22:
508:
469:
39:
343:, are known to reveal very interesting information. As often happens in number theory, the 'bad' primes play a rather active role in the theory.
569:
allow one to do this for the field of rational numbers. This generalises, but in some sense with loss of explicit information (as is typical of
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that singles out the richest class. These are special in their arithmetic. This is seen in their L-functions in rather favourable terms – the
86:
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Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic
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of such local functions; to understand the finite number of factors for the 'bad' primes one has to refer to the
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is posed. It is just one particularly interesting aspect of the general theory about values of L-functions L(
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Roessler, Damian (2005). "A note on the Manin-Mumford conjecture". In van der Geer, Gerard; Moonen, Ben;
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359:
190:
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336:
202:
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358:— cannot always be avoided. In the case of an elliptic curve there is an algorithm of
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both in terms of results and conjectures. Most of these can be posed for an abelian variety
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plays a prominent role in the arithmetic of abelian varieties. For instance, the canonical
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662:. Progress in Mathematics (in French). Vol. 35. Birkhäuser-Boston. pp. 327–352.
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with extra automorphisms, and more generally endomorphisms. In terms of the ring
414:(which was proven in 2001) was just a special case, so that's hardly surprising.
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546:). In those problems the special situation is more demanding than the general.
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697:. Progress in Mathematics. Vol. 239. Birkhäuser. pp. 311–318.
650:(1983). "Sous-variétés d'une variété abélienne et points de torsion". In
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can only contain a finite number of points that are of finite order (a
588:
329:
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available. To get an L-function for A itself, one takes a suitable
134:, or a family of abelian varieties. It goes back to the studies of
451:
case) the special role has been known of those abelian varieties
402:
action on it. In this way one gets a respectable definition of
286:
with remarkable properties that appear in the statement of the
249:
526:. That reflects a good understanding of their Tate modules as
260:, the latter (conjecturally finite) being difficult to study.
695:
Number fields and function fields — two parallel worlds
630:
which generalizes the statement to non-torsion points.
196:
429:, and there is much empirical evidence supporting it.
164:
626:. There are other more general versions, such as the
477:
457:
161:
or more general finitely-generated rings or fields).
46:. Unsourced material may be challenged and removed.
557:proposed, to use elliptic curves of CM-type to do
502:
463:
229:. A great deal of information about its possible
732:
169:There is some tension here between concepts:
335:. The 'bad' primes, for which the reduction
142:; and has become a very substantial area of
576:
439:Complex multiplication of abelian varieties
417:It is in terms of this L-function that the
406:for A. In general its properties, such as
237:is an elliptic curve. The question of the
534:to deal with in terms of the conjectural
432:
106:Learn how and when to remove this message
688:
646:
522:type, rather than needing more general
419:conjecture of Birch and Swinnerton-Dyer
346:Here a refined theory of (in effect) a
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549:In the case of elliptic curves, the
288:Birch and Swinnerton-Dyer conjecture
197:Rational points on abelian varieties
44:adding citations to reliable sources
15:
187:Siegel's theorem on integral points
165:Integer points on abelian varieties
13:
486:
483:
480:
317:— to get an abelian variety
293:
14:
757:
55:"Arithmetic of abelian varieties"
587:The Manin–Mumford conjecture of
376:For abelian varieties such as A
301:Reduction of an abelian variety
227:finitely-generated abelian group
20:
241:is thought to be bound up with
124:arithmetic of abelian varieties
31:needs additional citations for
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503:{\displaystyle {\rm {End}}(A)}
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410:, are still conjectural – the
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138:on what are now recognized as
1:
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394:of A, which is (dual to) the
185:. The basic results, such as
313:— say, a prime number
7:
524:automorphic representations
510:, there is a definition of
412:Taniyama–Shimura conjecture
382:, there is a definition of
10:
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563:imaginary quadratic fields
512:abelian variety of CM-type
436:
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217:), the group of points on
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571:several complex variables
252:theory here leads to the
191:diophantine approximation
181:is inherently defined in
157:; or more generally (for
577:Manin–Mumford conjecture
372:Hasse–Weil zeta function
233:is known, at least when
518:required is all of the
425:) at integer values of
599:, states that a curve
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465:
433:Complex multiplication
258:Tate–Shafarevich group
201:The basic result, the
173:belongs in a sense to
583:André–Oort conjecture
551:Kronecker Jugendtraum
530:. It also makes them
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404:Hasse–Weil L-function
309:of (the integers of)
746:Diophantine geometry
628:Bogomolov conjecture
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455:
445:Carl Friedrich Gauss
398:group H(A), and the
207:Diophantine geometry
203:Mordell–Weil theorem
126:is the study of the
40:improve this article
449:lemniscate function
408:functional equation
384:local zeta-function
183:projective geometry
144:arithmetic geometry
565:– in the way that
559:class field theory
553:was the programme
536:algebraic geometry
520:Pontryagin duality
500:
461:
443:Since the time of
328:, is possible for
741:Abelian varieties
555:Leopold Kronecker
516:harmonic analysis
464:{\displaystyle A}
447:(who knew of the
350:to reduction mod
280:Néron–Tate height
231:torsion subgroups
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605:Jacobian variety
540:Hodge conjecture
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396:Ă©tale cohomology
136:Pierre de Fermat
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561:explicitly for
544:Tate conjecture
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362:describing it.
341:singular points
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179:abelian variety
175:affine geometry
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140:elliptic curves
132:abelian variety
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567:roots of unity
528:Galois modules
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437:Main article:
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370:Main article:
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294:Reduction mod
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284:quadratic form
274:The theory of
268:Main article:
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388:Euler product
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348:right adjoint
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339:by acquiring
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245:(see below).
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171:integer point
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128:number theory
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57: –
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51:Find sources:
45:
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29:This article
27:
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18:
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694:
691:Schoof, René
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595:, proved by
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400:Galois group
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354:— the
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326:finite field
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254:Selmer group
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152:number field
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96:October 2013
93:
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69:
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38:Please help
33:verification
30:
392:Tate module
366:L-functions
356:NĂ©ron model
337:degenerates
307:prime ideal
243:L-functions
120:mathematics
735:Categories
721:1098.14030
676:0581.14031
656:Tate, John
634:References
589:Yuri Manin
581:See also:
330:almost all
66:newspapers
618:, unless
360:John Tate
305:modulo a
693:(eds.).
658:(eds.).
177:, while
713:2176757
668:0717600
603:in its
324:over a
276:heights
264:Heights
225:, is a
150:over a
80:scholar
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532:harder
250:torsor
130:of an
122:, the
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614:) in
282:is a
221:over
87:JSTOR
73:books
699:ISBN
591:and
542:and
256:and
248:The
239:rank
59:news
717:Zbl
672:Zbl
573:).
205:in
118:In
42:by
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709:MR
707:.
670:.
664:MR
654:;
622:=
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193:.
723:.
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624:J
620:C
616:J
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538:(
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492:(
487:d
484:n
481:E
459:A
427:s
423:s
379:p
352:p
333:p
321:p
319:A
315:p
311:K
303:A
296:p
235:A
223:K
219:A
215:K
213:(
211:A
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