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Inmathematics, particularly inalgebraic geometry,complex analysis andalgebraic number theory, anabelian variety is a smoothprojective algebraic variety that is also analgebraic group, i.e., has agroup law that can be defined byregular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for research on other topics in algebraic geometry and number theory.

An abelian variety can be defined by equations having coefficients in anyfield; the variety is then said to be definedover that field. Historically the first abelian varieties to be studied were those defined over the field ofcomplex numbers. Such abelian varieties turn out to be exactly thosecomplex tori that can beholomorphically embedded into a complexprojective space.

Abelian varieties defined overalgebraic number fields are a special case, which is important also from the viewpoint of number theory.Localization techniques lead naturally from abelian varieties defined over number fields to ones defined overfinite fields and variouslocal fields. Since a number field is the fraction field of aDedekind domain, for any nonzero prime of your Dedekind domain, there is a map from the Dedekind domain to the quotient of the Dedekind domain by the prime, which is a finite field for all finite primes. This induces a map from the fraction field to any such finite field. Given a curve with equation defined over the number field, we can apply this map to the coefficients to get a curve defined over some finite field, where the choices of finite field correspond to the finite primes of the number field.

Abelian varieties appear naturally asJacobian varieties (the connected components of zero inPicard varieties) andAlbanese varieties of other algebraic varieties. The group law of an abelian variety is necessarilycommutative and the variety isnon-singular. Anelliptic curve is an abelian variety of dimension 1. Abelian varieties haveKodaira dimension 0.

In the early nineteenth century, the theory ofelliptic functions succeeded in giving a basis for the theory ofelliptic integrals, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved thesquare roots ofcubic andquartic polynomials. When those were replaced by polynomials of higher degree, sayquintics, what would happen?

In the work ofNiels Abel andCarl Jacobi, the answer was formulated: this would involve functions oftwo complex variables, having four independentperiods (i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (anabelian surface): what would now be called theJacobian of ahyperelliptic curve of genus 2.

After Abel and Jacobi, some of the most important contributors to the theory of abelian functions wereRiemann,Weierstrass,Frobenius,Poincaré, andPicard. The subject was very popular at the time, already having a large literature.

By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions. Eventually, in the 1920s,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 wasAndré Weil in the 1940s who gave the subject its modern foundations in the language of algebraic geometry.

Today, abelian varieties form an important tool in number theory, indynamical systems (more specifically in the study ofHamiltonian systems), and in algebraic geometry (especiallyPicard varieties andAlbanese varieties).

Acomplex torus of dimensiong is atorus of real dimension 2g that carries the structure of acomplex manifold. It can always be obtained as thequotient of ag-dimensional complexvector space by alattice of rank 2g. A complex abelian variety of dimensiong is a complex torus of dimensiong that is also a projectivealgebraic variety over the field of complex numbers. By invoking theKodaira embedding theorem andChow's theorem, one may equivalently define a complex abelian variety of dimensiong to be a complex torus of dimensiong that admits a positive line bundle. Since they are complex tori, abelian varieties carry the structure of agroup. Amorphism of abelian varieties is a morphism of the underlying algebraic varieties that preserves theidentity element for the group structure. Anisogeny is a finite-to-one morphism.

When a complex torus carries the structure of an algebraic variety, this structure is necessarily unique. In the case${\displaystyle g=1}$, the notion of abelian variety is the same as that ofelliptic curve, and every complex torus gives rise to such a curve; for${\displaystyle g>1}$ it has been known sinceRiemann that the algebraic variety condition imposes extra constraints on a complex torus.

The following criterion by Riemann decides whether or not a givencomplex torus is an abelian variety, i.e., whether or not it can be holomorphically embedded into a projective space. LetX be ag-dimensional torus given as${\displaystyle X=V/L}$ whereV is a complex vector space of dimensiong andL is a lattice inV. ThenX is an abelian variety if and only if there exists apositive definitehermitian form onV whoseimaginary part takesintegral values on${\displaystyle L\times L}$. Such a form onX is usually called a (non-degenerate)Riemann form. Choosing a basis forV andL, one can make this condition more explicit. There are several equivalent formulations of this; all of them are known as the Riemann conditions.

Every algebraic curveC ofgenus${\displaystyle g\ge 1}$ is associated with an abelian varietyJ of dimensiong, by means of an analytic map ofC intoJ. As a torus,J carries a commutativegroup structure, and the image ofC generatesJ as a group. More accurately,J is covered by${\displaystyle C^g}$:^[1] any point inJ comes from ag-tuple of points inC. The study of differential forms onC, which give rise to theabelian integrals with which the theory started, can be derived from the simpler, translation-invariant theory of differentials onJ. The abelian varietyJ is called theJacobian variety ofC, for any non-singular curveC over the complex numbers. From the point of view ofbirational geometry, itsfunction field is the fixed field of thesymmetric group ong letters acting on the function field of${\displaystyle C^g}$.

Anabelian function is ameromorphic function on an abelian variety, which may be regarded therefore as a periodic function ofn complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety. For example, in the nineteenth century there was much interest inhyperelliptic integrals that may be expressed in terms of elliptic integrals. This comes down to asking thatJ is a product of elliptic curves,up to an isogeny.

One important structure theorem of abelian varieties isMatsusaka's theorem. It states that over an algebraically closed field every abelian variety${\displaystyle A}$ is the quotient of the Jacobian of some curve; that is, there is some surjection of abelian varieties${\displaystyle J\to A}$ where${\displaystyle J}$ is a Jacobian. This theorem remains true if the ground field is infinite.^[2]

Two equivalent definitions of abelian variety over a general fieldk are commonly in use:

• aconnected andcompletealgebraic group overk
• aconnected andprojectivealgebraic group overk.

When the base is the field${\displaystyle \mathbb{C}}$ of complex numbers, these notions coincide with the previous definition. Over all bases,elliptic curves are abelian varieties of dimension 1.

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 theRiemann hypothesis forcurves overfinite fields that he had announced in 1940 work, he had to introduce the notion of anabstract variety and to rewrite the foundations of algebraic geometry to work with varieties without projective embeddings (see also the history section in theAlgebraic Geometry article).

By the definitions, an abelian variety is a group variety. Its group of points can be proven to becommutative.

For the field${\displaystyle \mathbb{C}}$, and hence by theLefschetz principle for everyalgebraically closed field ofcharacteristic zero, thetorsion group of an abelian variety of dimensiong isisomorphic to${\displaystyle (\mathbb{Q}/\mathbb{Z}{)}^{2g}}$. Hence, itsn-torsion part is isomorphic to${\displaystyle (\mathbb{Z}/n\mathbb{Z}{)}^{2g}}$, i.e., the product of 2g copies of thecyclic group of ordern.

When the base field is an algebraically closed field of characteristicp, then-torsion is still isomorphic to${\displaystyle (\mathbb{Z}/n\mathbb{Z}{)}^{2g}}$ whenn andp arecoprime. Whenn andp are not coprime, the same result can be recovered provided one interprets it as saying that then-torsion defines a finite flat group scheme of rank 2g. If instead of looking at the full scheme structure on then-torsion, one considers only the geometric points, one obtains a new invariant for varieties in characteristicp (the so-calledp-rank when${\displaystyle n=p}$).

The group ofk-rational points for aglobal fieldk isfinitely generated by theMordell-Weil theorem. Hence, by the structure theorem forfinitely generated abelian groups, it is isomorphic to a product of afree abelian group${\displaystyle {\mathbb{Z}}^r}$ and a finite commutative group for some non-negative integerr called therank of the abelian variety. Similar results hold for some other classes of fieldsk.

The product of an abelian varietyA of dimensionm, and an abelian varietyB of dimensionn, over the same field, is an abelian variety of dimension${\displaystyle m+n}$. An abelian variety issimple if it is notisogenous to a product of abelian varieties of lower dimension. Any abelian variety is isogenous to a product of simple abelian varieties.

To an abelian varietyA over a fieldk, one associates adual abelian variety${\displaystyle A^{\vee }}$ (over the same field), which is the solution to the followingmoduli problem. A family of degree 0 line bundles parametrised by ak-varietyT is defined to be aline bundleL on${\displaystyle A\times T}$ such that

1. for allt inT, the restriction ofL to${\displaystyle A\times \{t\}}$ is a degree 0 line bundle,
2. the restriction ofL to${\displaystyle \{0\}\times T}$ is a trivial line bundle (here 0 is the identity ofA).

Then there is a variety${\displaystyle A^{\vee }}$ and a family of degree 0 line bundlesP, the Poincaré bundle, parametrised by${\displaystyle A^{\vee }}$ such that a familyL onT is associated a unique morphism${\displaystyle f:T\to A^{\vee }}$ so thatL is isomorphic to the pullback ofP along the morphism${\displaystyle 1_A\times f:A\times T\to A\times A^{\vee }}$. Applying this to the case whenT is a point, we see that the points of${\displaystyle A^{\vee }}$ correspond to line bundles of degree 0 onA, so there is a natural group operation on${\displaystyle A^{\vee }}$ given by tensor product of line bundles, which makes it into an abelian variety.

This association is a duality in the sense that it iscontravariant functorial, i.e., it associates to all morphisms${\displaystyle f:A\to B}$ dual morphisms${\displaystyle f^{\vee }:B^{\vee }\to A^{\vee }}$ in a compatible way, and there is anatural isomorphism between the double dual${\displaystyle (A^{\vee }{)}^{\vee }}$ and${\displaystyle A}$ (defined via the Poincaré bundle). Then-torsion of an abelian variety and then-torsion of its dual aredual to each other whenn is coprime to the characteristic of the base. In general — for alln — then-torsiongroup schemes of dual abelian varieties areCartier duals of each other. This generalises theWeil pairing for elliptic curves.

Apolarisation of an abelian variety is anisogeny from an abelian variety to its dual that is symmetric with respect todouble-duality for abelian varieties and for which the pullback of the Poincaré bundle along the associated graph morphism isample (so it is analogous to apositive definite quadratic form). Polarised abelian varieties have finiteautomorphism groups. Aprincipal 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${\displaystyle >1}$. Not all principally polarised abelian varieties are Jacobians of curves; see theSchottky problem. A polarisation induces aRosati involution on theendomorphism ring${\displaystyle \mathrm{E}\mathrm{n}\mathrm{d}(A)\otimes \mathbb{Q}}$ ofA.

Over the complex numbers, apolarised abelian variety can be defined as an abelian varietyA together with a choice of aRiemann formH. Two Riemann forms${\displaystyle H_1}$ and${\displaystyle H_2}$ are calledequivalent if there are positive integersn andm such that${\displaystyle n{H}_1=m{H}_2}$. A choice of an equivalence class of Riemann forms onA is called apolarisation ofA; over the complex number this is equivalent to the definition of polarisation given above. A morphism of polarised abelian varieties is a morphism${\displaystyle A\to B}$ of abelian varieties such that thepullback of the Riemann form onB toA is equivalent to the given form onA.

One can also define abelian varietiesscheme-theoretically andrelative to a base. This allows for a uniform treatment of phenomena such as reduction modp of abelian varieties (seeArithmetic of abelian varieties), and parameter-families of abelian varieties. Anabelian scheme over a base schemeS of relative dimensiong is aproper,smoothgroup scheme overS whosegeometric fibers areconnected and of dimensiong. 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.

For an abelian scheme${\displaystyle A/S}$, the group ofn-torsion points forms afinite flat group scheme. The union of the${\displaystyle p^n}$-torsion points, for alln, forms ap-divisible group.Deformations of abelian schemes are, according to theSerre–Tate theorem, governed by the deformation properties of the associatedp-divisible groups.

Let${\displaystyle A,B\in \mathbb{Z}}$ be such that${\displaystyle x^3+Ax+B}$ has no repeated complex roots. Then the discriminant${\displaystyle \mathrm{\Delta }=-16(4A^3+27B^2)}$ is nonzero. Let${\displaystyle R=\mathbb{Z}[1/\mathrm{\Delta }]}$, so${\displaystyle \mathrm{Spec}R}$ is an open subscheme of${\displaystyle \mathrm{Spec}\mathbb{Z}}$. Then${\displaystyle \mathrm{Proj}R[x,y,z]/(y^{2}z-x^3-Ax{z}^2-B{z}^3)}$ is an abelian scheme over${\displaystyle \mathrm{Spec}R}$. It can be extended to aNéron model over${\displaystyle \mathrm{Spec}\mathbb{Z}}$, which is a smooth group scheme over${\displaystyle \mathrm{Spec}\mathbb{Z}}$, but the Néron model is not proper and hence is not an abelian scheme over${\displaystyle \mathrm{Spec}\mathbb{Z}}$.

Viktor Abrashkin [ru]^[3] andJean-Marc Fontaine^[4] independently proved that there are no nonzero abelian varieties over${\displaystyle \mathbb{Q}}$ with good reduction at all primes. Equivalently, there are no nonzero abelian schemes over${\displaystyle \mathrm{Spec}\mathbb{Z}}$. The proof involves showing that the coordinates of${\displaystyle p^n}$-torsion points generate number fields with very little ramification and hence of small discriminant, while, on the other hand, there are lower bounds on discriminants of number fields.^[5]

Asemiabelian variety is a commutative group variety which is an extension of an abelian variety by atorus.^[6]

• Motives
• Timeline of abelian varieties
• Moduli of abelian varieties
• Equations defining abelian varieties
• Horrocks–Mumford bundle

• Birkenhake, Christina; Lange, H. (1992),Complex Abelian Varieties, Berlin, New York:Springer-Verlag,ISBN 978-0-387-54747-3. A comprehensive treatment of the complex theory, with an overview of the history of the subject.
• Dolgachev, I.V. (2001) [1994],"Abelian scheme",Encyclopedia of Mathematics,EMS Press
• Faltings, Gerd; Chai, Ching-Li (1990),Degeneration of Abelian Varieties,Springer Verlag,ISBN 3-540-52015-5
• Milne, James,Abelian Varieties, retrieved6 October 2016. Online course notes.
• Mumford, David (2008) [1970],Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, vol. 5, Providence, R.I.:American Mathematical Society,ISBN 978-81-85931-86-9,MR 0282985,OCLC 138290
• Venkov, B.B.; Parshin, A.N. (2001) [1994],"Abelian_variety",Encyclopedia of Mathematics,EMS Press
• Bruin, N; Flynn, E.V.,N-COVERS OF HYPERELLIPTIC CURVES(PDF), Oxford: Mathematical Institute, University of Oxford. Description of the Jacobian of the Covering Curves

• Abelian varieties
• Algebraic curves
• Geometry of divisors
• Algebraic surfaces
• Niels Henrik Abel

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