842:
838:
831:
823:
819:
817:
812:
810:
804:
800:
796:
789:
785:
783:
777:
773:
762:
758:
747:
739:
735:
727:
720:
715:
702:
698:
694:
690:
683:complex tori
674:
659:
657:
648:Weil pairing
635:
631:
627:
619:
615:
611:
607:
603:
599:
595:
591:
583:
579:
573:
568:
564:
560:
556:
552:
548:
544:
543:and to each
540:
536:
532:
526:
521:
517:
513:
509:
505:
501:
497:
493:
489:
481:
477:
473:
469:
465:
461:
457:
453:
410:
408:
402:
398:
394:
387:
383:
379:
345:
341:
337:
333:
329:
321:
318:
314:
312:
307:
303:
302:is called a
127:
69:. We define
66:
62:
60:
46:
39:
32:
26:
2566:: 153–175.
2560:Nagoya Math
1548:, and that
348:such that
29:mathematics
2680:Categories
2666:PlanetMath
2603:References
2486:, pp.74-80
2002:Note that
875:stabilizer
788:that take
687:André Weil
677:, and its
547:-morphism
130:such that
57:Definition
2495:Mumford,
2482:Mumford,
2409:^
2400:→
2381:Φ
2354:−
2328:→
2319:ι
2293:−
2285:∗
2281:ι
2277:≅
2269:^
2260:→
2241:Φ
2237:∘
2229:→
2223:^
2201:Φ
2177:^
2168:×
2096:→
2082:Φ
2035:∗
1965:⋅
1957:∗
1940:⊗
1924:∗
1907:⋅
1896:→
1882:Φ
1842:→
1812:→
1798:Φ
1768:×
1749:∈
1657:^
1635:≅
1569:^
1564:^
1504:∗
1492:^
1479:×
1470:≅
1455:∗
1437:×
1431:^
1399:^
1390:→
1384:^
1369:^
1261:∗
1249:^
1236:×
1207:^
1198:×
1192:→
1186:^
1177:×
1164:^
1151:×
1122:^
1073:^
1064:→
1058:^
1043:^
1014:^
985:^
976:→
970:^
955:^
902:→
638:-torsion
535:-variety
438:∨
430:×
424:→
363:∈
332:-variety
281:
248:→
236:×
175:∗
167:⊗
159:∗
151:≅
143:∗
105:
99:⊂
87:
2628:(1985).
2311:, where
2192:. Then,
2120:Theorem:
1983:, where
1675:, where
1414:so that
476:so that
352:for all
192:, where
919:isogeny
841:×
752:isogeny
705:), the
654:History
397:to {0}Ă—
2636:
2537:
1285:where
917:be an
634:- the
2656:(PDF)
2616:(PDF)
2592:(PDF)
2516:(PDF)
2470:(PDF)
2450:(PDF)
2430:Notes
857:uses
723:of a
670:of a
44:field
2634:ISBN
2535:ISBN
2130:and
2122:Let
1995:and
1987:and
1730:and
1222:and
885:Let
624:dual
582:and
340:on
61:Let
31:, a
2568:doi
1715:on
826:of
761:on
734:of
699:Pic
571:).
382:to
313:To
306:on
272:Pic
102:Pic
78:Pic
27:In
2682::
2564:81
2562:.
2558:.
1597:.
1590:.
1352:.
1137:.
815:.
713:.
610:→
606::
598:→
594::
567:,
557:T'
555:→
551::
500:→
492::
472:→
468::
405:).
386:Ă—{
310:.
2672:.
2658:.
2642:.
2618:.
2574:.
2570::
2543:.
2406:A
2397:A
2390:A
2386:P
2360:]
2357:g
2351:[
2331:A
2325:A
2322::
2299:]
2296:g
2290:[
2266:A
2257:A
2250:A
2246:P
2232:A
2220:A
2210:A
2206:P
2174:A
2165:A
2143:A
2139:P
2128:g
2124:A
2099:X
2093:Y
2087:K
2059:K
2031:p
2010:p
1997:Y
1993:X
1989:q
1985:p
1971:)
1968:)
1962:(
1953:p
1949:L
1944:L
1934:K
1929:(
1920:q
1916:R
1913:=
1910:)
1904:(
1899:Y
1893:X
1887:K
1861:)
1858:Y
1855:(
1850:b
1846:D
1839:)
1836:X
1833:(
1828:b
1824:D
1820::
1815:Y
1809:X
1803:K
1774:)
1771:Y
1765:X
1762:(
1757:b
1753:D
1744:K
1732:Y
1728:X
1717:X
1699:)
1696:X
1693:(
1688:b
1684:D
1663:)
1654:A
1648:(
1643:b
1639:D
1632:)
1629:A
1626:(
1621:b
1617:D
1578:f
1575:=
1561:f
1536:f
1514:B
1510:P
1500:)
1489:B
1483:1
1476:f
1473:(
1465:A
1461:P
1451:)
1445:A
1441:1
1428:f
1422:(
1396:A
1381:B
1375::
1366:f
1340:A
1320:B
1298:B
1294:P
1271:B
1267:P
1257:)
1246:B
1240:1
1233:f
1230:(
1204:B
1195:B
1183:B
1174:A
1171::
1161:B
1155:1
1148:f
1119:B
1096:A
1070:A
1055:B
1049::
1040:f
1011:A
982:A
967:B
961::
952:f
929:f
905:B
899:A
896::
893:f
867:L
865:(
863:K
855:p
851:K
845:.
843:Ă‚
839:A
824:K
820:Ă‚
813:Ă‚
807:)
805:L
803:(
801:K
799:/
797:A
790:L
786:L
780:)
778:L
776:(
774:K
763:A
759:L
748:A
740:J
736:J
728:C
721:J
703:A
701:(
695:A
691:A
675:V
660:K
636:n
632:n
628:n
620:n
616:n
612:A
608:B
604:f
600:B
596:A
592:f
584:A
580:A
569:P
565:A
561:f
553:T
549:f
545:k
541:T
537:T
533:k
522:A
518:A
514:A
510:T
506:A
504:Ă—
502:A
498:T
496:Ă—
494:A
490:f
488:Ă—
486:A
482:P
478:L
474:A
470:T
466:f
462:T
458:L
454:A
434:A
427:A
421:P
411:A
403:A
399:T
395:L
388:t
384:A
380:L
366:T
360:t
346:T
344:Ă—
342:A
338:L
334:T
330:k
322:A
315:A
308:A
290:)
287:A
284:(
276:0
251:A
245:A
240:k
232:A
212:q
209:,
206:p
203:,
200:m
180:L
171:q
164:L
155:p
148:L
139:m
128:L
114:)
111:A
108:(
96:)
93:A
90:(
82:0
67:k
63:A
47:k
40:A
20:)
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