27:
607:
702:
771:
262:
181:
139:
927:
823:
645:
434:
1047:
365:
312:
1074:
1018:
994:
947:
907:
887:
863:
843:
803:
722:
454:
385:
332:
282:
240:
213:
2005:
1841:
90:
1425:(1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields].
1575:
2428:
1385:
Arithmetic geometry. Papers from the conference held at the
University of Connecticut, Storrs, Connecticut, July 30 – August 10, 1984
2621:
488:
showed that
Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.
100:
2520:
2157:
2117:
1998:
2616:
2586:
2208:
2107:
1111:"Faltings relates the two notions of height by means of the Siegel moduli space.... It is the main idea of the proof."
2576:
1777:
1676:
1633:
1548:
1539:
Faltings, Gerd (1994). "The general case of S. Lang's conjecture". In
Cristante, Valentino; Messing, William (eds.).
1400:
954:
2286:
1991:
392:
2433:
2354:
2344:
2281:
2611:
2031:
1889:
1617:
2251:
2147:
973:
86:
1860:
583:
2510:
2474:
2173:
2086:
1884:
1837:
1541:
Barsotti
Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991
950:
1879:
561:
Faltings's 1983 paper had as consequences a number of statements which had previously been conjectured:
2484:
2122:
654:
404:
2530:
1821:
531:
216:
2443:
2423:
2359:
2276:
2178:
2137:
1668:
1470:
1427:
997:
782:
730:
618:
400:
245:
164:
122:
2334:
2142:
1380:
1347:
507:
569:
that a curve of genus greater than 1 over a number field has only finitely many rational points;
2127:
1902:
515:
473:
2241:
511:
2505:
2152:
2041:
912:
808:
624:
1656:
413:
2581:
2453:
2112:
1975:
1910:
1856:
1749:
1721:
1643:
1604:
1558:
1531:
1493:
1456:
1436:
1410:
1371:
1335:
1084:
2364:
1023:
785:, Faltings's theorem can be reformulated as a statement about the intersection of a curve
8:
2418:
2296:
2261:
2218:
2198:
1822:"On the rational solutions of the indeterminate equation of the third and fourth degrees"
1657:
866:
546:
481:
344:
291:
220:
116:
112:
40:
1914:
1571:"Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper"
1440:
1319:
2548:
2339:
2319:
2132:
1963:
1699:
1570:
1519:
1133:
1059:
1003:
979:
932:
892:
872:
848:
828:
788:
707:
499:
439:
370:
317:
267:
225:
198:
2291:
2448:
2395:
2266:
2081:
2076:
1783:
1773:
1737:
1672:
1629:
1592:
1544:
1396:
1359:
1343:
1092:
1922:
494:
proved
Shafarevich's finiteness conjecture using a known reduction to a case of the
2438:
2324:
2301:
1955:
1930:
1918:
1817:
1805:
1765:
1709:
1621:
1584:
1511:
1479:
1444:
1388:
1125:
1050:
542:
465:
1137:
2553:
2369:
2311:
2213:
2036:
2015:
1971:
1897:
1852:
1745:
1664:
1639:
1600:
1554:
1527:
1489:
1452:
1406:
1367:
1331:
1315:
535:
495:
485:
469:
146:
142:
16:
Curves of genus > 1 over the rationals have only finitely many rational points
2236:
2061:
2046:
2023:
1851:. Vol. Tome 1. Nice: Gauthier-Villars (published 1971). pp. 467–471.
1756:
Manin, Yu. (1966). "Rational points on algebraic curves over function fields".
1713:
958:
477:
388:
1769:
1625:
1392:
503:
2605:
2568:
2349:
2329:
2256:
2051:
1983:
1947:
1787:
1741:
1596:
1503:
1422:
1363:
1113:
966:
552:, borrowing also some of the easier ingredients of Faltings's original proof.
491:
335:
158:
150:
68:
50:
2515:
2489:
2479:
2469:
2271:
2091:
1566:
1088:
725:
184:
2390:
2228:
1501:
Faltings, Gerd (1991). "Diophantine approximation on abelian varieties".
577:
1796:
McQuillan, Michael (1995). "Division points on semi-abelian varieties".
2385:
1967:
1942:
1809:
1652:
1588:
1523:
1484:
1465:
1448:
1129:
1077:
972:
Another higher-dimensional generalization of
Faltings's theorem is the
962:
527:
647:
there are at most finitely many primitive integer solutions (pairwise
2246:
1686:
Lawrence, Brian; Venkatesh, Akshay (2020). "Diophantine problems and
1466:"Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern"
1053:
468:
conjectured that there are only finitely many isomorphism classes of
1959:
1725:
1515:
26:
1704:
1543:. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc.
2558:
2543:
648:
610:
617:
A sample application of
Faltings's theorem is to a weak form of
314:, there are either no points or infinitely many. In such cases,
2538:
1167:
1348:"Manin's proof of the Mordell conjecture over function fields"
1842:"Quelques conjectures de finitude en géométrie diophantienne"
1243:
1935:
1726:"Rational points on algebraic curves over function fields"
1083:
The
Mordell conjecture for function fields was proved by
506:. The main idea of Faltings's proof is the comparison of
1076:. Even more general conjectures have been put forth by
1183:
1000:(i.e., a variety of general type) over a number field
1730:
Izvestiya
Akademii Nauk SSSR. Seriya Matematicheskaya
1255:
1062:
1026:
1006:
982:
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657:
627:
586:
442:
416:
373:
347:
320:
294:
270:
248:
228:
201:
167:
125:
161:. The conjecture was later generalized by replacing
1900:(1968). "Algebraic curves over function fields I".
1291:
1279:
1231:
1207:
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1155:
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448:
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256:
234:
207:
175:
133:
1849:Actes du Congrès International des Mathématiciens
1685:
1267:
1249:
538:found a more elementary variant of Vojta's proof.
2603:
1945:(1991). "Siegel's theorem in the compact case".
1378:
1219:
407:restricts the structure of the torsion subgroup.
1611:
1116:(1984). "The Proof of the Mordell Conjecture".
2013:
1999:
1648:→ Gives Vojta's proof of Faltings's Theorem.
1620:. Vol. 201. New York: Springer-Verlag.
456:has only a finite number of rational points.
1929:
1612:Hindry, Marc; Silverman, Joseph H. (2000).
1189:
2006:
1992:
1758:American Mathematical Society Translations
25:
1795:
1703:
1483:
1261:
589:
250:
169:
127:
1538:
1500:
1463:
1421:
1416:
1314:
1237:
1213:
1177:
1173:
1095:found and fixed a gap in Manin's proof.
929:by an arbitrary finite-rank subgroup of
1896:
1877:
1816:
1565:
1342:
1297:
1285:
1201:
1161:
576:that abelian varieties with isomorphic
2604:
2429:Clifford's theorem on special divisors
1836:
476:degree over a fixed number field with
1987:
1941:
1755:
1720:
1415:→ Contains an English translation of
1273:
1225:
1112:
264:. Then the set of rational points on
1651:
1576:Publications Mathématiques de l'IHÉS
602:{\displaystyle \mathbb {Q} _{\ell }}
1933:(1963). "Algebraic number fields".
1324:Ann. Scuola Norm. Sup. Pisa Cl. Sci
805:with a finitely generated subgroup
436:, according to Faltings's theorem,
101:Siegel's theorem on integral points
13:
2587:Vector bundles on algebraic curves
2521:Weber's theorem (Algebraic curves)
2118:Hasse's theorem on elliptic curves
2108:Counting points on elliptic curves
1320:"The Mordell conjecture revisited"
916:
812:
776:
149:. This was conjectured in 1922 by
14:
2633:
697:{\displaystyle a^{n}+b^{n}=c^{n}}
609:-modules with Galois action) are
393:finitely generated abelian group
367:, if there are any points, then
115:, according to which a curve of
2209:Hurwitz's automorphisms theorem
1923:10.1070/IM1968v002n05ABEH000723
556:
521:
391:and its rational points form a
2622:Theorems in algebraic geometry
2434:Gonality of an algebraic curve
2345:Differential of the first kind
1903:Izv. Akad. Nauk SSSR Ser. Mat.
1659:Survey of Diophantine geometry
1118:The Mathematical Intelligencer
1105:
1036:
1030:
953:, which was proved in 1995 by
889:by an arbitrary subvariety of
480:outside a fixed finite set of
284:may be determined as follows:
119:greater than 1 over the field
1:
2577:Birkhoff–Grothendieck theorem
2287:Nagata's conjecture on curves
2158:Schoof–Elkies–Atkin algorithm
2032:Five points determine a conic
1618:Graduate Texts in Mathematics
1387:. New York: Springer-Verlag.
1307:
1250:Lawrence & Venkatesh 2020
766:{\displaystyle x^{n}+y^{n}=1}
472:of fixed dimension and fixed
190:
2148:Supersingular elliptic curve
1148:
845:. Generalizing by replacing
257:{\displaystyle \mathbb {Q} }
176:{\displaystyle \mathbb {Q} }
134:{\displaystyle \mathbb {Q} }
7:
2355:Riemann's existence theorem
2282:Hilbert's sixteenth problem
2174:Elliptic curve cryptography
2087:Fundamental pair of periods
1885:Encyclopedia of Mathematics
1826:Proc. Cambridge Philos. Soc
1352:L'Enseignement Mathématique
957:following work of Laurent,
498:, together with tools from
399:, later generalized to the
10:
2638:
2485:Moduli of algebraic curves
1714:10.1007/s00222-020-00966-7
773:has genus greater than 1.
502:, including the theory of
2617:Theorems in number theory
2567:
2529:
2498:
2462:
2411:
2404:
2378:
2310:
2227:
2191:
2166:
2100:
2069:
2060:
2022:
1626:10.1007/978-1-4612-1210-2
1393:10.1007/978-1-4613-8655-1
532:Diophantine approximation
460:
96:
82:
74:
64:
56:
46:
36:
24:
2252:Cayley–Bacharach theorem
2179:Elliptic curve primality
1878:Parshin, A. N. (2001) .
1690:-adic period mappings".
1471:Inventiones Mathematicae
1428:Inventiones Mathematicae
1098:
998:pseudo-canonical variety
974:Bombieri–Lang conjecture
516:Siegel modular varieties
157:until its 1983 proof by
87:Bombieri–Lang conjecture
2511:Riemann–Hurwitz formula
2475:Gromov–Witten invariant
2335:Compact Riemann surface
2123:Mazur's torsion theorem
1464:Faltings, Gerd (1984).
951:Mordell–Lang conjecture
922:{\displaystyle \Gamma }
818:{\displaystyle \Gamma }
640:{\displaystyle n\geq 4}
405:Mazur's torsion theorem
145:has only finitely many
91:Mordell–Lang conjecture
2128:Modular elliptic curve
1070:
1043:
1014:
990:
943:
923:
903:
883:
859:
839:
825:of an abelian variety
819:
799:
767:
718:
698:
641:
603:
545:gave a proof based on
530:gave a proof based on
450:
430:
429:{\displaystyle g>1}
381:
361:
328:
308:
278:
258:
236:
209:
177:
135:
2042:Rational normal curve
1770:10.1090/trans2/050/11
1071:
1044:
1015:
991:
944:
924:
904:
884:
860:
840:
820:
800:
768:
719:
699:
642:
619:Fermat's Last Theorem
604:
451:
431:
382:
362:
329:
309:
279:
259:
237:
210:
178:
136:
2612:Diophantine geometry
2582:Stable vector bundle
2454:Weil reciprocity law
2444:Riemann–Roch theorem
2424:Brill–Noether theory
2360:Riemann–Roch theorem
2277:Genus–degree formula
2138:Mordell–Weil theorem
2113:Division polynomials
1880:"Mordell conjecture"
1614:Diophantine geometry
1381:Silverman, Joseph H.
1085:Yuri Ivanovich Manin
1060:
1042:{\displaystyle X(k)}
1024:
1004:
980:
933:
913:
893:
873:
849:
829:
809:
789:
783:Mordell–Weil theorem
731:
708:
655:
625:
584:
541:Brian Lawrence and
440:
414:
401:Mordell–Weil theorem
371:
345:
334:may be handled as a
318:
292:
268:
246:
226:
199:
165:
123:
2405:Structure of curves
2297:Quartic plane curve
2219:Hyperelliptic curve
2199:De Franchis theorem
2143:Nagell–Lutz theorem
1915:1968IzMat...2.1145P
1441:1983InMat..73..349F
867:semiabelian variety
360:{\displaystyle g=1}
307:{\displaystyle g=0}
219:algebraic curve of
153:, and known as the
113:arithmetic geometry
41:Arithmetic geometry
21:
2412:Divisors on curves
2204:Faltings's theorem
2153:Schoof's algorithm
2133:Modularity theorem
1931:Shafarevich, I. R.
1810:10.1007/BF01241125
1589:10.1007/BF02684399
1485:10.1007/BF01388572
1449:10.1007/BF01388432
1344:Coleman, Robert F.
1130:10.1007/BF03024155
1066:
1039:
1010:
986:
939:
919:
899:
879:
855:
835:
815:
795:
763:
714:
694:
637:
599:
567:Mordell conjecture
550:-adic Hodge theory
500:algebraic geometry
446:
426:
377:
357:
324:
304:
274:
254:
232:
205:
173:
155:Mordell conjecture
131:
109:Faltings's theorem
20:Faltings's theorem
19:
2599:
2598:
2595:
2594:
2506:Hasse–Witt matrix
2449:Weierstrass point
2396:Smooth completion
2365:TeichmĂĽller space
2267:Cubic plane curve
2187:
2186:
2101:Arithmetic theory
2082:Elliptic integral
2077:Elliptic function
1818:Mordell, Louis J.
1093:Robert F. Coleman
1069:{\displaystyle X}
1013:{\displaystyle k}
989:{\displaystyle X}
942:{\displaystyle A}
902:{\displaystyle A}
882:{\displaystyle C}
858:{\displaystyle A}
838:{\displaystyle A}
798:{\displaystyle C}
717:{\displaystyle n}
704:, since for such
470:abelian varieties
449:{\displaystyle C}
397:Mordell's Theorem
380:{\displaystyle C}
327:{\displaystyle C}
277:{\displaystyle C}
235:{\displaystyle g}
208:{\displaystyle C}
106:
105:
2629:
2439:Jacobian variety
2409:
2408:
2312:Riemann surfaces
2302:Real plane curve
2262:Cramer's paradox
2242:BĂ©zout's theorem
2067:
2066:
2016:algebraic curves
2008:
2001:
1994:
1985:
1984:
1979:
1938:
1926:
1909:(5): 1191–1219.
1893:
1874:
1872:
1871:
1865:
1859:. Archived from
1846:
1833:
1813:
1791:
1753:
1717:
1707:
1689:
1682:
1662:
1647:
1608:
1562:
1535:
1497:
1487:
1460:
1414:
1375:
1339:
1316:Bombieri, Enrico
1301:
1295:
1289:
1283:
1277:
1271:
1265:
1259:
1253:
1247:
1241:
1235:
1229:
1223:
1217:
1211:
1205:
1199:
1193:
1190:Shafarevich 1963
1187:
1181:
1171:
1165:
1159:
1142:
1141:
1109:
1075:
1073:
1072:
1067:
1048:
1046:
1045:
1040:
1019:
1017:
1016:
1011:
995:
993:
992:
987:
948:
946:
945:
940:
928:
926:
925:
920:
908:
906:
905:
900:
888:
886:
885:
880:
864:
862:
861:
856:
844:
842:
841:
836:
824:
822:
821:
816:
804:
802:
801:
796:
772:
770:
769:
764:
756:
755:
743:
742:
723:
721:
720:
715:
703:
701:
700:
695:
693:
692:
680:
679:
667:
666:
646:
644:
643:
638:
621:: for any fixed
608:
606:
605:
600:
598:
597:
592:
549:
543:Akshay Venkatesh
508:Faltings heights
466:Igor Shafarevich
455:
453:
452:
447:
435:
433:
432:
427:
386:
384:
383:
378:
366:
364:
363:
358:
333:
331:
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325:
313:
311:
310:
305:
283:
281:
280:
275:
263:
261:
260:
255:
253:
241:
239:
238:
233:
214:
212:
211:
206:
182:
180:
179:
174:
172:
143:rational numbers
140:
138:
137:
132:
130:
29:
22:
18:
2637:
2636:
2632:
2631:
2630:
2628:
2627:
2626:
2602:
2601:
2600:
2591:
2563:
2554:Delta invariant
2525:
2494:
2458:
2419:Abel–Jacobi map
2400:
2374:
2370:Torelli theorem
2340:Dessin d'enfant
2320:Belyi's theorem
2306:
2292:PlĂĽcker formula
2223:
2214:Hurwitz surface
2183:
2162:
2096:
2070:Analytic theory
2062:Elliptic curves
2056:
2037:Projective line
2024:Rational curves
2018:
2012:
1982:
1960:10.2307/2944318
1869:
1867:
1863:
1844:
1780:
1687:
1679:
1665:Springer-Verlag
1636:
1583:(25): 131–149.
1551:
1516:10.2307/2944319
1417:Faltings (1983)
1403:
1383:, eds. (1986).
1379:Cornell, Gary;
1310:
1305:
1304:
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1244:
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1224:
1220:
1212:
1208:
1200:
1196:
1188:
1184:
1172:
1168:
1160:
1156:
1151:
1146:
1145:
1110:
1106:
1101:
1061:
1058:
1057:
1025:
1022:
1021:
1005:
1002:
1001:
981:
978:
977:
934:
931:
930:
914:
911:
910:
894:
891:
890:
874:
871:
870:
850:
847:
846:
830:
827:
826:
810:
807:
806:
790:
787:
786:
781:Because of the
779:
777:Generalizations
751:
747:
738:
734:
732:
729:
728:
709:
706:
705:
688:
684:
675:
671:
662:
658:
656:
653:
652:
626:
623:
622:
593:
588:
587:
585:
582:
581:
574:Isogeny theorem
559:
547:
536:Enrico Bombieri
524:
496:Tate conjecture
486:Aleksei Parshin
463:
441:
438:
437:
415:
412:
411:
372:
369:
368:
346:
343:
342:
319:
316:
315:
293:
290:
289:
269:
266:
265:
249:
247:
244:
243:
227:
224:
223:
200:
197:
196:
193:
168:
166:
163:
162:
147:rational points
126:
124:
121:
120:
111:is a result in
89:
83:Generalizations
32:
17:
12:
11:
5:
2635:
2625:
2624:
2619:
2614:
2597:
2596:
2593:
2592:
2590:
2589:
2584:
2579:
2573:
2571:
2569:Vector bundles
2565:
2564:
2562:
2561:
2556:
2551:
2546:
2541:
2535:
2533:
2527:
2526:
2524:
2523:
2518:
2513:
2508:
2502:
2500:
2496:
2495:
2493:
2492:
2487:
2482:
2477:
2472:
2466:
2464:
2460:
2459:
2457:
2456:
2451:
2446:
2441:
2436:
2431:
2426:
2421:
2415:
2413:
2406:
2402:
2401:
2399:
2398:
2393:
2388:
2382:
2380:
2376:
2375:
2373:
2372:
2367:
2362:
2357:
2352:
2347:
2342:
2337:
2332:
2327:
2322:
2316:
2314:
2308:
2307:
2305:
2304:
2299:
2294:
2289:
2284:
2279:
2274:
2269:
2264:
2259:
2254:
2249:
2244:
2239:
2233:
2231:
2225:
2224:
2222:
2221:
2216:
2211:
2206:
2201:
2195:
2193:
2189:
2188:
2185:
2184:
2182:
2181:
2176:
2170:
2168:
2164:
2163:
2161:
2160:
2155:
2150:
2145:
2140:
2135:
2130:
2125:
2120:
2115:
2110:
2104:
2102:
2098:
2097:
2095:
2094:
2089:
2084:
2079:
2073:
2071:
2064:
2058:
2057:
2055:
2054:
2049:
2047:Riemann sphere
2044:
2039:
2034:
2028:
2026:
2020:
2019:
2011:
2010:
2003:
1996:
1988:
1981:
1980:
1954:(3): 509–548.
1939:
1927:
1898:Parshin, A. N.
1894:
1875:
1834:
1814:
1804:(1): 143–159.
1793:
1778:
1754:(Translation:
1732:(in Russian).
1718:
1698:(3): 893–999.
1683:
1677:
1649:
1634:
1609:
1563:
1549:
1536:
1510:(3): 549–576.
1498:
1461:
1435:(3): 349–366.
1423:Faltings, Gerd
1419:
1401:
1376:
1358:(3): 393–427.
1340:
1330:(4): 615–640.
1311:
1309:
1306:
1303:
1302:
1290:
1278:
1266:
1262:McQuillan 1995
1254:
1242:
1230:
1218:
1206:
1194:
1182:
1166:
1153:
1152:
1150:
1147:
1144:
1143:
1114:Bloch, Spencer
1103:
1102:
1100:
1097:
1065:
1038:
1035:
1032:
1029:
1009:
985:
938:
918:
898:
878:
854:
834:
814:
794:
778:
775:
762:
759:
754:
750:
746:
741:
737:
713:
691:
687:
683:
678:
674:
670:
665:
661:
651:solutions) to
636:
633:
630:
615:
614:
596:
591:
570:
558:
555:
554:
553:
539:
523:
520:
478:good reduction
462:
459:
458:
457:
445:
425:
422:
419:
408:
389:elliptic curve
376:
356:
353:
350:
339:
323:
303:
300:
297:
273:
252:
231:
204:
192:
189:
171:
129:
104:
103:
98:
94:
93:
84:
80:
79:
76:
75:First proof in
72:
71:
66:
65:First proof by
62:
61:
58:
57:Conjectured in
54:
53:
48:
47:Conjectured by
44:
43:
38:
34:
33:
30:
15:
9:
6:
4:
3:
2:
2634:
2623:
2620:
2618:
2615:
2613:
2610:
2609:
2607:
2588:
2585:
2583:
2580:
2578:
2575:
2574:
2572:
2570:
2566:
2560:
2557:
2555:
2552:
2550:
2547:
2545:
2542:
2540:
2537:
2536:
2534:
2532:
2531:Singularities
2528:
2522:
2519:
2517:
2514:
2512:
2509:
2507:
2504:
2503:
2501:
2497:
2491:
2488:
2486:
2483:
2481:
2478:
2476:
2473:
2471:
2468:
2467:
2465:
2461:
2455:
2452:
2450:
2447:
2445:
2442:
2440:
2437:
2435:
2432:
2430:
2427:
2425:
2422:
2420:
2417:
2416:
2414:
2410:
2407:
2403:
2397:
2394:
2392:
2389:
2387:
2384:
2383:
2381:
2379:Constructions
2377:
2371:
2368:
2366:
2363:
2361:
2358:
2356:
2353:
2351:
2350:Klein quartic
2348:
2346:
2343:
2341:
2338:
2336:
2333:
2331:
2330:Bolza surface
2328:
2326:
2325:Bring's curve
2323:
2321:
2318:
2317:
2315:
2313:
2309:
2303:
2300:
2298:
2295:
2293:
2290:
2288:
2285:
2283:
2280:
2278:
2275:
2273:
2270:
2268:
2265:
2263:
2260:
2258:
2257:Conic section
2255:
2253:
2250:
2248:
2245:
2243:
2240:
2238:
2237:AF+BG theorem
2235:
2234:
2232:
2230:
2226:
2220:
2217:
2215:
2212:
2210:
2207:
2205:
2202:
2200:
2197:
2196:
2194:
2190:
2180:
2177:
2175:
2172:
2171:
2169:
2165:
2159:
2156:
2154:
2151:
2149:
2146:
2144:
2141:
2139:
2136:
2134:
2131:
2129:
2126:
2124:
2121:
2119:
2116:
2114:
2111:
2109:
2106:
2105:
2103:
2099:
2093:
2090:
2088:
2085:
2083:
2080:
2078:
2075:
2074:
2072:
2068:
2065:
2063:
2059:
2053:
2052:Twisted cubic
2050:
2048:
2045:
2043:
2040:
2038:
2035:
2033:
2030:
2029:
2027:
2025:
2021:
2017:
2009:
2004:
2002:
1997:
1995:
1990:
1989:
1986:
1977:
1973:
1969:
1965:
1961:
1957:
1953:
1950:
1949:
1948:Ann. of Math.
1944:
1940:
1936:
1932:
1928:
1924:
1920:
1916:
1912:
1908:
1905:
1904:
1899:
1895:
1891:
1887:
1886:
1881:
1876:
1866:on 2016-09-24
1862:
1858:
1854:
1850:
1843:
1839:
1838:Paršin, A. N.
1835:
1831:
1827:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1794:
1789:
1785:
1781:
1779:9780821817506
1775:
1771:
1767:
1763:
1759:
1751:
1747:
1743:
1739:
1736:: 1395–1440.
1735:
1731:
1727:
1723:
1722:Manin, Ju. I.
1719:
1715:
1711:
1706:
1701:
1697:
1693:
1684:
1680:
1678:3-540-61223-8
1674:
1670:
1666:
1661:
1660:
1654:
1650:
1645:
1641:
1637:
1635:0-387-98981-1
1631:
1627:
1623:
1619:
1615:
1610:
1606:
1602:
1598:
1594:
1590:
1586:
1582:
1578:
1577:
1572:
1568:
1567:Grauert, Hans
1564:
1560:
1556:
1552:
1550:0-12-197270-4
1546:
1542:
1537:
1533:
1529:
1525:
1521:
1517:
1513:
1509:
1506:
1505:
1504:Ann. of Math.
1499:
1495:
1491:
1486:
1481:
1477:
1474:(in German).
1473:
1472:
1467:
1462:
1458:
1454:
1450:
1446:
1442:
1438:
1434:
1431:(in German).
1430:
1429:
1424:
1420:
1418:
1412:
1408:
1404:
1402:0-387-96311-1
1398:
1394:
1390:
1386:
1382:
1377:
1373:
1369:
1365:
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1312:
1299:
1294:
1287:
1282:
1275:
1270:
1263:
1258:
1251:
1246:
1239:
1238:Bombieri 1990
1234:
1227:
1222:
1215:
1214:Faltings 1983
1210:
1203:
1198:
1191:
1186:
1179:
1178:Faltings 1984
1175:
1174:Faltings 1983
1170:
1163:
1158:
1154:
1139:
1135:
1131:
1127:
1123:
1119:
1115:
1108:
1104:
1096:
1094:
1090:
1086:
1081:
1079:
1063:
1055:
1052:
1033:
1027:
1007:
999:
983:
975:
970:
968:
964:
960:
956:
952:
949:leads to the
936:
896:
876:
868:
852:
832:
792:
784:
774:
760:
757:
752:
748:
744:
739:
735:
727:
711:
689:
685:
681:
676:
672:
668:
663:
659:
650:
634:
631:
628:
620:
612:
594:
579:
575:
571:
568:
564:
563:
562:
551:
544:
540:
537:
533:
529:
526:
525:
519:
517:
513:
512:naive heights
509:
505:
501:
497:
493:
492:Gerd Faltings
489:
487:
483:
479:
475:
471:
467:
443:
423:
420:
417:
409:
406:
403:.) Moreover,
402:
398:
394:
390:
374:
354:
351:
348:
340:
337:
336:conic section
321:
301:
298:
295:
287:
286:
285:
271:
229:
222:
218:
202:
188:
186:
160:
159:Gerd Faltings
156:
152:
151:Louis Mordell
148:
144:
118:
114:
110:
102:
99:
95:
92:
88:
85:
81:
77:
73:
70:
69:Gerd Faltings
67:
63:
59:
55:
52:
51:Louis Mordell
49:
45:
42:
39:
35:
31:Gerd Faltings
28:
23:
2516:Prym variety
2490:Stable curve
2480:Hodge bundle
2470:ELSV formula
2272:Fermat curve
2229:Plane curves
2203:
2192:Higher genus
2167:Applications
2092:Modular form
1951:
1946:
1934:
1906:
1901:
1883:
1868:. Retrieved
1861:the original
1848:
1829:
1825:
1801:
1798:Invent. Math
1797:
1761:
1760:. Series 2.
1757:
1733:
1729:
1695:
1692:Invent. Math
1691:
1658:
1613:
1580:
1574:
1540:
1507:
1502:
1475:
1469:
1432:
1426:
1384:
1355:
1354:. 2e SĂ©rie.
1351:
1327:
1323:
1298:Coleman 1990
1293:
1286:Grauert 1965
1281:
1269:
1257:
1245:
1233:
1221:
1209:
1202:Parshin 1968
1197:
1185:
1169:
1162:Mordell 1922
1157:
1121:
1117:
1107:
1089:Hans Grauert
1082:
971:
780:
726:Fermat curve
616:
578:Tate modules
573:
566:
560:
557:Consequences
522:Later proofs
504:NĂ©ron models
490:
474:polarization
464:
396:
217:non-singular
194:
185:number field
154:
108:
107:
97:Consequences
2391:Polar curve
1943:Vojta, Paul
1764:: 189–234.
1667:. pp.
1653:Lang, Serge
1091:. In 1990,
395:. (This is
2606:Categories
2386:Dual curve
2014:Topics in
1937:: 163–176.
1870:2016-06-11
1832:: 179–192.
1705:1807.02721
1478:(2): 381.
1308:References
1274:Manin 1963
1226:Vojta 1991
1078:Paul Vojta
961:, Hindry,
528:Paul Vojta
191:Background
2499:Morphisms
2247:Bitangent
1890:EMS Press
1788:0065-9290
1742:0373-2436
1597:1618-1913
1364:0013-8584
1149:Citations
1124:(2): 44.
955:McQuillan
917:Γ
813:Γ
632:≥
611:isogenous
595:ℓ
1840:(1970).
1820:(1922).
1724:(1963).
1655:(1997).
1569:(1965).
1346:(1990).
1318:(1990).
976:that if
967:Faltings
2559:Tacnode
2544:Crunode
1976:1109352
1968:2944318
1911:Bibcode
1857:0427323
1750:0157971
1644:1745599
1605:0222087
1559:1307396
1532:1109353
1524:2944319
1494:0732554
1457:0718935
1437:Bibcode
1411:0861969
1372:1096426
1336:1093712
1087:and by
1051:Zariski
1049:is not
1020:, then
959:Raynaud
649:coprime
183:by any
2539:Acnode
2463:Moduli
1974:
1966:
1855:
1786:
1776:
1748:
1740:
1675:
1671:–122.
1642:
1632:
1603:
1595:
1557:
1547:
1530:
1522:
1492:
1455:
1409:
1399:
1370:
1362:
1334:
1138:306251
1136:
965:, and
909:, and
482:places
461:Proofs
387:is an
1964:JSTOR
1864:(PDF)
1845:(PDF)
1700:arXiv
1520:JSTOR
1134:S2CID
1099:Notes
1054:dense
996:is a
963:Vojta
865:by a
410:When
341:When
288:When
242:over
221:genus
215:be a
117:genus
37:Field
2549:Cusp
1784:ISSN
1774:ISBN
1738:ISSN
1673:ISBN
1630:ISBN
1593:ISSN
1545:ISBN
1397:ISBN
1360:ISSN
724:the
580:(as
572:The
565:The
514:via
510:and
421:>
195:Let
78:1983
60:1922
1956:doi
1952:133
1919:doi
1806:doi
1802:120
1766:doi
1710:doi
1696:221
1669:101
1622:doi
1585:doi
1512:doi
1508:133
1480:doi
1445:doi
1389:doi
1126:doi
1056:in
141:of
2608::
1972:MR
1970:.
1962:.
1917:.
1907:32
1888:.
1882:.
1853:MR
1847:.
1830:21
1828:.
1824:.
1800:.
1782:.
1772:.
1762:59
1746:MR
1744:.
1734:27
1728:.
1708:.
1694:.
1663:.
1640:MR
1638:.
1628:.
1616:.
1601:MR
1599:.
1591:.
1581:25
1579:.
1573:.
1555:MR
1553:.
1528:MR
1526:.
1518:.
1490:MR
1488:.
1476:75
1468:.
1453:MR
1451:.
1443:.
1433:73
1407:MR
1405:.
1395:.
1368:MR
1366:.
1356:36
1350:.
1332:MR
1328:17
1326:.
1322:.
1176:;
1132:.
1120:.
1080:.
969:.
869:,
534:.
518:.
484:.
187:.
2007:e
2000:t
1993:v
1978:.
1958::
1925:.
1921::
1913::
1892:.
1873:.
1812:.
1808::
1792:)
1790:.
1768::
1752:.
1716:.
1712::
1702::
1688:p
1681:.
1646:.
1624::
1607:.
1587::
1561:.
1534:.
1514::
1496:.
1482::
1459:.
1447::
1439::
1413:.
1391::
1374:.
1338:.
1300:.
1288:.
1276:.
1264:.
1252:.
1240:.
1228:.
1216:.
1204:.
1192:.
1180:.
1164:.
1140:.
1128::
1122:6
1064:X
1037:)
1034:k
1031:(
1028:X
1008:k
984:X
937:A
897:A
877:C
853:A
833:A
793:C
761:1
758:=
753:n
749:y
745:+
740:n
736:x
712:n
690:n
686:c
682:=
677:n
673:b
669:+
664:n
660:a
635:4
629:n
613:.
590:Q
548:p
444:C
424:1
418:g
375:C
355:1
352:=
349:g
338:.
322:C
302:0
299:=
296:g
272:C
251:Q
230:g
203:C
170:Q
128:Q
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