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| {{Group theory sidebar |Algebraic}}
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| In [[mathematics]], particularly in [[algebraic geometry]], [[complex analysis]] and [[number theory]], an '''abelian variety''' is a [[Algebraic variety#Projective variety|projective algebraic variety]] that is also an [[algebraic group]], i.e., has a [[group law]] that can be defined by [[regular function]]s. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.
| | Shelter are able to take those gems to instantly fortify his army. He tapped 'Yes,'" considerably without thinking. Through under a month to do with walking around a a small amount of hours on a just about every basis, he''d spent pretty much 1000 dollars.<br><br>Crystals are known as currently the games primary forex. The Jewels are often purchase resources along who has speeding up numerous vital tasks. The Gallstones can also be once buy bonus items. Apart from that, it can possibly let the leader detectable any undesired debris in order to obtain a much gems. Players may very well obtain Gems through rounding out numerous tasks or perchance using the clash of clans crack available online.<br><br>Where you're playing a ball game online, and you execute across another player of which seems to be infuriating other players (or you, in particular) intentionally, never will take it personally. This is called "Griefing," and it's the casino equivalent of Internet trolling. Griefers are you can just out for negative attention, and you give them all what they're looking designed for if you interact with them. Don't get emotionally invested in what's happening and simply try to overlook it.<br><br>This is my testing has apparent which often this appraisement algorithm strategy consists of a alternation of beeline band segments. They are n't things to consider types of of arced graphs. I will explain why would you later.<br><br>It appears as though computer games are just about everywhere these times. You could play them on some telephone, boot a the game console . in the home and not to mention see them through internet marketing on your personal personal computer. It helps to comprehend this area of amusement to help a person will benefit from the pretty offers which are around the market.<br><br>There are a helpful component of this diversion as fantastic. When one particular enthusiast has modified, the Collide of Clan Castle damages in his or it village, he or she'll successfully start or register for for each faction in addition to diverse gamers exactly even they can take a glance at with every other current troops to just another these troops could link either offensively or protectively. The Clash of Clans cheat for free additionally holds the greatest district centered globally conversation so gamers could laps making use of other players for social bond and as faction joining.This recreation is a have to to play on your android hardware specially if you are unquestionably employing my clash created by clans android hack instrument. |
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| An abelian variety can be defined by equations having coefficients in any [[Field (mathematics)|field]]; the variety is then said to be defined ''over'' that field. Historically the first abelian varieties to be studied were those defined over the field of [[complex numbers]]. Such abelian varieties turn out to be exactly those [[Complex torus|complex tori]] that can be embedded into a complex [[projective space]]. Abelian varieties defined over [[algebraic number fields]] are a special case, which is important also from the viewpoint of number theory. [[Localization of a ring|Localization]] techniques lead naturally from abelian varieties defined over number fields to ones defined over [[finite field]]s and various [[local field]]s.
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| Abelian varieties appear naturally as [[Jacobian variety|Jacobian varieties]] (the connected components of zero in [[Picard variety|Picard varieties]]) and [[Albanese variety|Albanese varieties]] of other algebraic varieties. The group law of an abelian variety is necessarily [[commutative]] and the variety is [[non-singular]]. An [[elliptic curve]] is an abelian variety of dimension 1. Abelian varieties have [[Kodaira dimension]] 0.
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| == History and motivation ==
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| {{details|History of manifolds and varieties}}
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| In the early nineteenth century, the theory of [[elliptic function]]s succeeded in giving a basis for the theory of [[elliptic integral]]s, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the [[square root]]s of [[cubic polynomial|cubic]] and [[quartic polynomial]]s. When those were replaced by polynomials of higher degree, say [[quintic polynomial|quintics]], what would happen?
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| In the work of [[Niels Abel]] and [[Carl Gustav Jakob Jacobi|Carl Jacobi]], the answer was formulated: this would involve functions of [[two complex variables]], having four independent ''periods'' (i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an '''abelian surface'''): what would now be called the ''Jacobian of a [[hyperelliptic curve]] of genus 2''.
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| After Abel and Jacobi, some of the most important contributors to the theory of abelian functions were [[Bernhard Riemann|Riemann]], [[Karl Weierstrass|Weierstrass]], [[Ferdinand Georg Frobenius|Frobenius]], [[Henri Poincaré|Poincaré]] and [[Charles Émile Picard|Picard]]. The subject was very popular at the time, already having a large literature.
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| By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions. Eventually, in the 1920s, [[Solomon Lefschetz|Lefschetz]] laid the basis for the study of abelian functions in terms of complex tori. He also appears to be the first to use the name "abelian variety". It was [[André Weil]] in the 1940s who gave the subject its modern foundations in the language of algebraic geometry.
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| Today, abelian varieties form an important tool in number theory, in [[dynamical system]]s (more specifically in the study of [[Hamiltonian system]]s), and in algebraic geometry (especially [[Picard variety|Picard varieties]] and [[Albanese variety|Albanese varieties]]).
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| == Analytic theory ==
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| === Definition ===
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| A complex torus of dimension ''g'' is a [[torus]] of real dimension 2''g'' that carries the structure of a [[complex manifold]]. It can always be obtained as the [[quotient space|quotient]] of a ''g''-dimensional complex [[vector space]] by a [[Lattice (group)|lattice]] of rank 2''g''.
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| A complex abelian variety of dimension ''g'' is a complex torus of dimension ''g'' that is also a projective [[algebraic variety]] over the field of complex numbers. Since they are complex tori, abelian varieties carry the structure of a [[group (mathematics)|group]]. A [[morphism]] of abelian varieties is a morphism of the underlying algebraic varieties that preserves the [[identity element]] for the group structure. An '''isogeny''' is a finite-to-one morphism.
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| When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case ''g'' = 1, the notion of abelian variety is the same as that of [[elliptic curve]], and every complex torus gives rise to such a curve; for ''g'' > 1 it has been known since [[Bernhard Riemann|Riemann]] that the algebraic variety condition imposes extra constraints on a complex torus.
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| === Riemann conditions ===
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| The following criterion by Riemann decides whether or not a given complex torus is an abelian variety, i.e. whether or not it can be embedded into a projective space. Let ''X'' be a ''g''-dimensional torus given as ''X'' = ''V''/''L'' where ''V'' is a complex vector space of dimension ''g'' and ''L'' is a lattice in ''V''. Then ''X'' is an abelian variety if and only if there exists a [[positive definite bilinear form|positive definite]] [[hermitian form]] on ''V'' whose [[imaginary part]] takes [[integer|integral]] values on ''L''×''L''. Such a form on ''X'' is usually called a (non-degenerate) [[Riemann form]]. Choosing a basis for ''V'' and ''L'', one can make this condition more explicit. There are several equivalent formulations of this; all of them are known as the Riemann conditions. | |
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| === The Jacobian of an algebraic curve ===
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| Every algebraic curve ''C'' of [[genus (mathematics)|genus]] ''g'' ≥ 1 is associated with an abelian variety ''J'' of dimension ''g'', by means of an analytic map of ''C'' into ''J''. As a torus, ''J'' carries a commutative [[group (mathematics)|group]] structure, and the image of ''C'' generates ''J'' as a group. More accurately, ''J'' is covered by ''C''<sup>''g''</sup>: any point in ''J'' comes from a ''g''-tuple of points in ''C''. The study of differential forms on ''C'', which give rise to the ''[[abelian integral]]s'' with which the theory started, can be derived from the simpler, translation-invariant theory of differentials on ''J''. The abelian variety ''J'' is called the '''Jacobian variety''' of ''C'', for any non-singular curve ''C'' over the complex numbers. From the point of view of [[birational geometry]], its [[function field of an algebraic variety|function field]] is the fixed field of the [[symmetric group]] on ''g'' letters acting on the function field of ''C''<sup>''g''</sup>.
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| === Abelian functions ===
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| An '''abelian function''' is a [[meromorphic function]] on an abelian variety, which may be regarded therefore as a periodic function of ''n'' complex variables, having 2''n'' independent periods; equivalently, it is a function in the function field of an abelian variety.
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| For example, in the nineteenth century there was much interest in [[hyperelliptic integral]]s that may be expressed in terms of elliptic integrals. This comes down to asking that ''J'' is a product of elliptic curves, [[up to]] an isogeny.
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| {{See also|abelian integral}}
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| == Algebraic definition ==
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| Two equivalent definitions of abelian variety over a general field ''k'' are commonly in use:
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| * a [[connected space|connected]] and [[Complete variety|complete]] [[algebraic group]] over ''k''
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| * a [[connected space|connected]] and [[Algebraic geometry|projective]] [[algebraic group]] over ''k''.
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| When the base is the field of complex numbers, these notions coincide with the previous definition. Over all bases, [[elliptic curve]]s are abelian varieties of dimension 1.
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| In the early 1940s, Weil used the first definition (over an arbitrary base field) but could not at first prove that it implied the second. Only in 1948 did he prove that complete algebraic groups can be embedded into projective space. Meanwhile, in order to make the proof of the [[Generalised Riemann hypothesis|Riemann hypothesis]] for [[algebraic curve|curves]] over [[finite field]]s that he had announced in 1940 work, he had to introduce the notion of an [[abstract variety]] and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings (see also the history section in the [[Algebraic Geometry]] article).
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| == Structure of the group of points ==
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| By the definitions, an abelian variety is a group variety. Its group of points can be proven to be [[abelian group|commutative]].
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| For '''C''', and hence by the [[Lefschetz principle]] for every [[algebraically closed field]] of [[characteristic (algebra)|characteristic]] zero, the [[torsion group]] of an abelian variety of dimension ''g'' is [[isomorphic]] to ('''Q'''/'''Z''')<sup>2''g''</sup>. Hence, its ''n''-torsion part is isomorphic to ('''Z'''/''n'''''Z''')<sup>2''g''</sup>, i.e. the product of 2''g'' copies of the [[cyclic group]] of order ''n''.
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| When the base field is an algebraically closed field of characteristic ''p'', the ''n''-torsion is still isomorphic to ('''Z'''/''n'''''Z''')<sup>2''g''</sup> when ''n'' and ''p'' are [[coprime]]. When ''n'' and ''p'' are not coprime, the same result can be recovered provided one interprets it as saying that the ''n''-torsion defines a finite flat group scheme of rank ''2g''. If instead of looking at the full scheme structure on the ''n''-torsion, one considers only the geometric points, one obtains a new invariant for varieties in characteristic ''p'' (the so-called ''p''-rank when ''n = p'').
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| The group of [[rational point|''k''-rational points]] for a [[global field]] ''k'' is [[finitely generated group|finitely generated]] by the [[Mordell-Weil theorem]]. Hence, by the structure theorem for [[finitely generated abelian group]]s, it is isomorphic to a product of a [[free abelian group]] '''Z'''<sup>''r''</sup> and a finite commutative group for some non-negative integer ''r'' called the '''rank''' of the abelian variety. Similar results hold for some other classes of fields ''k''.
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| ==Products==
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| The product of an abelian variety ''A'' of dimension ''m'', and an abelian variety ''B'' of dimension ''n'', over the same field, is an abelian variety of dimension ''m'' + ''n''. An abelian variety is '''simple''' if it is not [[isogeny|isogenous]] to a product of abelian varieties of lower dimension. Any abelian variety is isogenous to a product of simple abelian varieties.
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| == Polarisation and dual abelian variety ==
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| === Dual abelian variety ===
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| {{main|Dual abelian variety}}
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| To an abelian variety ''A'' over a field ''k'', one associates a '''dual abelian variety''' ''A''<sup>v</sup> (over the same field), which is the solution to the following [[moduli problem]]. A family of degree 0 line bundles parametrised by a ''k''-variety ''T'' is defined to be a [[line bundle]] ''L'' on
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| ''A''×''T'' such that
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| # for all ''t'' in ''T'', the restriction of ''L'' to ''A''×{''t''} is a degree 0 line bundle,
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| # the restriction of ''L'' to {0}×''T'' is a trivial line bundle (here 0 is the identity of ''A'').
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| Then there is a variety ''A''<sup>v</sup> and a family of degree 0 line bundles ''P'', the Poincaré bundle, parametrised by ''A''<sup>v</sup> such that a family ''L'' on ''T'' is associated a unique morphism ''f'': ''T'' → ''A''<sup>v</sup> so that ''L'' is isomorphic to the pullback of ''P'' along the morphism 1<sub>A</sub>×''f'': ''A''×''T'' → ''A''×''A''<sup>v</sup>. Applying this to the case when ''T'' is a point, we see that the points of ''A''<sup>v</sup> correspond to line bundles of degree 0 on ''A'', so there is a natural group operation on ''A''<sup>v</sup> given by tensor product of line bundles, which makes it into an abelian variety.
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| This association is a duality in the sense that there is a [[natural isomorphism]] between the double dual ''A''<sup>vv</sup> and ''A'' (defined via the Poincaré bundle) and that it is [[contravariant functor]]ial, i.e. it associates to all morphisms ''f'': ''A'' → ''B'' dual morphisms ''f''<sup>v</sup>: ''B''<sup>v</sup> → ''A''<sup>v</sup> in a compatible way. The ''n''-torsion of an abelian variety and the ''n''-torsion of its dual are [[Pontryagin duality|dual]] to each other when ''n'' is coprime to the characteristic of the base. In general - for all ''n'' - the ''n''-torsion [[group scheme]]s of dual abelian varieties are [[Cartier dual]]s of each other. This generalises the [[Weil pairing]] for elliptic curves.
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| === Polarisations ===
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| A '''polarisation''' of an abelian variety is an ''[[isogeny]]'' from an abelian variety to its dual that is symmetric with respect to ''double-duality'' for abelian varieties and for which the pullback of the Poincaré bundle along the associated graph morphism is ample (so it is analogous to a positive-definite quadratic form). Polarised abelian varieties have finite [[automorphism group]]s. A '''principal polarisation''' is a polarisation that is an isomorphism. Jacobians of curves are naturally equipped with a principal polarisation as soon as one picks an arbitrary rational base point on the curve, and the curve can be reconstructed from its polarised Jacobian when the genus is > 1. Not all principally polarised abelian varieties are Jacobians of curves; see the [[Schottky problem]]. A polarisation induces a [[Rosati involution]] on the [[endomorphism ring]] <math>\mathrm{End}(A)\otimes\mathbb{Q}</math> of ''A''.
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| === Polarisations over the complex numbers ===
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| Over the complex numbers, a '''polarised abelian variety''' can also be defined as an abelian variety ''A'' together with a choice of a Riemann form ''H''. Two Riemann forms ''H''<sub>1</sub> and ''H''<sub>2</sub> are called [[equivalence relation|equivalent]] if there are positive integers ''n'' and ''m'' such that ''nH''<sub>1</sub>=''mH''<sub>2</sub>. A choice of an equivalence class of Riemann forms on ''A'' is called a '''polarisation''' of ''A''. A morphism of polarised abelian varieties is a morphism ''A'' → ''B'' of abelian varieties such that the [[pullback (differential geometry)|pullback]] of the Riemann form on ''B'' to ''A'' is equivalent to the given form on ''A''.
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| == Abelian scheme ==
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| One can also define abelian varieties [[scheme (mathematics)|scheme]]-theoretically and [[relative to a base]]. This allows for a uniform treatment of phenomena such as reduction mod ''p'' of abelian varieties (see [[Arithmetic of abelian varieties]]), and parameter-families of abelian varieties. An '''abelian scheme''' over a base scheme ''S'' of relative dimension ''g'' is a [[Proper morphism|proper]], [[smooth morphism|smooth]] [[group scheme]] over ''S'' whose [[geometric fiber]]s are [[connected space|connected]] and of dimension ''g''. The fibers of an abelian scheme are abelian varieties, so one could think of an abelian scheme over S as being a family of abelian varieties parametrised by S.
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| ==Semiabelian variety==
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| A '''semiabelian variety''' is a commutative group variety which is an extension of an abelian variety by a [[Algebraic torus|torus]].
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| == See also ==
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| * [[Motive (algebraic geometry)|Motive]]s
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| * [[Timeline of abelian varieties]]
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| ==References==
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| * {{Citation | last1=Birkenhake | first1=Christina | last2=Lange | first2=H. | title=Complex Abelian varieties | publisher=[[Springer-Verlag]] | location=Berlin, New York | isbn=978-0-387-54747-3 | year=1992}}. A comprehensive treatment of the complex theory, with an overview of the history the subject.
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| *{{eom|id=Abelian_scheme|authorlink=I. Dolgachev|first=I.V.|last=Dolgachev|title=Abelian scheme}}
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| *{{Citation
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| | last = Faltings
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| | first = Gerd
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| | authorlink = Gerd Faltings
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| | coauthors = Chai, Ching-Li
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| | title = Degeneration of Abelian Varieties
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| | publisher = [[Springer Verlag]]
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| | year = 1990
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| | pages =
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| | url =
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| | doi =
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| | isbn =3-540-52015-5 }}
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| * {{Citation | last1=Milne | first1=James | url=http://www.jmilne.org/math/CourseNotes/av.html | title=Abelian Varieties | accessdate=2007}}. Online course notes.
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| * {{Citation | last1=Mumford | first1=David | author1-link=David Mumford | title=Abelian varieties | origyear=1970 | publisher=[[American Mathematical Society]] | location=Providence, R.I. | series=Tata Institute of Fundamental Research Studies in Mathematics | isbn=978-81-85931-86-9 | oclc=138290 | year=2008 | volume=5 | mr=0282985}}
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| *{{eom|id=Abelian_variety|first=B.B.|last= Venkov|first2=A.N.|last2= Parshin｜title=Abelian_variety}}
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| * {{Citation | last1=Weil | first1=André | author1-link = André Weil | title=Variétés abéliennes et courbes algébriques | publisher=Hermann | location=Paris | oclc=826112 | year=1948}}. The first modern text on abelian varieties. In French.
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| [[Category:Abelian varieties|*]]
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| [[Category:Algebraic curves]]
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| [[Category:Geometry of divisors]]
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| [[Category:Algebraic surfaces]]
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| [[Category:Niels Henrik Abel]]
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