Information AboutAbelian Variety |
| CATEGORIES ABOUT ABELIAN VARIETY | |
| abelian varieties | |
| algebraic curves | |
| geometry of divisors | |
| algebraic surfaces | |
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HISTORY AND MOTIVATION In the early Nineteenth Century of the theory of Elliptic Function s succeded 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 and Quartic Polynomial s. When those were replaced by polynomials of higher degree, say Quintics , 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 Riemann , Weierstrass , Frobenius , Poincaré and 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 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 Dynamical System s (more specifically in the study of Hamiltonian System s), and in algebraic geometry (especially Picard Varieties and Albanese Varieties ). ANALYTIC THEORY Definition 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 of a ''g''-dimensional complex Vector Space by a Lattice of rank 2''g''. 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 . 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 ''n'' = 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 ''n'' > 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 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 Hermitian Bilinear Form on ''V'' whose Imaginary Part takes 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. The Jacobian of an algebraic curve Every algebraic curve ''C'' of , 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 2''n'' 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 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. See also: Abelian Integral . ALGEBRAIC DEFINITION Two equivalent definitions of abelian variety over a general field are commonly in use:
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. In the early 40s, Weil used the former definition 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 Riemann Hypothesis for 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). 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 . By the Lefschetz Principle , for every Algebraically Closed Field of Characteristic zero (and in particular, for C), the Torsion Group of an abelian variety of dimension ''g'' is Isomorphic to ('''Q'''/'''Z''')2''g''. Hence, its ''n''-torsion part is isomorphic to ('''Z'''/''n'''''Z''')2''g'', i.e. the product of 2''g'' copies of the Cyclic Group of order ''n''. Its free part is Uncountable , since every algebraically closed field of characteristic zero is uncountable. When the base field is an algebraically closed field of characteristic ''p'', the ''n''-torsion is still isomorphic to (Z/''n''Z)2''g'' when ''n'' and ''p'' are Coprime . The group of ''k''-rational points for a Number Field ''k'' is 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''r'' and a finite commutative group for some positive integer ''r'' called the '''rank''' of the abelian variety. Similar results hold for some other classes of fields ''k''. POLARIZATION 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). This association is a duality in the sense that there is a 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 generalizes the Weil Pairing for elliptic curves. Polarizations A polarization of an abelian variety is an ''isogeny'' from an abelian variety to its dual. Polarized abelian varieties have finite Automorphism Group s. A '''principal polarization''' is an ''isomorphism'' between an abelian variety and its dual. Jacobians of curves are naturally equipped with a principal polarization and the curve can be reconstructed from its polarized Jacobian. Not all principally polarized abelian varieties are Jacobians of curves; see the Schottky Problem . Polarizations over the complex numbers Over the complex numbers, a polarized abelian variety can also be defined as an abelian variety ''A'' together with a choice of a Riemann form ''H''. Two Riemann forms ''H''1 and ''H''2 are called Equivalent if there are positive integers ''n'' and ''m'' such that ''nH''1=''mH''2. A choice of an equivalence class of Riemann forms on ''A'' is called a '''polarization''' of ''A''. A morphism of polarized 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 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, sometimes called an abelian variety, over a base scheme ''S'' of relative dimension ''g'' is a Proper , Smooth Group Scheme over ''S'' whose Geometric Fiber s are Connected and of dimension ''g''. The fibers of an abelian scheme are abelian varieties. SEE ALSO
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