38:
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2009:
1436:(1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields].
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Arithmetic geometry. Papers from the conference held at the
University of Connecticut, Storrs, Connecticut, July 30 – August 10, 1984
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2002:
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showed that
Shafarevich's finiteness conjecture would imply the Mordell conjecture, using what is now called Parshin's trick.
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2355:
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2042:
1900:
1628:
2262:
984:
97:
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2485:
1122:"Faltings relates the two notions of height by means of the Siegel moduli space.... It is the main idea of the proof."
961:
2184:
2097:
1895:
1848:
1550:
Faltings, Gerd (1994). "The general case of S. Lang's conjecture". In
Cristante, Valentino; Messing, William (eds.).
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Barsotti
Symposium in Algebraic Geometry. Papers from the symposium held in Abano Terme, June 24–27, 1991
2052:
1890:
572:
Faltings's 1983 paper had as consequences a number of statements which had previously been conjectured:
2163:
665:
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2464:
2375:
2123:
1921:
1832:
1732:
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127:
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51:
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175:
133:
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2143:
1974:
1710:
1530:
1358:
1144:
510:
2350:
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2302:
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that a curve of genus greater than 1 over a number field has only finitely many rational points;
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1407:
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1966:
1941:
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1986:
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796:, Faltings's theorem can be reformulated as a statement about the intersection of a curve
546:
522:
518:
506:
496:
484:
480:
157:
153:
1833:"On the rational solutions of the indeterminate equation of the third and fourth degrees"
1668:
355:
302:
2247:
1925:
1582:"Mordells Vermutung über rationale Punkte auf algebraischen Kurven und Funktionenkörper"
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1330:
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1958:
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502:
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169:
161:
79:
61:
1933:
505:
proved
Shafarevich's finiteness conjecture using a known reduction to a case of the
2526:
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2102:
1577:
1099:
736:
195:
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17:
27:
Curves of genus > 1 over the rationals have only finitely many rational points
2401:
2239:
588:
37:
1862:. Vol. Tome 1. Nice: Gauthier-Villars (published 1971). pp. 467–471.
1767:
Manin, Yu. (1966). "Rational points on algebraic curves over function fields".
1724:
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1953:
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1403:
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973:
538:
1994:
1798:
1752:
1607:
1374:
563:, borrowing also some of the easier ingredients of Faltings's original proof.
2257:
1064:
1512:
Faltings, Gerd (1991). "Diophantine approximation on abelian varieties".
1807:
McQuillan, Michael (1995). "Division points on semi-abelian varieties".
2569:
2554:
1978:
1820:
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1534:
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983:
Another higher-dimensional generalization of
Faltings's theorem is the
659:
621:
658:
there are at most finitely many primitive integer solutions (pairwise
2549:
1697:
Lawrence, Brian; Venkatesh, Akshay (2020). "Diophantine problems and
1477:"Erratum: Endlichkeitssätze für abelsche Varietäten über Zahlkörpern"
479:
conjectured that there are only finitely many isomorphism classes of
1970:
1736:
1526:
1715:
1554:. Perspectives in Mathematics. San Diego, CA: Academic Press, Inc.
628:
A sample application of
Faltings's theorem is to a weak form of
325:, there are either no points or infinitely many. In such cases,
1998:
1359:"Manin's proof of the Mordell conjecture over function fields"
1853:"Quelques conjectures de finitude en géométrie diophantienne"
1946:
1737:"Rational points on algebraic curves over function fields"
1094:
The
Mordell conjecture for function fields was proved by
517:. The main idea of Faltings's proof is the comparison of
1087:. Even more general conjectures have been put forth by
1011:(i.e., a variety of general type) over a number field
1741:
Izvestiya
Akademii Nauk SSSR. Seriya Matematicheskaya
1073:
1037:
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638:
597:
453:
427:
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331:
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281:
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239:
212:
178:
136:
172:. The conjecture was later generalized by replacing
2578:
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2509:
2473:
2422:
2415:
2389:
2321:
2238:
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2177:
2111:
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2033:
1911:(1968). "Algebraic curves over function fields I".
107:
93:
85:
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57:
47:
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612:
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390:
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337:
317:
287:
267:
245:
218:
186:
144:
1860:Actes du Congrès International des Mathématiciens
1260:
549:found a more elementary variant of Vojta's proof.
1956:(1991). "Siegel's theorem in the compact case".
418:restricts the structure of the torsion subgroup.
1127:(1984). "The Proof of the Mordell Conjecture".
2010:
8:
1659:→ Gives Vojta's proof of Faltings's Theorem.
1631:. Vol. 201. New York: Springer-Verlag.
467:has only a finite number of rational points.
30:
1623:Hindry, Marc; Silverman, Joseph H. (2000).
1200:
2419:
2077:
2017:
2003:
1995:
1769:American Mathematical Society Translations
36:
29:
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1016:
992:
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841:
821:
801:
762:
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600:
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452:
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383:
357:
330:
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280:
261:
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211:
180:
179:
177:
138:
137:
135:
1427:
1248:
1224:
1188:
1184:
1106:found and fixed a gap in Manin's proof.
940:by an arbitrary finite-rank subgroup of
1308:
1296:
1212:
1172:
1165:
1115:
587:that abelian varieties with isomorphic
2440:Clifford's theorem on special divisors
487:degree over a fixed number field with
1426:→ Contains an English translation of
1284:
1236:
275:. Then the set of rational points on
7:
1587:Publications Mathématiques de l'IHÉS
613:{\displaystyle \mathbb {Q} _{\ell }}
1944:(1963). "Algebraic number fields".
1335:Ann. Scuola Norm. Sup. Pisa Cl. Sci
816:with a finitely generated subgroup
447:, according to Faltings's theorem,
112:Siegel's theorem on integral points
2598:Vector bundles on algebraic curves
2532:Weber's theorem (Algebraic curves)
2129:Hasse's theorem on elliptic curves
2119:Counting points on elliptic curves
1331:"The Mordell conjecture revisited"
927:
823:
160:. This was conjectured in 1922 by
25:
708:{\displaystyle a^{n}+b^{n}=c^{n}}
620:-modules with Galois action) are
404:finitely generated abelian group
378:, if there are any points, then
126:, according to which a curve of
2220:Hurwitz's automorphisms theorem
1934:10.1070/IM1968v002n05ABEH000723
402:and its rational points form a
2633:Theorems in algebraic geometry
2445:Gonality of an algebraic curve
2356:Differential of the first kind
1914:Izv. Akad. Nauk SSSR Ser. Mat.
1670:Survey of Diophantine geometry
1129:The Mathematical Intelligencer
1047:
1041:
964:, which was proved in 1995 by
900:by an arbitrary subvariety of
491:outside a fixed finite set of
295:may be determined as follows:
130:greater than 1 over the field
1:
2588:Birkhoff–Grothendieck theorem
2298:Nagata's conjecture on curves
2169:Schoof–Elkies–Atkin algorithm
2043:Five points determine a conic
1629:Graduate Texts in Mathematics
1398:. New York: Springer-Verlag.
1261:Lawrence & Venkatesh 2020
777:{\displaystyle x^{n}+y^{n}=1}
483:of fixed dimension and fixed
2159:Supersingular elliptic curve
856:. Generalizing by replacing
268:{\displaystyle \mathbb {Q} }
187:{\displaystyle \mathbb {Q} }
145:{\displaystyle \mathbb {Q} }
2366:Riemann's existence theorem
2293:Hilbert's sixteenth problem
2185:Elliptic curve cryptography
2098:Fundamental pair of periods
1896:Encyclopedia of Mathematics
1837:Proc. Cambridge Philos. Soc
1363:L'Enseignement Mathématique
968:following work of Laurent,
509:, together with tools from
410:, later generalized to the
2649:
2496:Moduli of algebraic curves
1725:10.1007/s00222-020-00966-7
784:has genus greater than 1.
513:, including the theory of
2628:Theorems in number theory
1637:10.1007/978-1-4612-1210-2
1404:10.1007/978-1-4613-8655-1
543:Diophantine approximation
35:
2263:Cayley–Bacharach theorem
2190:Elliptic curve primality
1889:Parshin, A. N. (2001) .
1701:-adic period mappings".
1482:Inventiones Mathematicae
1439:Inventiones Mathematicae
1009:pseudo-canonical variety
985:Bombieri–Lang conjecture
527:Siegel modular varieties
168:until its 1983 proof by
98:Bombieri–Lang conjecture
2522:Riemann–Hurwitz formula
2486:Gromov–Witten invariant
2346:Compact Riemann surface
2134:Mazur's torsion theorem
1475:Faltings, Gerd (1984).
962:Mordell–Lang conjecture
933:{\displaystyle \Gamma }
829:{\displaystyle \Gamma }
651:{\displaystyle n\geq 4}
416:Mazur's torsion theorem
156:has only finitely many
102:Mordell–Lang conjecture
2139:Modular elliptic curve
1081:
1054:
1025:
1001:
954:
934:
914:
894:
870:
850:
836:of an abelian variety
830:
810:
778:
729:
709:
652:
614:
556:gave a proof based on
541:gave a proof based on
461:
441:
440:{\displaystyle g>1}
392:
372:
339:
319:
289:
269:
247:
220:
188:
146:
2053:Rational normal curve
1781:10.1090/trans2/050/11
1082:
1055:
1026:
1002:
955:
935:
915:
895:
871:
851:
831:
811:
779:
730:
710:
653:
630:Fermat's Last Theorem
615:
462:
442:
393:
373:
340:
320:
290:
270:
248:
221:
189:
147:
2623:Diophantine geometry
2593:Stable vector bundle
2465:Weil reciprocity law
2455:Riemann–Roch theorem
2435:Brill–Noether theory
2371:Riemann–Roch theorem
2288:Genus–degree formula
2149:Mordell–Weil theorem
2124:Division polynomials
1891:"Mordell conjecture"
1625:Diophantine geometry
1392:Silverman, Joseph H.
1096:Yuri Ivanovich Manin
1071:
1053:{\displaystyle X(k)}
1035:
1015:
991:
944:
924:
904:
884:
860:
840:
820:
800:
794:Mordell–Weil theorem
742:
719:
666:
636:
595:
552:Brian Lawrence and
451:
425:
412:Mordell–Weil theorem
382:
356:
345:may be handled as a
329:
303:
279:
257:
237:
210:
176:
134:
2416:Structure of curves
2308:Quartic plane curve
2230:Hyperelliptic curve
2210:De Franchis theorem
2154:Nagell–Lutz theorem
1926:1968IzMat...2.1145P
1452:1983InMat..73..349F
878:semiabelian variety
371:{\displaystyle g=1}
318:{\displaystyle g=0}
230:algebraic curve of
164:, and known as the
124:arithmetic geometry
52:Arithmetic geometry
32:
2423:Divisors on curves
2215:Faltings's theorem
2164:Schoof's algorithm
2144:Modularity theorem
1942:Shafarevich, I. R.
1821:10.1007/BF01241125
1600:10.1007/BF02684399
1496:10.1007/BF01388572
1460:10.1007/BF01388432
1355:Coleman, Robert F.
1141:10.1007/BF03024155
1077:
1050:
1021:
997:
950:
930:
910:
890:
866:
846:
826:
806:
774:
725:
705:
648:
610:
578:Mordell conjecture
561:-adic Hodge theory
511:algebraic geometry
457:
437:
388:
368:
335:
315:
285:
265:
243:
216:
184:
166:Mordell conjecture
142:
120:Faltings's theorem
31:Faltings's theorem
18:Mordell conjecture
2610:
2609:
2606:
2605:
2517:Hasse–Witt matrix
2460:Weierstrass point
2407:Smooth completion
2376:TeichmĂĽller space
2278:Cubic plane curve
2198:
2197:
2112:Arithmetic theory
2093:Elliptic integral
2088:Elliptic function
1829:Mordell, Louis J.
1104:Robert F. Coleman
1080:{\displaystyle X}
1024:{\displaystyle k}
1000:{\displaystyle X}
953:{\displaystyle A}
913:{\displaystyle A}
893:{\displaystyle C}
869:{\displaystyle A}
849:{\displaystyle A}
809:{\displaystyle C}
728:{\displaystyle n}
715:, since for such
481:abelian varieties
460:{\displaystyle C}
408:Mordell's Theorem
391:{\displaystyle C}
338:{\displaystyle C}
288:{\displaystyle C}
246:{\displaystyle g}
219:{\displaystyle C}
117:
116:
16:(Redirected from
2640:
2450:Jacobian variety
2420:
2323:Riemann surfaces
2313:Real plane curve
2273:Cramer's paradox
2253:BĂ©zout's theorem
2078:
2027:algebraic curves
2019:
2012:
2005:
1996:
1990:
1949:
1937:
1920:(5): 1191–1219.
1904:
1885:
1883:
1882:
1876:
1870:. Archived from
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1844:
1824:
1802:
1764:
1728:
1718:
1700:
1693:
1673:
1658:
1619:
1573:
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1508:
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1471:
1425:
1386:
1350:
1327:Bombieri, Enrico
1312:
1306:
1300:
1294:
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1282:
1276:
1270:
1264:
1258:
1252:
1246:
1240:
1234:
1228:
1222:
1216:
1210:
1204:
1201:Shafarevich 1963
1198:
1192:
1182:
1176:
1170:
1153:
1152:
1120:
1086:
1084:
1083:
1078:
1059:
1057:
1056:
1051:
1030:
1028:
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1022:
1006:
1004:
1003:
998:
959:
957:
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951:
939:
937:
936:
931:
919:
917:
916:
911:
899:
897:
896:
891:
875:
873:
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867:
855:
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852:
847:
835:
833:
832:
827:
815:
813:
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807:
783:
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775:
767:
766:
754:
753:
734:
732:
731:
726:
714:
712:
711:
706:
704:
703:
691:
690:
678:
677:
657:
655:
654:
649:
632:: for any fixed
619:
617:
616:
611:
609:
608:
603:
560:
554:Akshay Venkatesh
519:Faltings heights
477:Igor Shafarevich
466:
464:
463:
458:
446:
444:
443:
438:
397:
395:
394:
389:
377:
375:
374:
369:
344:
342:
341:
336:
324:
322:
321:
316:
294:
292:
291:
286:
274:
272:
271:
266:
264:
252:
250:
249:
244:
225:
223:
222:
217:
193:
191:
190:
185:
183:
154:rational numbers
151:
149:
148:
143:
141:
40:
33:
21:
2648:
2647:
2643:
2642:
2641:
2639:
2638:
2637:
2613:
2612:
2611:
2602:
2574:
2565:Delta invariant
2536:
2505:
2469:
2430:Abel–Jacobi map
2411:
2385:
2381:Torelli theorem
2351:Dessin d'enfant
2331:Belyi's theorem
2317:
2303:PlĂĽcker formula
2234:
2225:Hurwitz surface
2194:
2173:
2107:
2081:Analytic theory
2073:Elliptic curves
2067:
2048:Projective line
2035:Rational curves
2029:
2023:
1993:
1971:10.2307/2944318
1952:
1940:
1907:
1888:
1880:
1878:
1874:
1855:
1847:
1827:
1806:
1791:
1766:
1731:
1698:
1696:
1690:
1676:Springer-Verlag
1662:
1647:
1622:
1594:(25): 131–149.
1576:
1562:
1549:
1527:10.2307/2944319
1511:
1474:
1432:
1428:Faltings (1983)
1414:
1394:, eds. (1986).
1390:Cornell, Gary;
1389:
1353:
1325:
1321:
1316:
1315:
1307:
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1211:
1207:
1199:
1195:
1183:
1179:
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1157:
1156:
1123:
1121:
1117:
1112:
1069:
1068:
1033:
1032:
1013:
1012:
989:
988:
942:
941:
922:
921:
902:
901:
882:
881:
858:
857:
838:
837:
818:
817:
798:
797:
792:Because of the
790:
788:Generalizations
758:
745:
740:
739:
717:
716:
695:
682:
669:
664:
663:
634:
633:
598:
593:
592:
585:Isogeny theorem
570:
558:
547:Enrico Bombieri
535:
507:Tate conjecture
497:Aleksei Parshin
474:
449:
448:
423:
422:
380:
379:
354:
353:
327:
326:
301:
300:
277:
276:
255:
254:
235:
234:
208:
207:
204:
174:
173:
158:rational points
132:
131:
122:is a result in
100:
94:Generalizations
43:
28:
23:
22:
15:
12:
11:
5:
2646:
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2625:
2615:
2614:
2608:
2607:
2604:
2603:
2601:
2600:
2595:
2590:
2584:
2582:
2580:Vector bundles
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2417:
2413:
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2410:
2409:
2404:
2399:
2393:
2391:
2387:
2386:
2384:
2383:
2378:
2373:
2368:
2363:
2358:
2353:
2348:
2343:
2338:
2333:
2327:
2325:
2319:
2318:
2316:
2315:
2310:
2305:
2300:
2295:
2290:
2285:
2280:
2275:
2270:
2265:
2260:
2255:
2250:
2244:
2242:
2236:
2235:
2233:
2232:
2227:
2222:
2217:
2212:
2206:
2204:
2200:
2199:
2196:
2195:
2193:
2192:
2187:
2181:
2179:
2175:
2174:
2172:
2171:
2166:
2161:
2156:
2151:
2146:
2141:
2136:
2131:
2126:
2121:
2115:
2113:
2109:
2108:
2106:
2105:
2100:
2095:
2090:
2084:
2082:
2075:
2069:
2068:
2066:
2065:
2060:
2058:Riemann sphere
2055:
2050:
2045:
2039:
2037:
2031:
2030:
2024:
2022:
2021:
2014:
2007:
1999:
1992:
1991:
1965:(3): 509–548.
1950:
1938:
1909:Parshin, A. N.
1905:
1886:
1845:
1825:
1815:(1): 143–159.
1804:
1789:
1765:(Translation:
1743:(in Russian).
1729:
1709:(3): 893–999.
1694:
1688:
1660:
1645:
1620:
1574:
1560:
1547:
1521:(3): 549–576.
1509:
1472:
1446:(3): 349–366.
1434:Faltings, Gerd
1430:
1412:
1387:
1369:(3): 393–427.
1351:
1341:(4): 615–640.
1322:
1320:
1317:
1314:
1313:
1301:
1289:
1277:
1273:McQuillan 1995
1265:
1253:
1241:
1229:
1217:
1205:
1193:
1177:
1164:
1163:
1161:
1158:
1155:
1154:
1125:Bloch, Spencer
1114:
1113:
1111:
1108:
1076:
1049:
1046:
1043:
1040:
1020:
996:
949:
929:
909:
889:
865:
845:
825:
805:
789:
786:
773:
770:
765:
761:
757:
752:
748:
724:
702:
698:
694:
689:
685:
681:
676:
672:
662:solutions) to
647:
644:
641:
626:
625:
607:
602:
581:
569:
566:
565:
564:
550:
534:
531:
489:good reduction
473:
470:
469:
468:
456:
436:
433:
430:
419:
400:elliptic curve
387:
367:
364:
361:
350:
334:
314:
311:
308:
284:
263:
242:
215:
203:
200:
182:
140:
115:
114:
109:
105:
104:
95:
91:
90:
87:
86:First proof in
83:
82:
77:
76:First proof by
73:
72:
69:
68:Conjectured in
65:
64:
59:
58:Conjectured by
55:
54:
49:
45:
44:
41:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
2645:
2634:
2631:
2629:
2626:
2624:
2621:
2620:
2618:
2599:
2596:
2594:
2591:
2589:
2586:
2585:
2583:
2581:
2577:
2571:
2568:
2566:
2563:
2561:
2558:
2556:
2553:
2551:
2548:
2547:
2545:
2543:
2542:Singularities
2539:
2533:
2530:
2528:
2525:
2523:
2520:
2518:
2515:
2514:
2512:
2508:
2502:
2499:
2497:
2494:
2492:
2489:
2487:
2484:
2482:
2479:
2478:
2476:
2472:
2466:
2463:
2461:
2458:
2456:
2453:
2451:
2448:
2446:
2443:
2441:
2438:
2436:
2433:
2431:
2428:
2427:
2425:
2421:
2418:
2414:
2408:
2405:
2403:
2400:
2398:
2395:
2394:
2392:
2390:Constructions
2388:
2382:
2379:
2377:
2374:
2372:
2369:
2367:
2364:
2362:
2361:Klein quartic
2359:
2357:
2354:
2352:
2349:
2347:
2344:
2342:
2341:Bolza surface
2339:
2337:
2336:Bring's curve
2334:
2332:
2329:
2328:
2326:
2324:
2320:
2314:
2311:
2309:
2306:
2304:
2301:
2299:
2296:
2294:
2291:
2289:
2286:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2268:Conic section
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2248:AF+BG theorem
2246:
2245:
2243:
2241:
2237:
2231:
2228:
2226:
2223:
2221:
2218:
2216:
2213:
2211:
2208:
2207:
2205:
2201:
2191:
2188:
2186:
2183:
2182:
2180:
2176:
2170:
2167:
2165:
2162:
2160:
2157:
2155:
2152:
2150:
2147:
2145:
2142:
2140:
2137:
2135:
2132:
2130:
2127:
2125:
2122:
2120:
2117:
2116:
2114:
2110:
2104:
2101:
2099:
2096:
2094:
2091:
2089:
2086:
2085:
2083:
2079:
2076:
2074:
2070:
2064:
2063:Twisted cubic
2061:
2059:
2056:
2054:
2051:
2049:
2046:
2044:
2041:
2040:
2038:
2036:
2032:
2028:
2020:
2015:
2013:
2008:
2006:
2001:
2000:
1997:
1988:
1984:
1980:
1976:
1972:
1968:
1964:
1961:
1960:
1959:Ann. of Math.
1955:
1951:
1947:
1943:
1939:
1935:
1931:
1927:
1923:
1919:
1916:
1915:
1910:
1906:
1902:
1898:
1897:
1892:
1887:
1877:on 2016-09-24
1873:
1869:
1865:
1861:
1854:
1850:
1849:Paršin, A. N.
1846:
1842:
1838:
1834:
1830:
1826:
1822:
1818:
1814:
1810:
1805:
1800:
1796:
1792:
1790:9780821817506
1786:
1782:
1778:
1774:
1770:
1762:
1758:
1754:
1750:
1747:: 1395–1440.
1746:
1742:
1738:
1734:
1733:Manin, Ju. I.
1730:
1726:
1722:
1717:
1712:
1708:
1704:
1695:
1691:
1689:3-540-61223-8
1685:
1681:
1677:
1672:
1671:
1665:
1661:
1656:
1652:
1648:
1646:0-387-98981-1
1642:
1638:
1634:
1630:
1626:
1621:
1617:
1613:
1609:
1605:
1601:
1597:
1593:
1589:
1588:
1583:
1579:
1578:Grauert, Hans
1575:
1571:
1567:
1563:
1561:0-12-197270-4
1557:
1553:
1548:
1544:
1540:
1536:
1532:
1528:
1524:
1520:
1517:
1516:
1515:Ann. of Math.
1510:
1506:
1502:
1497:
1492:
1488:
1485:(in German).
1484:
1483:
1478:
1473:
1469:
1465:
1461:
1457:
1453:
1449:
1445:
1442:(in German).
1441:
1440:
1435:
1431:
1429:
1423:
1419:
1415:
1413:0-387-96311-1
1409:
1405:
1401:
1397:
1393:
1388:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1356:
1352:
1348:
1344:
1340:
1336:
1332:
1328:
1324:
1323:
1318:
1310:
1305:
1302:
1298:
1293:
1290:
1286:
1281:
1278:
1274:
1269:
1266:
1262:
1257:
1254:
1250:
1249:Bombieri 1990
1245:
1242:
1238:
1233:
1230:
1226:
1225:Faltings 1983
1221:
1218:
1214:
1209:
1206:
1202:
1197:
1194:
1190:
1189:Faltings 1984
1186:
1185:Faltings 1983
1181:
1178:
1174:
1169:
1166:
1159:
1150:
1146:
1142:
1138:
1134:
1130:
1126:
1119:
1116:
1109:
1107:
1105:
1101:
1097:
1092:
1090:
1074:
1066:
1063:
1044:
1038:
1018:
1010:
994:
986:
981:
979:
975:
971:
967:
963:
960:leads to the
947:
907:
887:
879:
863:
843:
803:
795:
787:
785:
771:
768:
763:
759:
755:
750:
746:
738:
722:
700:
696:
692:
687:
683:
679:
674:
670:
661:
645:
642:
639:
631:
623:
605:
590:
586:
582:
579:
575:
574:
573:
567:
562:
555:
551:
548:
544:
540:
537:
536:
532:
530:
528:
524:
523:naive heights
520:
516:
512:
508:
504:
503:Gerd Faltings
500:
498:
494:
490:
486:
482:
478:
471:
454:
434:
431:
428:
420:
417:
414:.) Moreover,
413:
409:
405:
401:
385:
365:
362:
359:
351:
348:
347:conic section
332:
312:
309:
306:
298:
297:
296:
282:
240:
233:
229:
213:
201:
199:
197:
171:
170:Gerd Faltings
167:
163:
162:Louis Mordell
159:
155:
129:
125:
121:
113:
110:
106:
103:
99:
96:
92:
88:
84:
81:
80:Gerd Faltings
78:
74:
70:
66:
63:
62:Louis Mordell
60:
56:
53:
50:
46:
42:Gerd Faltings
39:
34:
19:
2527:Prym variety
2501:Stable curve
2491:Hodge bundle
2481:ELSV formula
2283:Fermat curve
2240:Plane curves
2214:
2203:Higher genus
2178:Applications
2103:Modular form
1962:
1957:
1945:
1917:
1912:
1894:
1879:. Retrieved
1872:the original
1859:
1840:
1836:
1812:
1809:Invent. Math
1808:
1772:
1771:. Series 2.
1768:
1744:
1740:
1706:
1703:Invent. Math
1702:
1669:
1624:
1591:
1585:
1551:
1518:
1513:
1486:
1480:
1443:
1437:
1395:
1366:
1365:. 2e SĂ©rie.
1362:
1338:
1334:
1309:Coleman 1990
1304:
1297:Grauert 1965
1292:
1280:
1268:
1256:
1244:
1232:
1220:
1213:Parshin 1968
1208:
1196:
1180:
1173:Mordell 1922
1168:
1132:
1128:
1118:
1100:Hans Grauert
1093:
982:
791:
737:Fermat curve
627:
589:Tate modules
584:
577:
571:
568:Consequences
533:Later proofs
515:NĂ©ron models
501:
485:polarization
475:
407:
228:non-singular
205:
196:number field
165:
119:
118:
108:Consequences
2402:Polar curve
1954:Vojta, Paul
1775:: 189–234.
1678:. pp.
1664:Lang, Serge
1102:. In 1990,
406:. (This is
2617:Categories
2397:Dual curve
2025:Topics in
1948:: 163–176.
1881:2016-06-11
1843:: 179–192.
1716:1807.02721
1489:(2): 381.
1319:References
1285:Manin 1963
1237:Vojta 1991
1089:Paul Vojta
972:, Hindry,
539:Paul Vojta
202:Background
2510:Morphisms
2258:Bitangent
1901:EMS Press
1799:0065-9290
1753:0373-2436
1608:1618-1913
1375:0013-8584
1160:Citations
1135:(2): 44.
966:McQuillan
928:Γ
824:Γ
643:≥
622:isogenous
606:ℓ
1851:(1970).
1831:(1922).
1735:(1963).
1666:(1997).
1580:(1965).
1357:(1990).
1329:(1990).
987:that if
978:Faltings
2570:Tacnode
2555:Crunode
1987:1109352
1979:2944318
1922:Bibcode
1868:0427323
1761:0157971
1655:1745599
1616:0222087
1570:1307396
1543:1109353
1535:2944319
1505:0732554
1468:0718935
1448:Bibcode
1422:0861969
1383:1096426
1347:1093712
1098:and by
1062:Zariski
1060:is not
1031:, then
970:Raynaud
660:coprime
194:by any
2550:Acnode
2474:Moduli
1985:
1977:
1866:
1797:
1787:
1759:
1751:
1686:
1682:–122.
1653:
1643:
1614:
1606:
1568:
1558:
1541:
1533:
1503:
1466:
1420:
1410:
1381:
1373:
1345:
1149:306251
1147:
976:, and
920:, and
493:places
472:Proofs
398:is an
1975:JSTOR
1875:(PDF)
1856:(PDF)
1711:arXiv
1531:JSTOR
1145:S2CID
1110:Notes
1065:dense
1007:is a
974:Vojta
876:by a
421:When
352:When
299:When
253:over
232:genus
226:be a
128:genus
48:Field
2560:Cusp
1795:ISSN
1785:ISBN
1749:ISSN
1684:ISBN
1641:ISBN
1604:ISSN
1556:ISBN
1408:ISBN
1371:ISSN
735:the
591:(as
583:The
576:The
525:via
521:and
432:>
206:Let
89:1983
71:1922
1967:doi
1963:133
1930:doi
1817:doi
1813:120
1777:doi
1721:doi
1707:221
1680:101
1633:doi
1596:doi
1523:doi
1519:133
1491:doi
1456:doi
1400:doi
1137:doi
1067:in
152:of
2619::
1983:MR
1981:.
1973:.
1928:.
1918:32
1899:.
1893:.
1864:MR
1858:.
1841:21
1839:.
1835:.
1811:.
1793:.
1783:.
1773:59
1757:MR
1755:.
1745:27
1739:.
1719:.
1705:.
1674:.
1651:MR
1649:.
1639:.
1627:.
1612:MR
1610:.
1602:.
1592:25
1590:.
1584:.
1566:MR
1564:.
1539:MR
1537:.
1529:.
1501:MR
1499:.
1487:75
1479:.
1464:MR
1462:.
1454:.
1444:73
1418:MR
1416:.
1406:.
1379:MR
1377:.
1367:36
1361:.
1343:MR
1339:17
1337:.
1333:.
1187:;
1143:.
1131:.
1091:.
980:.
880:,
545:.
529:.
495:.
198:.
2018:e
2011:t
2004:v
1989:.
1969::
1936:.
1932::
1924::
1903:.
1884:.
1823:.
1819::
1803:)
1801:.
1779::
1763:.
1727:.
1723::
1713::
1699:p
1692:.
1657:.
1635::
1618:.
1598::
1572:.
1545:.
1525::
1507:.
1493::
1470:.
1458::
1450::
1424:.
1402::
1385:.
1349:.
1311:.
1299:.
1287:.
1275:.
1263:.
1251:.
1239:.
1227:.
1215:.
1203:.
1191:.
1175:.
1151:.
1139::
1133:6
1075:X
1048:)
1045:k
1042:(
1039:X
1019:k
995:X
948:A
908:A
888:C
864:A
844:A
804:C
772:1
769:=
764:n
760:y
756:+
751:n
747:x
723:n
701:n
697:c
693:=
688:n
684:b
680:+
675:n
671:a
646:4
640:n
624:.
601:Q
559:p
455:C
435:1
429:g
386:C
366:1
363:=
360:g
349:.
333:C
313:0
310:=
307:g
283:C
262:Q
241:g
214:C
181:Q
139:Q
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
