717:
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282:
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Tropical and non-Archimedean geometry. Bellairs workshop in number theory, tropical and non-Archimedean geometry, Bellairs
Research Institute, Holetown, Barbados, USA, May 6–13, 2011
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Chambert-Loir, Antoine (2013). "Diophantine geometry and analytic spaces". In Amini, Omid; Baker, Matthew; Faber, Xander (eds.).
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Ullmo, Emmanuel (1998), "Positivité et Discrétion des Points Algébriques des
Courbes",
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The original
Bogomolov conjecture was proved by Emmanuel Ullmo and Shou-Wu Zhang using
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771:
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Zhang, S.-W. (1998), "Equidistribution of small points on abelian varieties",
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553:{\displaystyle \{P\in X({\overline {K}}):{\hat {h}}(P)<\epsilon \}}
277:{\displaystyle \{P\in C({\overline {K}}):{\hat {h}}(P)<\epsilon \}}
651:
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The Manin-Mumford conjecture: a brief survey, by Pavlos
Tzermias
672:. Contemporary Mathematics. Vol. 605. Providence, RI:
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In 1998, Zhang proved the following generalization:
445:if it is the translate of an abelian subvariety of
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195:
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109:
453:is not a torsion subvariety, then there is an
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412:associated to an ample symmetric divisor. A
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340:, the Bogomolov conjecture generalises the
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56:. A further generalization to general
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731:. You can help Knowledge (XXG) by
60:was also proved by Zhang in 1998.
14:
788:Conjectures that have been proved
715:
325:{\displaystyle {\hat {h}}(P)=0}
110:{\displaystyle {\overline {K}}}
44:. The conjecture was proven by
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472:{\displaystyle \epsilon >0}
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196:{\displaystyle \epsilon >0}
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1:
674:American Mathematical Society
24:is a conjecture, named after
509:
408:be the NĂ©ron-Tate height on
233:
102:
83:at least two defined over a
809:
710:
434:{\displaystyle X\subset A}
401:{\displaystyle {\hat {h}}}
158:{\displaystyle {\hat {h}}}
793:Algebraic geometry stubs
342:Manin-Mumford conjecture
38:Manin-Mumford conjecture
449:by a torsion point. If
177:. Then there exists an
175:ample symmetric divisor
727:–related article is a
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473:
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278:
197:
159:
125:, fix an embedding of
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639:Annals of Mathematics
589:Annals of Mathematics
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36:that generalizes the
783:Diophantine geometry
676:. pp. 161–179.
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22:Bogomolov conjecture
42:arithmetic geometry
30:arithmetic geometry
725:algebraic geometry
550:
479:such that the set
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443:torsion subvariety
431:
398:
322:
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203:such that the set
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155:
107:
778:Abelian varieties
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683:978-1-4704-1021-6
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284: is finite.
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173:associated to an
167:NĂ©ron-Tate height
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119:algebraic closure
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58:abelian varieties
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34:algebraic curves
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354:Arakelov theory
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332:if and only if
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74:algebraic curve
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54:Arakelov theory
26:Fedor Bogomolov
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652:10.2307/120986
646:(1): 159–165,
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612:10.2307/120987
596:(1): 167–179,
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560: is not
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360:Generalization
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52:in 1998 using
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562:Zariski dense
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375:defined over
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338:torsion point
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50:Shou-Wu Zhang
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19:
733:expanding it
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450:
446:
442:
441:is called a
409:
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351:
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287:
170:
133:
126:
122:
87:
85:number field
80:
69:
67:
21:
15:
165:denote the
117:denote the
18:mathematics
772:Categories
692:1281.14002
620:0934.14013
573:References
414:subvariety
379:, and let
136:, and let
545:ϵ
527:^
510:¯
496:∈
461:ϵ
426:⊂
393:^
356:in 1998.
302:^
269:ϵ
251:^
234:¯
220:∈
185:ϵ
150:^
129:into its
103:¯
64:Statement
690:
680:
618:
371:be an
288:Since
90:, let
72:be an
32:about
20:, the
723:This
598:arXiv
348:Proof
336:is a
78:genus
28:, in
729:stub
678:ISBN
542:<
464:>
367:Let
266:<
188:>
68:Let
48:and
688:Zbl
648:doi
644:147
616:Zbl
608:doi
594:147
564:in
169:on
121:of
76:of
40:in
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642:,
628:^
614:,
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694:.
650::
623:.
610::
600::
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566:X
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533:(
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518::
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507:K
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499:X
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451:X
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429:A
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311:P
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260:P
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226:(
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