Editing 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 curve]]s; 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 field]]s or more general finitely-generated rings or fields).

==Integer points on abelian varieties==
There is some tension here between concepts: ''integer point'' belongs in a sense to [[affine geometry]], while ''abelian variety'' is inherently defined in [[projective geometry]]. The basic results, such as [[Siegel's theorem on integral points]], come from the theory of [[diophantine approximation]].

==Rational points on abelian varieties==
The basic result, the [[Mordell–Weil theorem]] in [[Diophantine geometry]], says that ''A''(''K''), the group of points on ''A'' over ''K'', is a [[finitely-generated abelian group]]. A great deal of information about its possible [[torsion subgroup]]s is known, at least when ''A'' is an elliptic curve. The question of the ''rank'' is thought to be bound up with [[L-function]]s (see below).

The [[torsor]] theory here leads to the [[Selmer group]] and [[Tate–Shafarevich group]], the latter (conjecturally finite) being difficult to study.

==Heights==
{{Main|Height function}}
The theory of [[height function|heights]] plays a prominent role in the arithmetic of abelian varieties. For instance, the canonical [[Néron–Tate height]] is a [[quadratic form]] with remarkable properties that appear in the statement of the [[Birch and Swinnerton-Dyer conjecture]].

==Reduction mod ''p''==
Reduction of an abelian variety ''A'' modulo a [[prime ideal]] of (the integers of) ''K'' &mdash; say, a prime number ''p'' &mdash; to get an abelian variety ''A<sub>p</sub>'' over a [[finite field]], is possible for [[almost all]] ''p''. The 'bad' primes, for which the reduction [[degeneracy (mathematics)|degenerates]] by acquiring [[singular point of an algebraic variety|singular points]], are known to reveal very interesting information. As often happens in number theory, the 'bad' primes play a rather active role in the theory.

Here a refined theory of (in effect) a [[right adjoint]] to reduction mod ''p'' &mdash; the [[Néron model]] &mdash; cannot always be avoided. In the case of an elliptic curve there is an algorithm of [[John Tate (mathematician)|John Tate]] describing it.

==L-functions==
{{Main|Hasse–Weil zeta function}}
For abelian varieties such as  A<sub>''p''</sub>, there is a definition of [[local zeta-function]] available. To get an L-function for A itself, one takes a suitable [[Euler product]] of such local functions; to understand the finite number of factors for the 'bad' primes one has to refer to the [[Tate module]] of A, which is (dual to) the [[étale cohomology]] group H<sup>1</sup>(A), and the [[Galois group]] action on it. In this way one gets a respectable definition of [[Hasse–Weil zeta function|Hasse–Weil L-function]] for A. In general its properties, such as [[functional equation]], are still conjectural – the [[Taniyama–Shimura conjecture]] (which was proven in 2001) was just a special case, so that's hardly surprising.

It is in terms of this L-function that the [[conjecture of Birch and Swinnerton-Dyer]] is posed. It is just one particularly interesting aspect of the general theory about values of L-functions L(''s'') at integer values of ''s'', and there is much empirical evidence supporting it.

==Complex multiplication==
{{Main|Complex multiplication of abelian varieties}}
Since the time of [[Carl Friedrich Gauss]] (who knew of the ''lemniscate function'' case) the special role has been known of those abelian varieties <math>A</math> with extra automorphisms, and more generally endomorphisms. In terms of the ring <math>{\rm End}(A)</math>, there is a definition of [[Complex multiplication of abelian varieties|abelian variety of CM-type]] that singles out the richest class. These are special in their arithmetic. This is seen in their L-functions in rather favourable terms – the [[harmonic analysis]] required is all of the [[Pontryagin duality]] type, rather than needing more general [[automorphic representation]]s. That reflects a good understanding of their Tate modules as [[Galois module]]s. It also makes them ''harder'' to deal with in terms of the conjectural [[algebraic geometry]] ([[Hodge conjecture]] and [[Tate conjecture]]). In those problems the special situation is more demanding than the general.

In the case of elliptic curves, the [[Hilbert's twelfth problem|Kronecker Jugendtraum]] was the programme [[Leopold Kronecker]] proposed, to use elliptic curves of CM-type to do [[class field theory]] explicitly for [[imaginary quadratic field]]s – in the way that [[root of unity|roots of unity]] 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 [[several complex variables]]).

==Manin–Mumford conjecture==
{{See also|André–Oort conjecture}}
The Manin–Mumford conjecture of [[Yuri Manin]] and [[David Mumford]], proved by [[Michel Raynaud]],<ref>{{cite encyclopedia | first=Michel | last=Raynaud | authorlink=Michel Raynaud | chapter=Sous-variétés d'une variété abélienne et points de torsion | language=French | editor1-last=Artin | editor1-first=Michael | editor1-link=Michael Artin | editor2-last=Tate | editor2-first=John | editor2-link=John Tate (mathematician) | title=Arithmetic and geometry. Papers dedicated to I. R. Shafarevich on the occasion of his sixtieth birthday. Vol. I: Arithmetic | series=Progress in Mathematics | volume=35 | publisher=Birkhäuser-Boston | year=1983 | pages= 327–352 | zbl=0581.14031 | mr=0717600}}</ref><ref>{{cite encyclopedia | zbl=1098.14030 | mr=2176757 | last=Roessler | first=Damian | chapter=A note on the Manin-Mumford conjecture | editor1-last=van der Geer | editor1-first=Gerard | editor2-last=Moonen | editor2-first=Ben | editor3-last=Schoof | editor3-first=René | editor3-link=René Schoof | title=Number fields and function fields — two parallel worlds | publisher=Birkhäuser | series=Progress in  Mathematics | volume=239 | pages=311–318 | year=2005 | isbn=0-8176-4397-4 }}</ref> states that a curve ''C'' in its [[Jacobian variety]] ''J'' can only contain a finite number of points that are of finite order (a [[Torsion (algebra)|torsion point]]) in ''J'', unless ''C'' = ''J''. There are other more general versions, such as the [[Bogomolov conjecture]] which generalizes the statement to non-torsion points.

==References==
{{reflist}}

[[Category:Abelian varieties]]
[[Category:Diophantine geometry]]