2710:
In much the same way, one can define a graph-theoretic analogue of Abel–Jacobi map as a piecewise-linear map from a finite graph into a flat torus (or a Cayley graph associated with a finite abelian group), which is closely related to asymptotic behaviors of random walks on crystal lattices, and can
910:
4052:
514:
2454:
3749:
2620:
747:
2012:
1268:
3878:
1358:
171:
4158:
2334:
1612:
398:
2345:
618:
4432:
3072:
2866:
3873:
310:
239:
4351:
1918:
676:
386:
1506:
1416:
2944:
3504:
1702:
3441:
4208:
3332:
3166:
1030:
2984:
1840:
2687:
1216:
3592:
4585:(divisors of degree 0 modulo equivalence). For higher-dimensional compact projective varieties the Albanese variety and the Picard variety are dual but need not be isomorphic.
3789:
1448:
2253:
1535:
1174:
2484:
4258:
3630:
2064:
3382:
3216:
3122:
548:
4715:
3625:
3288:
1873:
1644:
716:
4521:
1298:
1053:
739:
1073:
571:
2770:
4474:
2171:
1796:
1100:
940:
4287:
4080:
4568:
3259:
963:
4541:
4228:
3815:
3548:
3528:
3236:
3004:
2886:
2740:
2507:
2211:
2191:
2144:
2124:
2104:
2084:
2032:
1769:
1749:
1725:
1664:
1555:
1468:
1378:
1128:
983:
905:{\displaystyle {\begin{cases}u:C\to J(C)\\u(p)=\left(\int _{p_{0}}^{p}\omega _{1},\dots ,\int _{p_{0}}^{p}\omega _{g}\right){\bmod {\Lambda }}\end{cases}}}
2515:
5138:
1926:
4570:
This implies that the Abel-Jacobi map induces an injective map (of abelian groups) from the space of divisor classes of degree zero to the
Jacobian.
4867:
5230:
4827:
4708:
4612:
3751:
which is a well-defined holomorphic mapping with rank 1 (maximal rank). Then we can naturally extend this to a mapping of divisor classes;
1225:
5296:
4918:
4817:
4685:
4047:{\displaystyle \varphi (D)=\sum _{j=1}^{r}\varphi (P_{j})-\sum _{j=1}^{s}\varphi (Q_{j}),\quad D=P_{1}\cdots P_{r}/Q_{1}\cdots Q_{s}.}
5286:
1303:
509:{\displaystyle \Omega _{j}=\left(\int _{\gamma _{j}}\omega _{1},\ldots ,\int _{\gamma _{j}}\omega _{g}\right)\in \mathbb {C} ^{g}.}
111:
4085:
2264:
4996:
4701:
2699:. The Abel–Jacobi map of a Riemannian manifold shows up in the large time asymptotics of the heat kernel on a periodic manifold (
2449:{\displaystyle {\begin{cases}{\overline {A}}_{M}:{\overline {M}}\to E^{*}\\c\mapsto \left(h\mapsto \int _{c}h\right)\end{cases}}}
4574:
Jacobi proved that this map is also surjective (known as Jacobi inversion problem), so the two groups are naturally isomorphic.
1564:
5143:
5054:
576:
5064:
4991:
4366:
4741:
3017:
4961:
2775:
5326:
4857:
3820:
5220:
5184:
255:
4883:
4796:
195:
4303:
1878:
626:
345:
1473:
1383:
5331:
5194:
4832:
2891:
520:
87:
5336:
5321:
5240:
3446:
5153:
5133:
5069:
4986:
4847:
1669:
4888:
67:
4852:
3387:
4260:
is precisely the subgroup of principal divisors. Together with the Jacobi inversion problem, we can say that
4163:
4082:
then this map is independent of the choice of the base point so we can define the base point independent map
3293:
3127:
5044:
4289:
is isomorphic as a group to the group of divisors of degree zero modulo its subgroup of principal divisors.
988:
2949:
1801:
4837:
4625:; Sunada, Toshikazu (2000), "Albanese maps and an off diagonal long time asymptotic for the heat kernel",
2634:
5215:
4951:
1179:
4751:
3556:
4913:
4862:
3757:
1421:
2219:
1511:
1137:
5291:
5163:
5074:
4822:
4634:
3744:{\displaystyle \varphi (P)=\left(\int _{P_{0}}^{P}\zeta _{1},...,\int _{P_{0}}^{P}\zeta _{g}\right).}
2462:
2354:
756:
5006:
4971:
4928:
4908:
4544:
3793:
75:
51:
4236:
2695:
The Abel–Jacobi map is unique up to translations of the Jacobi torus. The map has applications in
2040:
5258:
4842:
4437:
and therefore speak of the value of the Abel–Jacobi map on divisors. The theorem is then that if
3337:
3171:
3077:
3007:
526:
241:
generating it. On the other hand, another more algebro-geometric way of saying that the genus of
39:
5049:
5029:
5001:
3597:
3264:
1845:
1620:
688:
4482:
1273:
1035:
721:
5158:
5105:
4976:
4791:
4786:
4681:
4608:
2696:
1728:
1058:
556:
331:
71:
63:
4581:
of a compact complex curve (dual of holomorphic 1-forms modulo periods) is isomorphic to its
5148:
5034:
5011:
4665:
4653:
4642:
4582:
4578:
2749:
320:
99:
47:
4452:
2149:
1774:
1078:
918:
5263:
5079:
5021:
4923:
4746:
4725:
4604:
4263:
2743:
2035:
551:
43:
4059:
4946:
4638:
4550:
3241:
2719:
We provide an analytic construction of the Abel-Jacobi map on compact
Riemann surfaces.
945:
4771:
4756:
4733:
4526:
4213:
3800:
3533:
3513:
3221:
2989:
2871:
2725:
2492:
2196:
2176:
2129:
2109:
2089:
2069:
2017:
1754:
1734:
1710:
1649:
1540:
1453:
1363:
1219:
1113:
968:
177:
5315:
5278:
5059:
5039:
4966:
4761:
4622:
2615:{\displaystyle J_{1}(M)=H_{1}(M,\mathbb {R} )/H_{1}(M,\mathbb {Z} )_{\mathbb {R} }.}
5225:
5199:
5189:
5179:
4981:
4801:
1558:
17:
4297:
The following theorem was proved by Abel (known as Abel's theorem): Suppose that
5100:
4938:
31:
2007:{\displaystyle M\to H_{1}(M,\mathbb {R} )/H_{1}(M,\mathbb {Z} )_{\mathbb {R} }}
5095:
4669:
4693:
4956:
4646:
1842:. By integrating an integral harmonic 1-form along paths from a basepoint
1131:
55:
5268:
5253:
4356:
is a divisor (meaning a formal integer-linear combination of points of
5248:
78:
if and only if they are indistinguishable under the Abel–Jacobi map.
1263:{\displaystyle \operatorname {tor} =\operatorname {tor} (\pi ^{ab})}
2014:
without choosing a basis for cohomology, we argue as follows. Let
4595:
E. Arbarello; M. Cornalba; P. Griffiths; J. Harris (1985). "1.3,
330:
By definition, this is the space of globally defined holomorphic
1055:
Thus the difference is erased in the passage to the quotient by
388:. Given forms and closed loops we can integrate, and we define 2
4697:
1353:{\displaystyle g:\pi ^{ab}\to \pi ^{ab}/\operatorname {tor} }
1102:
does change the map, but only by a translation of the torus.
886:
188:, or in other words, closed loops. Therefore, we can choose 2
166:{\displaystyle H_{1}(C,\mathbb {Z} )\cong \mathbb {Z} ^{2g}.}
4153:{\displaystyle \varphi _{0}:\mathrm {Div} ^{(0)}(M)\to J(M)}
2956:
2329:{\displaystyle {\tilde {M}}\to E^{*}=H_{1}(M,\mathbb {R} ),}
2442:
898:
1607:{\displaystyle \varphi =g\circ f:\pi \to \mathbb {Z} ^{b}}
4233:
The below Abel's theorem show that the kernel of the map
3006:
dimensional complex vector space consists of holomorphic
1704:
is called the universal (or maximal) free abelian cover.
2692:
is obtained from the map above by passing to quotients.
613:{\displaystyle \mathbb {C} ^{g}\cong \mathbb {R} ^{2g}}
94:
is constructed using path integration. Namely, suppose
4553:
4529:
4485:
4455:
4427:{\displaystyle u(D)=\sum \nolimits _{i}n_{i}u(p_{i})}
4369:
4306:
4266:
4239:
4216:
4166:
4088:
4062:
3881:
3823:
3803:
3760:
3633:
3600:
3559:
3536:
3516:
3449:
3390:
3340:
3296:
3267:
3244:
3224:
3174:
3130:
3080:
3020:
2992:
2952:
2894:
2874:
2778:
2752:
2728:
2637:
2518:
2495:
2465:
2348:
2267:
2222:
2199:
2179:
2152:
2132:
2112:
2092:
2072:
2043:
2020:
1929:
1881:
1848:
1804:
1777:
1757:
1737:
1713:
1672:
1652:
1623:
1567:
1543:
1514:
1476:
1456:
1424:
1386:
1366:
1306:
1276:
1228:
1182:
1140:
1116:
1081:
1061:
1038:
991:
971:
948:
921:
750:
724:
691:
629:
579:
559:
529:
401:
348:
258:
198:
114:
3067:{\displaystyle \int _{a_{k}}\zeta _{j}=\delta _{jk}}
915:
Although this is seemingly dependent on a path from
685:
is then defined as follows. We pick some base point
5277:
5239:
5208:
5172:
5121:
5114:
5088:
5020:
4937:
4901:
4876:
4810:
4779:
4770:
4732:
3124:. We can form a symmetric matrix whose entries are
2861:{\displaystyle \{a_{1},...,a_{g},b_{1},...,b_{g}\}}
4676:Farkas, Hershel M; Kra, Irwin (23 December 1991),
4656:(2012), "Lecture on topological crystallography",
4562:
4535:
4515:
4468:
4426:
4345:
4281:
4252:
4222:
4202:
4152:
4074:
4046:
3868:{\displaystyle \varphi :\mathrm {Div} (M)\to J(M)}
3867:
3809:
3783:
3743:
3619:
3586:
3542:
3522:
3498:
3435:
3376:
3326:
3282:
3253:
3230:
3210:
3160:
3116:
3066:
2998:
2978:
2938:
2880:
2860:
2764:
2734:
2681:
2614:
2501:
2478:
2448:
2328:
2247:
2205:
2185:
2165:
2138:
2118:
2098:
2078:
2058:
2026:
2006:
1912:
1867:
1834:
1790:
1763:
1743:
1719:
1696:
1658:
1638:
1606:
1549:
1529:
1500:
1462:
1442:
1410:
1372:
1352:
1292:
1262:
1210:
1168:
1122:
1094:
1067:
1047:
1024:
977:
957:
934:
904:
733:
710:
670:
612:
565:
542:
508:
380:
304:
233:
165:
4603:. Grundlehren der Mathematischen Wissenschaften.
2489:Definition. The Jacobi variety (Jacobi torus) of
305:{\displaystyle H^{0}(C,K)\cong \mathbb {C} ^{g},}
2715:The Abel–Jacobi map of a compact Riemann surface
234:{\displaystyle \gamma _{1},\ldots ,\gamma _{2g}}
4346:{\displaystyle D=\sum \nolimits _{i}n_{i}p_{i}}
1913:{\displaystyle \mathbb {R} /\mathbb {Z} =S^{1}}
671:{\displaystyle J(C)=\mathbb {C} ^{g}/\Lambda .}
381:{\displaystyle \omega _{1},\ldots ,\omega _{g}}
58:to its Jacobi torus. The name derives from the
1501:{\displaystyle \pi ^{ab}/\operatorname {tor} }
1411:{\displaystyle \pi ^{ab}/\operatorname {tor} }
54:, it is a more general construction mapping a
4709:
2939:{\displaystyle \{\zeta _{1},...,\zeta _{g}\}}
8:
3430:
3404:
2933:
2895:
2855:
2779:
2700:
1106:The Abel–Jacobi map of a Riemannian manifold
3499:{\displaystyle J(M)={\mathbb {C}}^{g}/L(M)}
1032:so integration over it gives an element of
965:any two such paths define a closed loop in
5118:
4776:
4716:
4702:
4694:
2711:be used for design of crystal structures.
1697:{\displaystyle \ker(\varphi )\subset \pi }
4577:The Abel–Jacobi theorem implies that the
4552:
4528:
4484:
4460:
4454:
4415:
4399:
4389:
4368:
4337:
4327:
4317:
4305:
4265:
4244:
4238:
4215:
4179:
4168:
4165:
4114:
4103:
4093:
4087:
4061:
4035:
4022:
4013:
4007:
3994:
3971:
3955:
3944:
3928:
3912:
3901:
3880:
3830:
3822:
3802:
3761:
3759:
3727:
3717:
3710:
3705:
3680:
3670:
3663:
3658:
3632:
3605:
3599:
3558:
3535:
3515:
3479:
3473:
3468:
3467:
3466:
3448:
3424:
3411:
3395:
3389:
3339:
3318:
3306:
3301:
3295:
3266:
3243:
3223:
3173:
3152:
3140:
3135:
3129:
3079:
3055:
3042:
3030:
3025:
3019:
2991:
2961:
2955:
2954:
2951:
2927:
2902:
2893:
2873:
2849:
2824:
2811:
2786:
2777:
2751:
2727:
2661:
2642:
2636:
2603:
2602:
2601:
2593:
2592:
2577:
2568:
2561:
2560:
2545:
2523:
2517:
2494:
2466:
2464:
2425:
2394:
2377:
2368:
2358:
2349:
2347:
2316:
2315:
2300:
2287:
2269:
2268:
2266:
2233:
2221:
2198:
2178:
2157:
2151:
2131:
2111:
2091:
2071:
2045:
2044:
2042:
2019:
1998:
1997:
1996:
1988:
1987:
1972:
1963:
1956:
1955:
1940:
1928:
1904:
1893:
1892:
1887:
1883:
1882:
1880:
1853:
1847:
1825:
1824:
1809:
1803:
1782:
1776:
1756:
1736:
1712:
1671:
1651:
1625:
1624:
1622:
1598:
1594:
1593:
1566:
1542:
1521:
1517:
1516:
1513:
1490:
1481:
1475:
1455:
1431:
1427:
1426:
1423:
1400:
1391:
1385:
1365:
1342:
1333:
1317:
1305:
1281:
1275:
1248:
1227:
1199:
1181:
1151:
1139:
1115:
1086:
1080:
1060:
1037:
1012:
1011:
996:
990:
970:
947:
926:
920:
889:
885:
874:
864:
857:
852:
833:
823:
816:
811:
751:
749:
723:
696:
690:
657:
651:
647:
646:
628:
601:
597:
596:
586:
582:
581:
578:
558:
534:
528:
497:
493:
492:
477:
465:
460:
441:
429:
424:
406:
400:
372:
353:
347:
293:
289:
288:
263:
257:
222:
203:
197:
151:
147:
146:
135:
134:
119:
113:
3436:{\displaystyle c_{k}\in \{a_{k},b_{k}\}}
718:and, nearly mimicking the definition of
59:
4210:denotes the divisors of degree zero of
4203:{\displaystyle \mathrm {Div} ^{(0)}(M)}
3327:{\displaystyle \int _{c_{k}}\zeta _{j}}
3161:{\displaystyle \int _{b_{k}}\zeta _{j}}
5139:Clifford's theorem on special divisors
2704:
2339:which, furthermore, descends to a map
1025:{\displaystyle H_{1}(C,\mathbb {Z} ),}
2979:{\displaystyle {\mathcal {H}}^{1}(M)}
2486:is the universal free abelian cover.
2173:to it. By integrating along the path
1835:{\displaystyle H_{1}(M,\mathbb {R} )}
7:
4601:Geometry of Algebraic Curves, Vol. 1
2682:{\displaystyle A_{M}:M\to J_{1}(M),}
1923:Similarly, in order to define a map
1751:be the space of harmonic 1-forms on
573:(that is, they are a real basis for
4386:
4314:
1211:{\displaystyle f:\pi \to \pi ^{ab}}
5297:Vector bundles on algebraic curves
5231:Weber's theorem (Algebraic curves)
4828:Hasse's theorem on elliptic curves
4818:Counting points on elliptic curves
4175:
4172:
4169:
4110:
4107:
4104:
3837:
3834:
3831:
3768:
3765:
3762:
3587:{\displaystyle \varphi :M\to J(M)}
1062:
1039:
890:
725:
662:
620:), and the Jacobian is defined by
560:
531:
403:
180:consists of (homology classes of)
27:Construction in algebraic geometry
25:
3784:{\displaystyle \mathrm {Div} (M)}
3290:matrix whose entries consists of
2868:be a canonical homology basis on
1508:is non-canonically isomorphic to
1443:{\displaystyle \mathbb {Z} ^{2g}}
1418:is non-canonically isomorphic to
105:, which means topologically that
4476:are all positive integers, then
3550:-dimensional complex Lie group.
3530:which is a compact, commutative
3238:be the lattice generated by the
2248:{\displaystyle h\to \int _{c}h.}
1875:, we obtain a map to the circle
1530:{\displaystyle \mathbb {Z} ^{b}}
1169:{\displaystyle \pi =\pi _{1}(M)}
4919:Hurwitz's automorphisms theorem
3983:
2479:{\displaystyle {\overline {M}}}
1614:be the composite homomorphism.
1360:be the quotient by torsion. If
5144:Gonality of an algebraic curve
5055:Differential of the first kind
4510:
4504:
4495:
4489:
4421:
4408:
4379:
4373:
4276:
4270:
4197:
4191:
4186:
4180:
4147:
4141:
4135:
4132:
4126:
4121:
4115:
3977:
3964:
3934:
3921:
3891:
3885:
3862:
3856:
3850:
3847:
3841:
3778:
3772:
3643:
3637:
3581:
3575:
3569:
3493:
3487:
3459:
3453:
2973:
2967:
2673:
2667:
2654:
2598:
2583:
2565:
2551:
2535:
2529:
2418:
2407:
2387:
2320:
2306:
2280:
2274:
2226:
2050:
1993:
1978:
1960:
1946:
1933:
1829:
1815:
1685:
1679:
1666:corresponding to the subgroup
1630:
1589:
1470:is the genus; more generally,
1326:
1257:
1241:
1192:
1176:be its fundamental group. Let
1163:
1157:
1016:
1002:
985:and, therefore, an element of
796:
790:
780:
774:
768:
639:
633:
281:
269:
139:
125:
1:
5287:Birkhoff–Grothendieck theorem
4997:Nagata's conjecture on curves
4868:Schoof–Elkies–Atkin algorithm
4742:Five points determine a conic
2193:, we obtain a linear form on
2106:is represented by a point of
4858:Supersingular elliptic curve
4253:{\displaystyle \varphi _{0}}
2471:
2382:
2363:
2059:{\displaystyle {\tilde {M}}}
1798:canonically identified with
5065:Riemann's existence theorem
4992:Hilbert's sixteenth problem
4884:Elliptic curve cryptography
4797:Fundamental pair of periods
4449:divisors, meaning that the
3377:{\displaystyle j,k=1,...,g}
3211:{\displaystyle j,k=1,...,g}
3117:{\displaystyle j,k=1,...,g}
1270:be the torsion subgroup of
543:{\displaystyle \Omega _{j}}
342:linearly independent forms
5353:
5195:Moduli of algebraic curves
3620:{\displaystyle P_{0}\in M}
3283:{\displaystyle g\times 2g}
2701:Kotani & Sunada (2000)
1868:{\displaystyle x_{0}\in M}
1639:{\displaystyle {\bar {M}}}
711:{\displaystyle p_{0}\in C}
521:Riemann bilinear relations
90:, the Jacobian of a curve
88:complex algebraic geometry
4670:10.1007/s11537-012-1144-4
4516:{\displaystyle u(D)=u(E)}
1293:{\displaystyle \pi ^{ab}}
1048:{\displaystyle \Lambda .}
734:{\displaystyle \Lambda ,}
550:generate a nondegenerate
4962:Cayley–Bacharach theorem
4889:Elliptic curve primality
1068:{\displaystyle \Lambda }
566:{\displaystyle \Lambda }
5221:Riemann–Hurwitz formula
5185:Gromov–Witten invariant
5045:Compact Riemann surface
4833:Mazur's torsion theorem
2258:This gives rise a map
82:Construction of the map
4838:Modular elliptic curve
4680:, New York: Springer,
4564:
4537:
4517:
4470:
4428:
4347:
4283:
4254:
4224:
4204:
4154:
4076:
4048:
3960:
3917:
3869:
3811:
3785:
3745:
3621:
3588:
3544:
3524:
3500:
3437:
3378:
3328:
3284:
3255:
3232:
3212:
3162:
3118:
3068:
3000:
2980:
2940:
2882:
2862:
2766:
2765:{\displaystyle g>0}
2736:
2683:
2616:
2503:
2480:
2450:
2330:
2249:
2207:
2187:
2167:
2140:
2120:
2100:
2080:
2060:
2028:
2008:
1914:
1869:
1836:
1792:
1765:
1745:
1721:
1698:
1660:
1640:
1617:Definition. The cover
1608:
1551:
1531:
1502:
1464:
1444:
1412:
1374:
1354:
1294:
1264:
1212:
1170:
1124:
1096:
1075:. Changing base-point
1069:
1049:
1026:
979:
959:
936:
906:
735:
712:
672:
614:
567:
544:
510:
382:
306:
235:
167:
4752:Rational normal curve
4647:10.1007/s002200050033
4565:
4538:
4518:
4471:
4469:{\displaystyle n_{i}}
4429:
4348:
4284:
4255:
4225:
4205:
4155:
4077:
4049:
3940:
3897:
3870:
3812:
3786:
3746:
3622:
3589:
3545:
3525:
3501:
3438:
3379:
3329:
3285:
3256:
3233:
3213:
3163:
3119:
3069:
3001:
2981:
2941:
2883:
2863:
2767:
2737:
2684:
2617:
2504:
2481:
2451:
2331:
2250:
2208:
2188:
2168:
2166:{\displaystyle x_{0}}
2141:
2126:together with a path
2121:
2101:
2081:
2061:
2029:
2009:
1915:
1870:
1837:
1793:
1791:{\displaystyle E^{*}}
1766:
1746:
1722:
1699:
1661:
1641:
1609:
1552:
1532:
1503:
1465:
1445:
1413:
1375:
1355:
1295:
1265:
1213:
1171:
1125:
1097:
1095:{\displaystyle p_{0}}
1070:
1050:
1027:
980:
960:
937:
935:{\displaystyle p_{0}}
907:
736:
713:
673:
615:
568:
545:
511:
383:
307:
236:
168:
38:is a construction of
5292:Stable vector bundle
5164:Weil reciprocity law
5154:Riemann–Roch theorem
5134:Brill–Noether theory
5070:Riemann–Roch theorem
4987:Genus–degree formula
4848:Mordell–Weil theorem
4823:Division polynomials
4551:
4527:
4483:
4453:
4367:
4304:
4282:{\displaystyle J(M)}
4264:
4237:
4214:
4164:
4086:
4060:
3879:
3821:
3801:
3758:
3631:
3598:
3594:by choosing a point
3557:
3553:We can define a map
3534:
3514:
3447:
3388:
3338:
3294:
3265:
3242:
3222:
3172:
3128:
3078:
3018:
2990:
2950:
2892:
2872:
2776:
2750:
2726:
2635:
2516:
2493:
2463:
2346:
2265:
2220:
2197:
2177:
2150:
2130:
2110:
2090:
2070:
2041:
2018:
1927:
1879:
1846:
1802:
1775:
1755:
1735:
1711:
1670:
1650:
1621:
1565:
1541:
1512:
1474:
1454:
1422:
1384:
1364:
1304:
1274:
1226:
1180:
1138:
1130:be a smooth compact
1114:
1079:
1059:
1036:
989:
969:
946:
919:
748:
722:
689:
627:
577:
557:
527:
519:It follows from the
399:
346:
256:
196:
176:Geometrically, this
112:
5327:Riemannian geometry
5115:Structure of curves
5007:Quartic plane curve
4929:Hyperelliptic curve
4909:De Franchis theorem
4853:Nagell–Lutz theorem
4639:2000CMaPh.209..633K
4545:linearly equivalent
4293:Abel–Jacobi theorem
4075:{\displaystyle r=s}
3794:divisor class group
3722:
3675:
2946:the dual basis for
869:
828:
338:, so we can choose
76:linearly equivalent
52:Riemannian geometry
18:Abel–Jacobi theorem
5122:Divisors on curves
4914:Faltings's theorem
4863:Schoof's algorithm
4843:Modularity theorem
4563:{\displaystyle E.}
4560:
4533:
4513:
4466:
4424:
4343:
4279:
4250:
4220:
4200:
4150:
4072:
4044:
3865:
3817:then define a map
3807:
3781:
3741:
3701:
3654:
3617:
3584:
3540:
3520:
3496:
3433:
3374:
3324:
3280:
3254:{\displaystyle 2g}
3251:
3228:
3208:
3158:
3114:
3064:
3008:differential forms
2996:
2976:
2936:
2878:
2858:
2762:
2742:denotes a compact
2732:
2679:
2612:
2499:
2476:
2446:
2441:
2326:
2245:
2203:
2183:
2163:
2136:
2116:
2096:
2076:
2056:
2034:be a point in the
2024:
2004:
1910:
1865:
1832:
1788:
1761:
1741:
1717:
1694:
1656:
1636:
1604:
1547:
1527:
1498:
1460:
1440:
1408:
1370:
1350:
1290:
1260:
1208:
1166:
1120:
1092:
1065:
1045:
1022:
975:
958:{\displaystyle p,}
955:
932:
902:
897:
848:
807:
731:
708:
668:
610:
563:
540:
506:
378:
332:differential forms
302:
231:
163:
72:effective divisors
40:algebraic geometry
5332:Niels Henrik Abel
5309:
5308:
5305:
5304:
5216:Hasse–Witt matrix
5159:Weierstrass point
5106:Smooth completion
5075:Teichmüller space
4977:Cubic plane curve
4897:
4896:
4811:Arithmetic theory
4792:Elliptic integral
4787:Elliptic function
4654:Sunada, Toshikazu
4627:Comm. Math. Phys.
4614:978-0-387-90997-4
4536:{\displaystyle D}
4360:). We can define
4223:{\displaystyle M}
3810:{\displaystyle M}
3543:{\displaystyle g}
3523:{\displaystyle M}
3231:{\displaystyle L}
2999:{\displaystyle g}
2881:{\displaystyle M}
2735:{\displaystyle M}
2697:Systolic geometry
2502:{\displaystyle M}
2474:
2385:
2366:
2277:
2206:{\displaystyle E}
2186:{\displaystyle c}
2139:{\displaystyle c}
2119:{\displaystyle M}
2099:{\displaystyle x}
2079:{\displaystyle M}
2053:
2027:{\displaystyle x}
1764:{\displaystyle M}
1744:{\displaystyle E}
1729:Riemannian metric
1720:{\displaystyle M}
1659:{\displaystyle M}
1633:
1550:{\displaystyle b}
1463:{\displaystyle g}
1373:{\displaystyle M}
1123:{\displaystyle M}
978:{\displaystyle C}
42:which relates an
16:(Redirected from
5344:
5337:Riemann surfaces
5322:Algebraic curves
5149:Jacobian variety
5119:
5022:Riemann surfaces
5012:Real plane curve
4972:Cramer's paradox
4952:Bézout's theorem
4777:
4726:algebraic curves
4718:
4711:
4704:
4695:
4690:
4678:Riemann surfaces
4672:
4649:
4618:
4583:Jacobian variety
4579:Albanese variety
4569:
4567:
4566:
4561:
4542:
4540:
4539:
4534:
4522:
4520:
4519:
4514:
4475:
4473:
4472:
4467:
4465:
4464:
4433:
4431:
4430:
4425:
4420:
4419:
4404:
4403:
4394:
4393:
4352:
4350:
4349:
4344:
4342:
4341:
4332:
4331:
4322:
4321:
4288:
4286:
4285:
4280:
4259:
4257:
4256:
4251:
4249:
4248:
4229:
4227:
4226:
4221:
4209:
4207:
4206:
4201:
4190:
4189:
4178:
4159:
4157:
4156:
4151:
4125:
4124:
4113:
4098:
4097:
4081:
4079:
4078:
4073:
4053:
4051:
4050:
4045:
4040:
4039:
4027:
4026:
4017:
4012:
4011:
3999:
3998:
3976:
3975:
3959:
3954:
3933:
3932:
3916:
3911:
3874:
3872:
3871:
3866:
3840:
3816:
3814:
3813:
3808:
3790:
3788:
3787:
3782:
3771:
3750:
3748:
3747:
3742:
3737:
3733:
3732:
3731:
3721:
3716:
3715:
3714:
3685:
3684:
3674:
3669:
3668:
3667:
3626:
3624:
3623:
3618:
3610:
3609:
3593:
3591:
3590:
3585:
3549:
3547:
3546:
3541:
3529:
3527:
3526:
3521:
3508:Jacobian variety
3505:
3503:
3502:
3497:
3483:
3478:
3477:
3472:
3471:
3442:
3440:
3439:
3434:
3429:
3428:
3416:
3415:
3400:
3399:
3383:
3381:
3380:
3375:
3333:
3331:
3330:
3325:
3323:
3322:
3313:
3312:
3311:
3310:
3289:
3287:
3286:
3281:
3261:-columns of the
3260:
3258:
3257:
3252:
3237:
3235:
3234:
3229:
3217:
3215:
3214:
3209:
3167:
3165:
3164:
3159:
3157:
3156:
3147:
3146:
3145:
3144:
3123:
3121:
3120:
3115:
3073:
3071:
3070:
3065:
3063:
3062:
3047:
3046:
3037:
3036:
3035:
3034:
3005:
3003:
3002:
2997:
2985:
2983:
2982:
2977:
2966:
2965:
2960:
2959:
2945:
2943:
2942:
2937:
2932:
2931:
2907:
2906:
2887:
2885:
2884:
2879:
2867:
2865:
2864:
2859:
2854:
2853:
2829:
2828:
2816:
2815:
2791:
2790:
2771:
2769:
2768:
2763:
2741:
2739:
2738:
2733:
2688:
2686:
2685:
2680:
2666:
2665:
2647:
2646:
2625:Definition. The
2621:
2619:
2618:
2613:
2608:
2607:
2606:
2596:
2582:
2581:
2572:
2564:
2550:
2549:
2528:
2527:
2508:
2506:
2505:
2500:
2485:
2483:
2482:
2477:
2475:
2467:
2455:
2453:
2452:
2447:
2445:
2444:
2438:
2434:
2430:
2429:
2399:
2398:
2386:
2378:
2373:
2372:
2367:
2359:
2335:
2333:
2332:
2327:
2319:
2305:
2304:
2292:
2291:
2279:
2278:
2270:
2254:
2252:
2251:
2246:
2238:
2237:
2212:
2210:
2209:
2204:
2192:
2190:
2189:
2184:
2172:
2170:
2169:
2164:
2162:
2161:
2145:
2143:
2142:
2137:
2125:
2123:
2122:
2117:
2105:
2103:
2102:
2097:
2085:
2083:
2082:
2077:
2065:
2063:
2062:
2057:
2055:
2054:
2046:
2033:
2031:
2030:
2025:
2013:
2011:
2010:
2005:
2003:
2002:
2001:
1991:
1977:
1976:
1967:
1959:
1945:
1944:
1919:
1917:
1916:
1911:
1909:
1908:
1896:
1891:
1886:
1874:
1872:
1871:
1866:
1858:
1857:
1841:
1839:
1838:
1833:
1828:
1814:
1813:
1797:
1795:
1794:
1789:
1787:
1786:
1770:
1768:
1767:
1762:
1750:
1748:
1747:
1742:
1726:
1724:
1723:
1718:
1703:
1701:
1700:
1695:
1665:
1663:
1662:
1657:
1646:of the manifold
1645:
1643:
1642:
1637:
1635:
1634:
1626:
1613:
1611:
1610:
1605:
1603:
1602:
1597:
1556:
1554:
1553:
1548:
1536:
1534:
1533:
1528:
1526:
1525:
1520:
1507:
1505:
1504:
1499:
1494:
1489:
1488:
1469:
1467:
1466:
1461:
1449:
1447:
1446:
1441:
1439:
1438:
1430:
1417:
1415:
1414:
1409:
1404:
1399:
1398:
1379:
1377:
1376:
1371:
1359:
1357:
1356:
1351:
1346:
1341:
1340:
1325:
1324:
1299:
1297:
1296:
1291:
1289:
1288:
1269:
1267:
1266:
1261:
1256:
1255:
1217:
1215:
1214:
1209:
1207:
1206:
1175:
1173:
1172:
1167:
1156:
1155:
1129:
1127:
1126:
1121:
1101:
1099:
1098:
1093:
1091:
1090:
1074:
1072:
1071:
1066:
1054:
1052:
1051:
1046:
1031:
1029:
1028:
1023:
1015:
1001:
1000:
984:
982:
981:
976:
964:
962:
961:
956:
941:
939:
938:
933:
931:
930:
911:
909:
908:
903:
901:
900:
894:
893:
884:
880:
879:
878:
868:
863:
862:
861:
838:
837:
827:
822:
821:
820:
740:
738:
737:
732:
717:
715:
714:
709:
701:
700:
677:
675:
674:
669:
661:
656:
655:
650:
619:
617:
616:
611:
609:
608:
600:
591:
590:
585:
572:
570:
569:
564:
549:
547:
546:
541:
539:
538:
515:
513:
512:
507:
502:
501:
496:
487:
483:
482:
481:
472:
471:
470:
469:
446:
445:
436:
435:
434:
433:
411:
410:
387:
385:
384:
379:
377:
376:
358:
357:
321:canonical bundle
311:
309:
308:
303:
298:
297:
292:
268:
267:
240:
238:
237:
232:
230:
229:
208:
207:
172:
170:
169:
164:
159:
158:
150:
138:
124:
123:
48:Jacobian variety
21:
5352:
5351:
5347:
5346:
5345:
5343:
5342:
5341:
5312:
5311:
5310:
5301:
5273:
5264:Delta invariant
5235:
5204:
5168:
5129:Abel–Jacobi map
5110:
5084:
5080:Torelli theorem
5050:Dessin d'enfant
5030:Belyi's theorem
5016:
5002:Plücker formula
4933:
4924:Hurwitz surface
4893:
4872:
4806:
4780:Analytic theory
4772:Elliptic curves
4766:
4747:Projective line
4734:Rational curves
4728:
4722:
4688:
4675:
4658:Japan. J. Math.
4652:
4621:
4615:
4605:Springer-Verlag
4594:
4591:
4549:
4548:
4525:
4524:
4523:if and only if
4481:
4480:
4456:
4451:
4450:
4411:
4395:
4385:
4365:
4364:
4333:
4323:
4313:
4302:
4301:
4295:
4262:
4261:
4240:
4235:
4234:
4212:
4211:
4167:
4162:
4161:
4102:
4089:
4084:
4083:
4058:
4057:
4031:
4018:
4003:
3990:
3967:
3924:
3877:
3876:
3819:
3818:
3799:
3798:
3756:
3755:
3723:
3706:
3676:
3659:
3653:
3649:
3629:
3628:
3601:
3596:
3595:
3555:
3554:
3532:
3531:
3512:
3511:
3465:
3445:
3444:
3420:
3407:
3391:
3386:
3385:
3336:
3335:
3314:
3302:
3297:
3292:
3291:
3263:
3262:
3240:
3239:
3220:
3219:
3170:
3169:
3148:
3136:
3131:
3126:
3125:
3076:
3075:
3051:
3038:
3026:
3021:
3016:
3015:
2988:
2987:
2953:
2948:
2947:
2923:
2898:
2890:
2889:
2870:
2869:
2845:
2820:
2807:
2782:
2774:
2773:
2748:
2747:
2744:Riemann surface
2724:
2723:
2717:
2657:
2638:
2633:
2632:
2627:Abel–Jacobi map
2597:
2573:
2541:
2519:
2514:
2513:
2491:
2490:
2461:
2460:
2440:
2439:
2421:
2414:
2410:
2401:
2400:
2390:
2357:
2350:
2344:
2343:
2296:
2283:
2263:
2262:
2229:
2218:
2217:
2195:
2194:
2175:
2174:
2153:
2148:
2147:
2128:
2127:
2108:
2107:
2088:
2087:
2068:
2067:
2039:
2038:
2036:universal cover
2016:
2015:
1992:
1968:
1936:
1925:
1924:
1900:
1877:
1876:
1849:
1844:
1843:
1805:
1800:
1799:
1778:
1773:
1772:
1753:
1752:
1733:
1732:
1709:
1708:
1668:
1667:
1648:
1647:
1619:
1618:
1592:
1563:
1562:
1539:
1538:
1515:
1510:
1509:
1477:
1472:
1471:
1452:
1451:
1425:
1420:
1419:
1387:
1382:
1381:
1362:
1361:
1329:
1313:
1302:
1301:
1277:
1272:
1271:
1244:
1224:
1223:
1195:
1178:
1177:
1147:
1136:
1135:
1112:
1111:
1108:
1082:
1077:
1076:
1057:
1056:
1034:
1033:
992:
987:
986:
967:
966:
944:
943:
922:
917:
916:
896:
895:
870:
853:
829:
812:
806:
802:
784:
783:
752:
746:
745:
741:define the map
720:
719:
692:
687:
686:
683:Abel–Jacobi map
645:
625:
624:
595:
580:
575:
574:
555:
554:
530:
525:
524:
491:
473:
461:
456:
437:
425:
420:
419:
415:
402:
397:
396:
368:
349:
344:
343:
287:
259:
254:
253:
218:
199:
194:
193:
145:
115:
110:
109:
84:
44:algebraic curve
36:Abel–Jacobi map
28:
23:
22:
15:
12:
11:
5:
5350:
5348:
5340:
5339:
5334:
5329:
5324:
5314:
5313:
5307:
5306:
5303:
5302:
5300:
5299:
5294:
5289:
5283:
5281:
5279:Vector bundles
5275:
5274:
5272:
5271:
5266:
5261:
5256:
5251:
5245:
5243:
5237:
5236:
5234:
5233:
5228:
5223:
5218:
5212:
5210:
5206:
5205:
5203:
5202:
5197:
5192:
5187:
5182:
5176:
5174:
5170:
5169:
5167:
5166:
5161:
5156:
5151:
5146:
5141:
5136:
5131:
5125:
5123:
5116:
5112:
5111:
5109:
5108:
5103:
5098:
5092:
5090:
5086:
5085:
5083:
5082:
5077:
5072:
5067:
5062:
5057:
5052:
5047:
5042:
5037:
5032:
5026:
5024:
5018:
5017:
5015:
5014:
5009:
5004:
4999:
4994:
4989:
4984:
4979:
4974:
4969:
4964:
4959:
4954:
4949:
4943:
4941:
4935:
4934:
4932:
4931:
4926:
4921:
4916:
4911:
4905:
4903:
4899:
4898:
4895:
4894:
4892:
4891:
4886:
4880:
4878:
4874:
4873:
4871:
4870:
4865:
4860:
4855:
4850:
4845:
4840:
4835:
4830:
4825:
4820:
4814:
4812:
4808:
4807:
4805:
4804:
4799:
4794:
4789:
4783:
4781:
4774:
4768:
4767:
4765:
4764:
4759:
4757:Riemann sphere
4754:
4749:
4744:
4738:
4736:
4730:
4729:
4723:
4721:
4720:
4713:
4706:
4698:
4692:
4691:
4687:978-0387977034
4686:
4673:
4650:
4623:Kotani, Motoko
4619:
4613:
4597:Abel's Theorem
4590:
4587:
4572:
4571:
4559:
4556:
4532:
4512:
4509:
4506:
4503:
4500:
4497:
4494:
4491:
4488:
4463:
4459:
4435:
4434:
4423:
4418:
4414:
4410:
4407:
4402:
4398:
4392:
4388:
4384:
4381:
4378:
4375:
4372:
4354:
4353:
4340:
4336:
4330:
4326:
4320:
4316:
4312:
4309:
4294:
4291:
4278:
4275:
4272:
4269:
4247:
4243:
4219:
4199:
4196:
4193:
4188:
4185:
4182:
4177:
4174:
4171:
4149:
4146:
4143:
4140:
4137:
4134:
4131:
4128:
4123:
4120:
4117:
4112:
4109:
4106:
4101:
4096:
4092:
4071:
4068:
4065:
4043:
4038:
4034:
4030:
4025:
4021:
4016:
4010:
4006:
4002:
3997:
3993:
3989:
3986:
3982:
3979:
3974:
3970:
3966:
3963:
3958:
3953:
3950:
3947:
3943:
3939:
3936:
3931:
3927:
3923:
3920:
3915:
3910:
3907:
3904:
3900:
3896:
3893:
3890:
3887:
3884:
3864:
3861:
3858:
3855:
3852:
3849:
3846:
3843:
3839:
3836:
3833:
3829:
3826:
3806:
3780:
3777:
3774:
3770:
3767:
3764:
3740:
3736:
3730:
3726:
3720:
3713:
3709:
3704:
3700:
3697:
3694:
3691:
3688:
3683:
3679:
3673:
3666:
3662:
3657:
3652:
3648:
3645:
3642:
3639:
3636:
3616:
3613:
3608:
3604:
3583:
3580:
3577:
3574:
3571:
3568:
3565:
3562:
3539:
3519:
3495:
3492:
3489:
3486:
3482:
3476:
3470:
3464:
3461:
3458:
3455:
3452:
3432:
3427:
3423:
3419:
3414:
3410:
3406:
3403:
3398:
3394:
3373:
3370:
3367:
3364:
3361:
3358:
3355:
3352:
3349:
3346:
3343:
3321:
3317:
3309:
3305:
3300:
3279:
3276:
3273:
3270:
3250:
3247:
3227:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3155:
3151:
3143:
3139:
3134:
3113:
3110:
3107:
3104:
3101:
3098:
3095:
3092:
3089:
3086:
3083:
3061:
3058:
3054:
3050:
3045:
3041:
3033:
3029:
3024:
2995:
2975:
2972:
2969:
2964:
2958:
2935:
2930:
2926:
2922:
2919:
2916:
2913:
2910:
2905:
2901:
2897:
2877:
2857:
2852:
2848:
2844:
2841:
2838:
2835:
2832:
2827:
2823:
2819:
2814:
2810:
2806:
2803:
2800:
2797:
2794:
2789:
2785:
2781:
2761:
2758:
2755:
2731:
2716:
2713:
2690:
2689:
2678:
2675:
2672:
2669:
2664:
2660:
2656:
2653:
2650:
2645:
2641:
2623:
2622:
2611:
2605:
2600:
2595:
2591:
2588:
2585:
2580:
2576:
2571:
2567:
2563:
2559:
2556:
2553:
2548:
2544:
2540:
2537:
2534:
2531:
2526:
2522:
2498:
2473:
2470:
2457:
2456:
2443:
2437:
2433:
2428:
2424:
2420:
2417:
2413:
2409:
2406:
2403:
2402:
2397:
2393:
2389:
2384:
2381:
2376:
2371:
2365:
2362:
2356:
2355:
2353:
2337:
2336:
2325:
2322:
2318:
2314:
2311:
2308:
2303:
2299:
2295:
2290:
2286:
2282:
2276:
2273:
2256:
2255:
2244:
2241:
2236:
2232:
2228:
2225:
2202:
2182:
2160:
2156:
2135:
2115:
2095:
2075:
2052:
2049:
2023:
2000:
1995:
1990:
1986:
1983:
1980:
1975:
1971:
1966:
1962:
1958:
1954:
1951:
1948:
1943:
1939:
1935:
1932:
1907:
1903:
1899:
1895:
1890:
1885:
1864:
1861:
1856:
1852:
1831:
1827:
1823:
1820:
1817:
1812:
1808:
1785:
1781:
1760:
1740:
1716:
1693:
1690:
1687:
1684:
1681:
1678:
1675:
1655:
1632:
1629:
1601:
1596:
1591:
1588:
1585:
1582:
1579:
1576:
1573:
1570:
1546:
1524:
1519:
1497:
1493:
1487:
1484:
1480:
1459:
1437:
1434:
1429:
1407:
1403:
1397:
1394:
1390:
1380:is a surface,
1369:
1349:
1345:
1339:
1336:
1332:
1328:
1323:
1320:
1316:
1312:
1309:
1287:
1284:
1280:
1259:
1254:
1251:
1247:
1243:
1240:
1237:
1234:
1231:
1220:abelianisation
1205:
1202:
1198:
1194:
1191:
1188:
1185:
1165:
1162:
1159:
1154:
1150:
1146:
1143:
1119:
1107:
1104:
1089:
1085:
1064:
1044:
1041:
1021:
1018:
1014:
1010:
1007:
1004:
999:
995:
974:
954:
951:
929:
925:
913:
912:
899:
892:
888:
883:
877:
873:
867:
860:
856:
851:
847:
844:
841:
836:
832:
826:
819:
815:
810:
805:
801:
798:
795:
792:
789:
786:
785:
782:
779:
776:
773:
770:
767:
764:
761:
758:
757:
755:
730:
727:
707:
704:
699:
695:
679:
678:
667:
664:
660:
654:
649:
644:
641:
638:
635:
632:
607:
604:
599:
594:
589:
584:
562:
537:
533:
517:
516:
505:
500:
495:
490:
486:
480:
476:
468:
464:
459:
455:
452:
449:
444:
440:
432:
428:
423:
418:
414:
409:
405:
375:
371:
367:
364:
361:
356:
352:
313:
312:
301:
296:
291:
286:
283:
280:
277:
274:
271:
266:
262:
228:
225:
221:
217:
214:
211:
206:
202:
178:homology group
174:
173:
162:
157:
154:
149:
144:
141:
137:
133:
130:
127:
122:
118:
83:
80:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5349:
5338:
5335:
5333:
5330:
5328:
5325:
5323:
5320:
5319:
5317:
5298:
5295:
5293:
5290:
5288:
5285:
5284:
5282:
5280:
5276:
5270:
5267:
5265:
5262:
5260:
5257:
5255:
5252:
5250:
5247:
5246:
5244:
5242:
5241:Singularities
5238:
5232:
5229:
5227:
5224:
5222:
5219:
5217:
5214:
5213:
5211:
5207:
5201:
5198:
5196:
5193:
5191:
5188:
5186:
5183:
5181:
5178:
5177:
5175:
5171:
5165:
5162:
5160:
5157:
5155:
5152:
5150:
5147:
5145:
5142:
5140:
5137:
5135:
5132:
5130:
5127:
5126:
5124:
5120:
5117:
5113:
5107:
5104:
5102:
5099:
5097:
5094:
5093:
5091:
5089:Constructions
5087:
5081:
5078:
5076:
5073:
5071:
5068:
5066:
5063:
5061:
5060:Klein quartic
5058:
5056:
5053:
5051:
5048:
5046:
5043:
5041:
5040:Bolza surface
5038:
5036:
5035:Bring's curve
5033:
5031:
5028:
5027:
5025:
5023:
5019:
5013:
5010:
5008:
5005:
5003:
5000:
4998:
4995:
4993:
4990:
4988:
4985:
4983:
4980:
4978:
4975:
4973:
4970:
4968:
4967:Conic section
4965:
4963:
4960:
4958:
4955:
4953:
4950:
4948:
4947:AF+BG theorem
4945:
4944:
4942:
4940:
4936:
4930:
4927:
4925:
4922:
4920:
4917:
4915:
4912:
4910:
4907:
4906:
4904:
4900:
4890:
4887:
4885:
4882:
4881:
4879:
4875:
4869:
4866:
4864:
4861:
4859:
4856:
4854:
4851:
4849:
4846:
4844:
4841:
4839:
4836:
4834:
4831:
4829:
4826:
4824:
4821:
4819:
4816:
4815:
4813:
4809:
4803:
4800:
4798:
4795:
4793:
4790:
4788:
4785:
4784:
4782:
4778:
4775:
4773:
4769:
4763:
4762:Twisted cubic
4760:
4758:
4755:
4753:
4750:
4748:
4745:
4743:
4740:
4739:
4737:
4735:
4731:
4727:
4719:
4714:
4712:
4707:
4705:
4700:
4699:
4696:
4689:
4683:
4679:
4674:
4671:
4667:
4663:
4659:
4655:
4651:
4648:
4644:
4640:
4636:
4632:
4628:
4624:
4620:
4616:
4610:
4606:
4602:
4598:
4593:
4592:
4588:
4586:
4584:
4580:
4575:
4557:
4554:
4546:
4530:
4507:
4501:
4498:
4492:
4486:
4479:
4478:
4477:
4461:
4457:
4448:
4444:
4440:
4416:
4412:
4405:
4400:
4396:
4390:
4382:
4376:
4370:
4363:
4362:
4361:
4359:
4338:
4334:
4328:
4324:
4318:
4310:
4307:
4300:
4299:
4298:
4292:
4290:
4273:
4267:
4245:
4241:
4231:
4217:
4194:
4183:
4144:
4138:
4129:
4118:
4099:
4094:
4090:
4069:
4066:
4063:
4056:Note that if
4054:
4041:
4036:
4032:
4028:
4023:
4019:
4014:
4008:
4004:
4000:
3995:
3991:
3987:
3984:
3980:
3972:
3968:
3961:
3956:
3951:
3948:
3945:
3941:
3937:
3929:
3925:
3918:
3913:
3908:
3905:
3902:
3898:
3894:
3888:
3882:
3859:
3853:
3844:
3827:
3824:
3804:
3796:
3795:
3775:
3754:If we denote
3752:
3738:
3734:
3728:
3724:
3718:
3711:
3707:
3702:
3698:
3695:
3692:
3689:
3686:
3681:
3677:
3671:
3664:
3660:
3655:
3650:
3646:
3640:
3634:
3614:
3611:
3606:
3602:
3578:
3572:
3566:
3563:
3560:
3551:
3537:
3517:
3509:
3490:
3484:
3480:
3474:
3462:
3456:
3450:
3425:
3421:
3417:
3412:
3408:
3401:
3396:
3392:
3371:
3368:
3365:
3362:
3359:
3356:
3353:
3350:
3347:
3344:
3341:
3319:
3315:
3307:
3303:
3298:
3277:
3274:
3271:
3268:
3248:
3245:
3225:
3205:
3202:
3199:
3196:
3193:
3190:
3187:
3184:
3181:
3178:
3175:
3153:
3149:
3141:
3137:
3132:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3059:
3056:
3052:
3048:
3043:
3039:
3031:
3027:
3022:
3013:
3009:
2993:
2986:, which is a
2970:
2962:
2928:
2924:
2920:
2917:
2914:
2911:
2908:
2903:
2899:
2875:
2850:
2846:
2842:
2839:
2836:
2833:
2830:
2825:
2821:
2817:
2812:
2808:
2804:
2801:
2798:
2795:
2792:
2787:
2783:
2759:
2756:
2753:
2745:
2729:
2720:
2714:
2712:
2708:
2706:
2705:Sunada (2012)
2702:
2698:
2693:
2676:
2670:
2662:
2658:
2651:
2648:
2643:
2639:
2631:
2630:
2629:
2628:
2609:
2589:
2586:
2578:
2574:
2569:
2557:
2554:
2546:
2542:
2538:
2532:
2524:
2520:
2512:
2511:
2510:
2509:is the torus
2496:
2487:
2468:
2435:
2431:
2426:
2422:
2415:
2411:
2404:
2395:
2391:
2379:
2374:
2369:
2360:
2351:
2342:
2341:
2340:
2323:
2312:
2309:
2301:
2297:
2293:
2288:
2284:
2271:
2261:
2260:
2259:
2242:
2239:
2234:
2230:
2223:
2216:
2215:
2214:
2200:
2180:
2158:
2154:
2133:
2113:
2093:
2073:
2047:
2037:
2021:
1984:
1981:
1973:
1969:
1964:
1952:
1949:
1941:
1937:
1930:
1921:
1905:
1901:
1897:
1888:
1862:
1859:
1854:
1850:
1821:
1818:
1810:
1806:
1783:
1779:
1758:
1738:
1730:
1714:
1705:
1691:
1688:
1682:
1676:
1673:
1653:
1627:
1615:
1599:
1586:
1583:
1580:
1577:
1574:
1571:
1568:
1560:
1557:is the first
1544:
1522:
1495:
1491:
1485:
1482:
1478:
1457:
1435:
1432:
1405:
1401:
1395:
1392:
1388:
1367:
1347:
1343:
1337:
1334:
1330:
1321:
1318:
1314:
1310:
1307:
1285:
1282:
1278:
1252:
1249:
1245:
1238:
1235:
1232:
1229:
1221:
1203:
1200:
1196:
1189:
1186:
1183:
1160:
1152:
1148:
1144:
1141:
1133:
1117:
1105:
1103:
1087:
1083:
1042:
1019:
1008:
1005:
997:
993:
972:
952:
949:
927:
923:
881:
875:
871:
865:
858:
854:
849:
845:
842:
839:
834:
830:
824:
817:
813:
808:
803:
799:
793:
787:
777:
771:
765:
762:
759:
753:
744:
743:
742:
728:
705:
702:
697:
693:
684:
665:
658:
652:
642:
636:
630:
623:
622:
621:
605:
602:
592:
587:
553:
535:
522:
503:
498:
488:
484:
478:
474:
466:
462:
457:
453:
450:
447:
442:
438:
430:
426:
421:
416:
412:
407:
395:
394:
393:
391:
373:
369:
365:
362:
359:
354:
350:
341:
337:
333:
328:
326:
322:
318:
299:
294:
284:
278:
275:
272:
264:
260:
252:
251:
250:
248:
244:
226:
223:
219:
215:
212:
209:
204:
200:
191:
187:
183:
179:
160:
155:
152:
142:
131:
128:
120:
116:
108:
107:
106:
104:
101:
97:
93:
89:
81:
79:
77:
73:
69:
65:
61:
57:
53:
49:
45:
41:
37:
33:
19:
5226:Prym variety
5200:Stable curve
5190:Hodge bundle
5180:ELSV formula
5128:
4982:Fermat curve
4939:Plane curves
4902:Higher genus
4877:Applications
4802:Modular form
4677:
4661:
4657:
4630:
4626:
4600:
4596:
4576:
4573:
4446:
4442:
4438:
4436:
4357:
4355:
4296:
4232:
4055:
3792:
3753:
3627:and setting
3552:
3507:
3011:
2721:
2718:
2709:
2694:
2691:
2626:
2624:
2488:
2458:
2338:
2257:
1922:
1771:, with dual
1706:
1616:
1559:Betti number
1109:
914:
682:
680:
518:
389:
339:
335:
329:
324:
316:
314:
246:
242:
189:
185:
181:
175:
102:
95:
91:
85:
35:
29:
5101:Polar curve
4633:: 633–670,
3875:by setting
1707:Now assume
32:mathematics
5316:Categories
5096:Dual curve
4724:Topics in
4589:References
3443:. We call
3012:Dual basis
5209:Morphisms
4957:Bitangent
4447:effective
4387:∑
4315:∑
4242:φ
4136:→
4091:φ
4029:⋯
4001:⋯
3962:φ
3942:∑
3938:−
3919:φ
3899:∑
3883:φ
3851:→
3825:φ
3725:ζ
3703:∫
3678:ζ
3656:∫
3635:φ
3612:∈
3570:→
3561:φ
3402:∈
3316:ζ
3299:∫
3272:×
3150:ζ
3133:∫
3053:δ
3040:ζ
3023:∫
2925:ζ
2900:ζ
2746:of genus
2655:→
2472:¯
2423:∫
2419:↦
2408:↦
2396:∗
2388:→
2383:¯
2364:¯
2289:∗
2281:→
2275:~
2231:∫
2227:→
2051:~
1934:→
1860:∈
1784:∗
1692:π
1689:⊂
1683:φ
1677:
1631:¯
1590:→
1587:π
1578:∘
1569:φ
1479:π
1389:π
1331:π
1327:→
1315:π
1279:π
1246:π
1239:
1222:map. Let
1197:π
1193:→
1190:π
1149:π
1142:π
1063:Λ
1040:Λ
891:Λ
872:ω
850:∫
843:…
831:ω
809:∫
769:→
726:Λ
703:∈
663:Λ
593:≅
561:Λ
532:Ω
523:that the
489:∈
475:ω
463:γ
458:∫
451:…
439:ω
427:γ
422:∫
404:Ω
370:ω
363:…
351:ω
285:≅
220:γ
213:…
201:γ
143:≅
70:that two
4664:: 1–39,
4445:are two
3014:we mean
1537:, where
1450:, where
1132:manifold
392:vectors
249:is that
56:manifold
5269:Tacnode
5254:Crunode
4635:Bibcode
2086:. Thus
1218:be its
552:lattice
319:is the
60:theorem
46:to its
5249:Acnode
5173:Moduli
4684:
4611:
4160:where
3384:where
3218:. Let
3168:, for
3074:, for
2888:, and
2772:. Let
2459:where
1731:. Let
1727:has a
1561:. Let
1300:. Let
1134:. Let
315:where
192:loops
182:cycles
68:Jacobi
34:, the
2146:from
100:genus
50:. In
5259:Cusp
4682:ISBN
4609:ISBN
4441:and
3791:the
3506:the
3334:for
2757:>
2722:Let
2703:and
1110:Let
681:The
98:has
74:are
66:and
64:Abel
4666:doi
4643:doi
4631:209
4599:".
4547:to
4543:is
3797:of
3510:of
2707:).
2213::
2066:of
1674:ker
1496:tor
1406:tor
1348:tor
1236:tor
1230:tor
942:to
887:mod
334:on
323:on
245:is
184:in
86:In
62:of
30:In
5318::
4660:,
4641:,
4629:,
4607:.
4230:.
3010:.
1920:.
327:.
4717:e
4710:t
4703:v
4668::
4662:7
4645::
4637::
4617:.
4558:.
4555:E
4531:D
4511:)
4508:E
4505:(
4502:u
4499:=
4496:)
4493:D
4490:(
4487:u
4462:i
4458:n
4443:E
4439:D
4422:)
4417:i
4413:p
4409:(
4406:u
4401:i
4397:n
4391:i
4383:=
4380:)
4377:D
4374:(
4371:u
4358:C
4339:i
4335:p
4329:i
4325:n
4319:i
4311:=
4308:D
4277:)
4274:M
4271:(
4268:J
4246:0
4218:M
4198:)
4195:M
4192:(
4187:)
4184:0
4181:(
4176:v
4173:i
4170:D
4148:)
4145:M
4142:(
4139:J
4133:)
4130:M
4127:(
4122:)
4119:0
4116:(
4111:v
4108:i
4105:D
4100::
4095:0
4070:s
4067:=
4064:r
4042:.
4037:s
4033:Q
4024:1
4020:Q
4015:/
4009:r
4005:P
3996:1
3992:P
3988:=
3985:D
3981:,
3978:)
3973:j
3969:Q
3965:(
3957:s
3952:1
3949:=
3946:j
3935:)
3930:j
3926:P
3922:(
3914:r
3909:1
3906:=
3903:j
3895:=
3892:)
3889:D
3886:(
3863:)
3860:M
3857:(
3854:J
3848:)
3845:M
3842:(
3838:v
3835:i
3832:D
3828::
3805:M
3779:)
3776:M
3773:(
3769:v
3766:i
3763:D
3739:.
3735:)
3729:g
3719:P
3712:0
3708:P
3699:,
3696:.
3693:.
3690:.
3687:,
3682:1
3672:P
3665:0
3661:P
3651:(
3647:=
3644:)
3641:P
3638:(
3615:M
3607:0
3603:P
3582:)
3579:M
3576:(
3573:J
3567:M
3564::
3538:g
3518:M
3494:)
3491:M
3488:(
3485:L
3481:/
3475:g
3469:C
3463:=
3460:)
3457:M
3454:(
3451:J
3431:}
3426:k
3422:b
3418:,
3413:k
3409:a
3405:{
3397:k
3393:c
3372:g
3369:,
3366:.
3363:.
3360:.
3357:,
3354:1
3351:=
3348:k
3345:,
3342:j
3320:j
3308:k
3304:c
3278:g
3275:2
3269:g
3249:g
3246:2
3226:L
3206:g
3203:,
3200:.
3197:.
3194:.
3191:,
3188:1
3185:=
3182:k
3179:,
3176:j
3154:j
3142:k
3138:b
3112:g
3109:,
3106:.
3103:.
3100:.
3097:,
3094:1
3091:=
3088:k
3085:,
3082:j
3060:k
3057:j
3049:=
3044:j
3032:k
3028:a
2994:g
2974:)
2971:M
2968:(
2963:1
2957:H
2934:}
2929:g
2921:,
2918:.
2915:.
2912:.
2909:,
2904:1
2896:{
2876:M
2856:}
2851:g
2847:b
2843:,
2840:.
2837:.
2834:.
2831:,
2826:1
2822:b
2818:,
2813:g
2809:a
2805:,
2802:.
2799:.
2796:.
2793:,
2788:1
2784:a
2780:{
2760:0
2754:g
2730:M
2677:,
2674:)
2671:M
2668:(
2663:1
2659:J
2652:M
2649::
2644:M
2640:A
2610:.
2604:R
2599:)
2594:Z
2590:,
2587:M
2584:(
2579:1
2575:H
2570:/
2566:)
2562:R
2558:,
2555:M
2552:(
2547:1
2543:H
2539:=
2536:)
2533:M
2530:(
2525:1
2521:J
2497:M
2469:M
2436:)
2432:h
2427:c
2416:h
2412:(
2405:c
2392:E
2380:M
2375::
2370:M
2361:A
2352:{
2324:,
2321:)
2317:R
2313:,
2310:M
2307:(
2302:1
2298:H
2294:=
2285:E
2272:M
2243:.
2240:h
2235:c
2224:h
2201:E
2181:c
2159:0
2155:x
2134:c
2114:M
2094:x
2074:M
2048:M
2022:x
1999:R
1994:)
1989:Z
1985:,
1982:M
1979:(
1974:1
1970:H
1965:/
1961:)
1957:R
1953:,
1950:M
1947:(
1942:1
1938:H
1931:M
1906:1
1902:S
1898:=
1894:Z
1889:/
1884:R
1863:M
1855:0
1851:x
1830:)
1826:R
1822:,
1819:M
1816:(
1811:1
1807:H
1780:E
1759:M
1739:E
1715:M
1686:)
1680:(
1654:M
1628:M
1600:b
1595:Z
1584::
1581:f
1575:g
1572:=
1545:b
1523:b
1518:Z
1492:/
1486:b
1483:a
1458:g
1436:g
1433:2
1428:Z
1402:/
1396:b
1393:a
1368:M
1344:/
1338:b
1335:a
1322:b
1319:a
1311::
1308:g
1286:b
1283:a
1258:)
1253:b
1250:a
1242:(
1233:=
1204:b
1201:a
1187::
1184:f
1164:)
1161:M
1158:(
1153:1
1145:=
1118:M
1088:0
1084:p
1043:.
1020:,
1017:)
1013:Z
1009:,
1006:C
1003:(
998:1
994:H
973:C
953:,
950:p
928:0
924:p
882:)
876:g
866:p
859:0
855:p
846:,
840:,
835:1
825:p
818:0
814:p
804:(
800:=
797:)
794:p
791:(
788:u
781:)
778:C
775:(
772:J
766:C
763::
760:u
754:{
729:,
706:C
698:0
694:p
666:.
659:/
653:g
648:C
643:=
640:)
637:C
634:(
631:J
606:g
603:2
598:R
588:g
583:C
536:j
504:.
499:g
494:C
485:)
479:g
467:j
454:,
448:,
443:1
431:j
417:(
413:=
408:j
390:g
374:g
366:,
360:,
355:1
340:g
336:C
325:C
317:K
300:,
295:g
290:C
282:)
279:K
276:,
273:C
270:(
265:0
261:H
247:g
243:C
227:g
224:2
216:,
210:,
205:1
190:g
186:C
161:.
156:g
153:2
148:Z
140:)
136:Z
132:,
129:C
126:(
121:1
117:H
103:g
96:C
92:C
20:)
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