538:
The analytic subgroup underlying the generalized
Jacobian can be described as follows. (This does not always determine the algebraic structure as two non-isomorphic commutative algebraic groups may be isomorphic as analytic groups.) Suppose that
93:
659:
77:
28:
531:, the algebraic structure of the generalized Jacobian determines an analytic structure of the generalized Jacobian making it a
581:) induced by the integral of a 1-form over a 1-cycle. The analytic generalized Jacobian is then the quotient group Ω(−
690:
685:
73:
35:
294:
627:
58:
42:
655:
532:
62:
619:
54:
17:
669:
639:
665:
635:
283:
50:
648:
552:
528:
223:
is the universal group with these properties, in the sense that any rational map from
679:
70:
69:. Generalized Jacobians of a curve are extensions of the Jacobian of the curve by a
66:
449:
is the group of invertible elements of the local ring modulo those that are 1 mod
287:
654:, Graduate Texts in Mathematics, vol. 117, New York: Springer-Verlag,
308:
a nonzero effective divisor the generalized
Jacobian is an extension of
631:
623:
610:
Rosenlicht, Maxwell (1954), "Generalized
Jacobian varieties.",
227:
to a group with the properties above factors uniquely through
510:−1, which in characteristic 0 is isomorphic to a product of
422:
of the underlying field. The product runs over the points
53:
associated to a curve with a divisor, generalizing the
133:
is a commutative algebraic group with a rational map
647:
312:by a connected commutative affine algebraic group
492:. It is the product of the multiplicative group
57:of a complete curve. They were introduced by
8:
245:does not depend on the choice of base point
34:For other generalizations of Jacobians, see
194:is the divisor of a rational function
543:is a curve with an effective divisor
261:Structure of the generalized Jacobian
7:
650:Algebraic groups and class fields.
573:)* of the complex vector space Ω(−
551:. There is a natural map from the
501:by a unipotent group of dimension
325:)−1. So we have an exact sequence
61:in 1954, and can be used to study
25:
76:, giving nontrivial examples of
1:
523:Complex generalized Jacobians
269:= 0 the generalized Jacobian
78:Chevalley's structure theorem
29:Clarke's generalized Jacobian
646:Serre, Jean-Pierre (1988) ,
413:by the multiplicative group
18:Generalized Jacobian variety
278:is just the usual Jacobian
124:. The generalized Jacobian
707:
33:
26:
577:) (1-forms with poles on
116:is a fixed base point on
65:of a curve, with abelian
100:an effective divisor on
27:Not to be confused with
404:of a product of groups
479:, the number of times
74:affine algebraic group
36:intermediate Jacobian
519:−1 additive groups.
47:generalized Jacobian
176:is regular outside
161:to the identity of
431:in the support of
257:by a translation.
249:, though changing
108:is the support of
63:ramified coverings
59:Maxwell Rosenlicht
43:algebraic geometry
569:) to the dual Ω(−
533:complex Lie group
321:of dimension deg(
253:changes that map
49:is a commutative
16:(Redirected from
698:
691:Algebraic curves
686:Algebraic groups
672:
653:
642:
435:, and the group
55:Jacobian variety
21:
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624:10.2307/1969715
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529:complex numbers
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284:abelian variety
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263:
244:
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222:
190:) = 0 whenever
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149:
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51:algebraic group
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23:
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15:
12:
11:
5:
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618:(3): 505–530,
605:
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589:
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553:homology group
524:
521:
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496:
483:
476:
472:has dimension
466:
462:
453:
443:
439:
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417:
408:
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385:
381:
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364:is a quotient
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342:
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316:
273:
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231:
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92:is a complete
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24:
14:
13:
10:
9:
6:
4:
3:
2:
703:
692:
689:
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684:
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671:
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663:
661:0-387-96648-X
657:
652:
651:
644:
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637:
633:
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621:
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613:
612:Ann. of Math.
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607:
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599:
596: −
595:
588:
584:
580:
576:
572:
568:
565: −
564:
557:
554:
550:
547:with support
546:
542:
536:
534:
530:
522:
520:
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508:
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56:
52:
48:
44:
37:
30:
19:
649:
615:
611:
597:
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582:
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574:
570:
566:
562:
555:
548:
544:
540:
537:
526:
515:
511:
506:
502:
497:
493:
489:
484:
480:
473:
467:
463:
459:
458:. The group
454:
450:
444:
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436:
432:
427:
423:
418:
414:
409:
405:
403:
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348:
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309:
305:
303:
298:
290:
279:
274:
270:
266:
264:
254:
250:
246:
241:
237:
236:. The group
232:
228:
224:
219:
215:
213:
207:
203:
199:
195:
191:
187:
183:
177:
173:
166:
162:
158:
154:
146:
142:
138:
134:
129:
125:
121:
117:
113:
109:
105:
101:
97:
89:
87:
67:Galois group
46:
40:
150:such that:
94:nonsingular
71:commutative
680:Categories
604:References
488:occurs in
355:The group
202:such that
84:Definition
527:Over the
288:dimension
214:Moreover
88:Suppose
670:0103191
640:0061422
632:1969715
206:≡1 mod
120:not in
96:curve,
668:
658:
638:
630:
293:, the
157:takes
112:, and
628:JSTOR
614:, 2,
295:genus
282:, an
137:from
656:ISBN
368:0 →
329:0 →
304:For
265:For
620:doi
600:).
585:)*/
400:→ 0
377:→ Π
351:→ 0
297:of
286:of
198:on
141:to
41:In
682::
666:MR
664:,
636:MR
634:,
626:,
616:59
535:.
391:→
347:→
338:→
301:.
104:,
80:.
45:a
622::
598:S
594:C
592:(
590:1
587:H
583:m
579:m
575:m
571:m
567:S
563:C
561:(
559:1
556:H
549:S
545:m
541:C
516:i
512:n
507:i
503:n
498:m
494:G
490:m
485:i
481:P
477:i
474:n
468:i
464:P
460:U
455:i
451:P
445:i
441:P
437:U
433:m
428:i
424:P
419:m
415:G
410:i
406:R
397:m
393:L
387:i
383:P
379:U
374:m
370:G
361:m
357:L
349:J
344:m
340:J
335:m
331:L
323:m
318:m
314:L
310:J
306:m
299:C
291:g
280:J
275:m
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267:m
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251:P
247:P
242:m
238:J
233:m
229:J
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216:J
210:.
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200:C
196:g
192:D
188:D
186:(
184:f
180:.
178:S
174:f
170:.
167:m
163:J
159:P
155:f
147:m
143:J
139:C
135:f
130:m
126:J
122:S
118:C
114:P
110:m
106:S
102:C
98:m
90:C
38:.
31:.
20:)
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