33:
274:
532:
955:
43:
684:
1047:
644:
525:
101:
73:
1124:
735:
634:
80:
1114:
813:
518:
87:
960:
58:
881:
808:
222:
69:
558:
503:
778:
674:
1037:
1001:
1154:
700:
613:
498:
429:
of translates of the
Weierstrass zeta function, plus (iii) a function with arbitrary poles but no residues at them.
1149:
1011:
649:
1057:
398:
380:
349:
970:
950:
886:
803:
664:
705:
669:
425:) elliptic function into pieces of 'three kinds' parallels the representation as (i) a constant, plus (ii) a
204:
The differentials of the first kind, when integrated along paths, give rise to integrals that generalise the
861:
330:
94:
654:
493:
410:
284:
180:
1032:
768:
568:
190:
The dimension of the space of differentials of the first kind, by means of this identification, is the
730:
679:
50:
1119:
980:
891:
639:
437:
945:
445:
823:
788:
745:
725:
468:
338:
296:
1086:
659:
453:
426:
866:
846:
818:
156:, a differential of the first kind ω is therefore the same thing as a 1-form that is everywhere
467:
is one where all poles are simple. There is a higher-dimensional analogue available, using the
975:
922:
793:
608:
603:
441:
406:
345:
292:
205:
161:
149:
145:
965:
851:
828:
480:
360:
356:
184:
157:
137:
1091:
896:
838:
740:
563:
542:
391:
334:
168:
141:
133:
763:
17:
1065:
588:
573:
550:
414:
209:
176:
172:
1143:
1106:
876:
856:
783:
578:
1042:
1016:
1006:
996:
798:
618:
449:
384:
191:
917:
755:
422:
418:
388:
123:
32:
436:, though the terminology is not completely consistent. In the algebraic group (
912:
510:
463:
has traditionally been one with residues at all poles being zero. One of the
773:
1096:
1081:
1076:
183:. In either case the definition has its origins in the theory of
379:. The idea behind this has been supported by modern theories of
514:
26:
459:
On the other hand, a meromorphic abelian differential of the
291:
has to be determined by analysis of the possible pole at the
287:
of any given degree > 4. The allowable power
371:
The traditional terminology also included differentials
54:
269:{\displaystyle \int {\frac {x^{k}\,dx}{\sqrt {Q(x)}}}}
299:. When this is done, one finds that the condition is
225:
329:
Quite generally, as this example illustrates, for a
1105:
1056:
1025:
989:
938:
931:
905:
837:
754:
718:
693:
627:
596:
587:
549:
322:5 or 6, at most 2 for degree 7 or 8, and so on (as
432:The same type of decomposition exists in general,
268:
421:, with integer residues. The decomposition of a (
348:, this is the quantity known classically as the
355:. It is also, in general, the dimension of the
132:is a traditional term used in the theories of
526:
8:
59:introducing citations to additional sources
935:
593:
533:
519:
511:
367:Differentials of the second and third kind
452:, and the decomposition is in terms of a
242:
236:
229:
224:
49:Relevant discussion may be found on the
956:Clifford's theorem on special divisors
387:, and through the use of morphisms to
7:
1125:Vector bundles on algebraic curves
1048:Weber's theorem (Algebraic curves)
645:Hasse's theorem on elliptic curves
635:Counting points on elliptic curves
25:
70:"Differential of the first kind"
42:relies largely or entirely on a
31:
736:Hurwitz's automorphisms theorem
359:, which takes the place of the
212:. They include for example the
961:Gonality of an algebraic curve
872:Differential of the first kind
260:
254:
129:differential of the first kind
1:
1115:Birkhoff–Grothendieck theorem
814:Nagata's conjecture on curves
685:Schoof–Elkies–Atkin algorithm
559:Five points determine a conic
440:) theory the three kinds are
383:, both from the side of more
675:Supersingular elliptic curve
381:algebraic differential forms
882:Riemann's existence theorem
809:Hilbert's sixteenth problem
701:Elliptic curve cryptography
614:Fundamental pair of periods
499:Encyclopedia of Mathematics
403:integral of the second kind
152:. Given a complex manifold
1171:
1012:Moduli of algebraic curves
337:, the Hodge number is the
148:), for everywhere-regular
399:Weierstrass zeta function
779:Cayley–Bacharach theorem
706:Elliptic curve primality
318:at most 1 for degree of
18:Holomorphic differential
1038:Riemann–Hurwitz formula
1002:Gromov–Witten invariant
862:Compact Riemann surface
650:Mazur's torsion theorem
331:compact Riemann surface
214:hyperelliptic integrals
208:to all curves over the
655:Modular elliptic curve
494:"Abelian differential"
411:logarithmic derivative
285:square-free polynomial
270:
569:Rational normal curve
295:on the corresponding
271:
1120:Stable vector bundle
981:Weil reciprocity law
971:Riemann–Roch theorem
951:Brill–Noether theory
887:Riemann–Roch theorem
804:Genus–degree formula
665:Mordell–Weil theorem
640:Division polynomials
438:generalized Jacobian
417:, and therefore has
223:
181:Kähler differentials
150:differential 1-forms
55:improve this article
932:Structure of curves
824:Quartic plane curve
746:Hyperelliptic curve
726:De Franchis theorem
670:Nagell–Lutz theorem
314:or in other words,
297:hyperelliptic curve
146:algebraic varieties
1155:Algebraic geometry
939:Divisors on curves
731:Faltings's theorem
680:Schoof's algorithm
660:Modularity theorem
454:composition series
427:linear combination
373:of the second kind
346:algebraic surfaces
344:. For the case of
266:
206:elliptic integrals
1150:Complex manifolds
1137:
1136:
1133:
1132:
1033:Hasse–Witt matrix
976:Weierstrass point
923:Smooth completion
892:TeichmĂĽller space
794:Cubic plane curve
714:
713:
628:Arithmetic theory
609:Elliptic integral
604:Elliptic function
442:abelian varieties
407:elliptic function
377:of the third kind
293:point at infinity
264:
263:
185:abelian integrals
162:algebraic variety
144:(more generally,
138:complex manifolds
136:(more generally,
120:
119:
105:
16:(Redirected from
1162:
966:Jacobian variety
936:
839:Riemann surfaces
829:Real plane curve
789:Cramer's paradox
769:BĂ©zout's theorem
594:
543:algebraic curves
535:
528:
521:
512:
507:
481:Logarithmic form
469:Poincaré residue
434:mutatis mutandis
409:theory; it is a
392:algebraic groups
361:Jacobian variety
357:Albanese variety
275:
273:
272:
267:
265:
250:
249:
241:
240:
230:
142:algebraic curves
134:Riemann surfaces
115:
112:
106:
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63:
35:
27:
21:
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1138:
1129:
1101:
1092:Delta invariant
1070:
1052:
1021:
985:
946:Abel–Jacobi map
927:
901:
897:Torelli theorem
867:Dessin d'enfant
847:Belyi's theorem
833:
819:PlĂĽcker formula
750:
741:Hurwitz surface
710:
689:
623:
597:Analytic theory
589:Elliptic curves
583:
564:Projective line
551:Rational curves
545:
539:
492:
489:
477:
369:
335:algebraic curve
232:
231:
221:
220:
210:complex numbers
116:
110:
107:
64:
62:
48:
36:
23:
22:
15:
12:
11:
5:
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1166:
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1142:
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1128:
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1107:Vector bundles
1103:
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1084:
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751:
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652:
647:
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631:
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622:
621:
616:
611:
606:
600:
598:
591:
585:
584:
582:
581:
576:
574:Riemann sphere
571:
566:
561:
555:
553:
547:
546:
540:
538:
537:
530:
523:
515:
509:
508:
488:
485:
484:
483:
476:
473:
446:algebraic tori
415:theta function
401:was called an
368:
365:
312:
311:
277:
276:
262:
259:
256:
253:
248:
245:
239:
235:
228:
202:
201:
177:coherent sheaf
173:global section
171:it would be a
118:
117:
53:. Please help
39:
37:
30:
24:
14:
13:
10:
9:
6:
4:
3:
2:
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1153:
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1113:
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1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1071:
1064:
1063:
1061:
1059:
1058:Singularities
1055:
1049:
1046:
1044:
1041:
1039:
1036:
1034:
1031:
1030:
1028:
1024:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
994:
992:
988:
982:
979:
977:
974:
972:
969:
967:
964:
962:
959:
957:
954:
952:
949:
947:
944:
943:
941:
937:
934:
930:
924:
921:
919:
916:
914:
911:
910:
908:
906:Constructions
904:
898:
895:
893:
890:
888:
885:
883:
880:
878:
877:Klein quartic
875:
873:
870:
868:
865:
863:
860:
858:
857:Bolza surface
855:
853:
852:Bring's curve
850:
848:
845:
844:
842:
840:
836:
830:
827:
825:
822:
820:
817:
815:
812:
810:
807:
805:
802:
800:
797:
795:
792:
790:
787:
785:
784:Conic section
782:
780:
777:
775:
772:
770:
767:
765:
764:AF+BG theorem
762:
761:
759:
757:
753:
747:
744:
742:
739:
737:
734:
732:
729:
727:
724:
723:
721:
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707:
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617:
615:
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610:
607:
605:
602:
601:
599:
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590:
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580:
579:Twisted cubic
577:
575:
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567:
565:
562:
560:
557:
556:
554:
552:
548:
544:
536:
531:
529:
524:
522:
517:
516:
513:
505:
501:
500:
495:
491:
490:
486:
482:
479:
478:
474:
472:
470:
466:
462:
457:
455:
451:
450:affine spaces
447:
443:
439:
435:
430:
428:
424:
420:
416:
412:
408:
404:
400:
395:
393:
390:
386:
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378:
374:
366:
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362:
358:
354:
351:
347:
343:
340:
336:
332:
327:
325:
321:
317:
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305:
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301:
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286:
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257:
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211:
207:
199:
196:
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188:
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182:
178:
174:
170:
166:
163:
159:
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130:
125:
114:
103:
100:
96:
93:
89:
86:
82:
79:
75:
72: –
71:
67:
66:Find sources:
60:
56:
52:
46:
45:
44:single source
40:This article
38:
34:
29:
28:
19:
1066:
1043:Prym variety
1017:Stable curve
1007:Hodge bundle
997:ELSV formula
871:
799:Fermat curve
756:Plane curves
719:Higher genus
694:Applications
619:Modular form
497:
464:
460:
458:
433:
431:
419:simple poles
402:
396:
385:Hodge theory
376:
372:
370:
352:
350:irregularity
341:
328:
323:
319:
315:
313:
307:
303:
288:
280:
278:
213:
203:
197:
192:Hodge number
189:
169:non-singular
164:
153:
128:
127:
121:
108:
98:
91:
84:
77:
65:
41:
1072:singularity
918:Polar curve
461:second kind
423:meromorphic
389:commutative
158:holomorphic
124:mathematics
111:August 2022
1144:Categories
913:Dual curve
541:Topics in
487:References
465:third kind
310:− 1,
81:newspapers
1026:Morphisms
774:Bitangent
504:EMS Press
227:∫
51:talk page
475:See also
216:of type
167:that is
160:; on an
1097:Tacnode
1082:Crunode
506:, 2001
175:of the
95:scholar
1077:Acnode
990:Moduli
448:, and
279:where
140:) and
97:
90:
83:
76:
68:
413:of a
339:genus
326:= ).
283:is a
179:Ω of
102:JSTOR
88:books
1087:Cusp
397:The
375:and
74:news
405:in
333:or
122:In
57:by
1146::
502:,
496:,
471:.
456:.
444:,
394:.
363:.
306:≤
187:.
126:,
1069:k
1067:A
534:e
527:t
520:v
353:q
342:g
324:g
320:Q
316:k
308:g
304:k
289:k
281:Q
261:)
258:x
255:(
252:Q
247:x
244:d
238:k
234:x
200:.
198:h
165:V
154:M
113:)
109:(
99:·
92:·
85:·
78:·
61:.
47:.
20:)
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