About: Arithmetic of abelian varieties

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In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields).

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rdf:type	Thing yago:WikicatConjectures yago:Abstraction100002137 yago:Assortment108398773 yago:Cognition100023271 yago:Collection107951464 yago:Concept105835747 yago:Content105809192 yago:Group100031264 yago:Hypothesis105888929 yago:Idea105833840 yago:PsychologicalFeature100023100 book yago:Speculation105891783 yago:WikicatAbelianVarieties
rdfs:label	Manin-Mumford-Vermutung (de) Arithmetic of abelian varieties (en) アーベル多様体の数論 (ja)
rdfs:comment	In mathematics, the arithmetic of abelian varieties is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety A over a number field K; or more generally (for global fields or more general finitely-generated rings or fields). (en) 数学では、アーベル多様体の数論(arithmetic of abelian varieties)とは、アーベル多様体、あるいはそれらの族の数論を研究することである。これは、現在は楕円曲線として認識されているピエール・ド・フェルマー(Pierre de Fermat)の研究に遡り、結果と予想の両面で非常に実績に富んだ分野となっている。これらの楕円曲線上で得られたうちの大半は、数体 K（あるいは、より一般的には、大域体や有限生成の環や体）の上のアーベル多様体 A に対して成立している。 (ja) In der Mathematik ist die Manin-Mumford-Vermutung ein von Michel Raynaud bewiesener Lehrsatz der arithmetischen Geometrie, der unabhängig von Juri Manin und David Mumford vermutet worden war. Er besagt, dass für eine über definierte abelsche Varietät und eine Kurve vom Geschlecht mindestens , es nur endlich viele Torsionspunkte in gibt. Insbesondere erhält man für die Jacobi-Varietät einer algebraischen Kurve , dass im Fall die Kurve nur endlich viele Torsionspunkte (für die Gruppenstruktur der Jacobi-Varietät als abelsche Varietät) enthalten kann. (Das war die ursprüngliche Formulierung der Vermutung.) (de)
rdfs:seeAlso	André–Oort conjecture
dct:subject	Diophantine geometry Abelian varieties
Wikipage page ID	384577 (xsd:integer)
Wikipage revision ID	1065426757 (xsd:integer)
Link from a Wikipage to another Wikipage	Carl Friedrich Gauss Prime ideal Projective geometry Root of unity Mordell–Weil theorem Birch and Swinnerton-Dyer conjecture David Mumford Hodge conjecture John Tate (mathematician) Degeneracy (mathematics) Jacobian variety L-function Néron model Mathematics Elliptic curve Arithmetic geometry Leopold Kronecker Local zeta-function Bogomolov conjecture Complex multiplication of abelian varieties Yuri Manin Étale cohomology Functional equation Harmonic analysis Automorphic representation Néron–Tate height Torsor Galois group Galois module Hasse–Weil zeta function Tate conjecture Algebraic geometry Almost all Euler product Finite field Diophantine geometry Number field Number theory Diophantine approximation Diophantine geometry Global field Hilbert's twelfth problem Quadratic form Height function Tate–Shafarevich group Finitely-generated abelian group Abelian variety Abelian varieties Affine geometry Torsion (algebra) Pierre de Fermat Pontryagin duality Class field theory Michel Raynaud Complex coordinate space Selmer group Torsion subgroup Siegel's theorem on integral points Singular point of an algebraic variety Tate module Taniyama–Shimura conjecture Right adjoint Conjecture of Birch and Swinnerton-Dyer Imaginary quadratic field
sameAs	Arithmetic of abelian varieties Arithmetic of abelian varieties Arithmetic of abelian varieties Arithmetic of abelian varieties Arithmetic of abelian varieties Arithmetic of abelian varieties
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