Abelian variety

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v t e

In

. 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 any

.

Abelian varieties defined over

, 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 as

0.

History and motivation

In the early nineteenth century, the theory of

, what would happen?

In the work of

of genus 2.

After Abel and Jacobi, some of the most important contributors to the theory of abelian functions were

Picard

. 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 was André 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, in

).

Analytic theory

Definition

A complex torus of dimension g is a torus of real dimension 2g that carries the structure of a complex manifold. It can always be obtained as the quotient of a g-dimensional complex vector space by a lattice of rank 2g. 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. By invoking the Kodaira embedding theorem and Chow's theorem one may equivalently define a complex abelian variety of dimension g to be a complex torus of dimension g that admits a positive line bundle. Since they are complex tori, abelian varieties carry the structure of a 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.

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 Riemann that the algebraic variety condition imposes extra constraints on a complex torus.

Riemann conditions

The following criterion by Riemann decides whether or not a given

. 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.

The Jacobian of an algebraic curve

Every algebraic curve C of 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 structure, and the image of C generates J as a group. More accurately, J is covered by C^g:^[1] 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 integrals 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 is the fixed field of the symmetric group on g letters acting on the function field of C^g.

Abelian functions

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 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 in

an isogeny.

Important theorems

One important structure theorem of abelian varieties is Matsusaka'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]

Algebraic definition

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

• a connected and complete algebraic group over k
• a connected and projective algebraic group over k.

When the base is the field 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 the

article).

Structure of the group of points

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

For C, and hence by the

of order n.

When the base field is an algebraically closed field of characteristic p, the n-torsion is still isomorphic to (Z/nZ)^2g 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).

The group of

Z^r 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.

Products

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 isogenous to a product of abelian varieties of lower dimension. Any abelian variety is isogenous to a product of simple abelian varieties.

Polarisation and dual abelian variety

Dual abelian variety

To an abelian variety A over a field k, one associates a dual abelian variety A^v (over the same field), which is the solution to the following

L on A×T such that

1. for all t in T, the restriction of L to A×{t} is a degree 0 line bundle,
2. the restriction of L to {0}×T is a trivial line bundle (here 0 is the identity of A).

Then there is a variety A^v and a family of degree 0 line bundles P, the Poincaré bundle, parametrised by A^v such that a family L on T is associated a unique morphism f: T → A^v so that L is isomorphic to the pullback of P along the morphism 1_A×f: A×T → A×A^v. Applying this to the case when T is a point, we see that the points of A^v correspond to line bundles of degree 0 on A, so there is a natural group operation on A^v given by tensor product of line bundles, which makes it into an abelian variety.

This association is a duality in the sense that + it is

for elliptic curves.

Polarisations

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 groups. 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 ${\displaystyle \mathrm {End} (A)\otimes \mathbb {Q} }$ of A.

Polarisations over the complex numbers

Over the complex numbers, a polarised abelian variety can be defined as an abelian variety A together with a choice of a Riemann form H. Two Riemann forms H₁ and H₂ are called equivalent if there are positive integers n and m such that nH₁=mH₂. A choice of an equivalence class of Riemann forms on A is called a polarisation of A; over the complex number this is equivalent to the definition of polarisation given above. A morphism of polarised abelian varieties is a morphism A → B of abelian varieties such that the pullback of the Riemann form on B to A is equivalent to the given form on A.

Abelian scheme

One can also define abelian varieties

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.

For an abelian scheme A / S, the group of n-torsion points forms a

, governed by the deformation properties of the associated p-divisible groups.

Example

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

Non-existence

V. A. Abrashkin^[3] and Jean-Marc Fontaine^[4] independently proved that there are no nonzero abelian varieties over Q with good reduction at all primes. Equivalently, there are no nonzero abelian schemes over Spec Z. The proof involves showing that the coordinates of pⁿ-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]

Semiabelian variety

A semiabelian variety is a commutative group variety which is an extension of an abelian variety by a torus.

See also

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

References

Sources

• Birkenhake, Christina; Lange, H. (1992), Complex Abelian Varieties, Berlin, New York:
  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
• ISBN 3-540-52015-5
• Milne, James, Abelian Varieties, retrieved 6 October 2016. Online course notes.
• 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