View source for Abelian variety

{{TEX|done}}
{{MSC|14}}

An Abelian variety is
an
[[Algebraic group|algebraic group]] that is a complete
[[Algebraic variety|algebraic variety]]. The completeness condition
implies severe restrictions on an Abelian variety. Thus, an Abelian
variety can be imbedded as a closed subvariety in a projective space;
each rational mapping of a non-singular variety into an Abelian
variety is regular; the group law on an Abelian variety is
commutative.

The theory of Abelian varieties over the field of complex numbers $\C$
is, in essence, equivalent to the theory of Abelian functions founded
by C.G.J. Jacobi, N.H. Abel and B. Riemann. If $\C^n$ denotes
$n$-dimension vector space, $\Gamma\subset\C^n$ is a lattice (cf.
[[Discrete subgroup|Discrete subgroup]]) of rank $2n$, then the
quotient group $X=\C^n/\Gamma$ is a
[[Complex torus|complex torus]]. Meromorphic functions on $X$ are the
same thing as meromorphic functions on $\C^n$ that are invariant with
respect to the period lattice $\Gamma$. If the field $K$ of meromorphic
functions on $X$ has transcendence degree $n$, then $X$ can be given
the structure of an algebraic group. This structure is unique by
virtue of the compactness of $X$, and it is such that the field of
rational functions of this structure coincides with $K$. The algebraic
groups formed in this way are Abelian varieties, and each Abelian
variety over the field $\C$ arises in this way. The matrix which
defines a basis of $\Gamma$ can be reduced to the form $(E|Z)$, where $E$ is
the identity matrix and $Z$ is a matrix of order $n\times n$. The complex
torus $X=\C^n/\Gamma$ is an Abelian variety if and only if $Z$ is symmetric and
has positive-definite imaginary part. It should be pointed out that,
as real Lie groups, all varieties $X$ are isomorphic, but this is not
true for their analytic or algebraic structures, which vary strongly
when deforming the lattice $\Gamma$. Inspection of the period matrix $Z$
shows that its variation has an analytic character, which results in
the construction of the moduli variety of all Abelian varieties of
given dimension $n$. The dimension of the moduli variety is $n(n+1)/2$ (cf.
[[Moduli problem|Moduli problem]]).

The theory of Abelian varieties over an arbitrary field $k$ is due to
A. Weil
{{Cite|We}},
{{Cite|We2}}. It has numerous applications both in algebraic
geometry itself and in other fields of mathematics, particularly in
number theory and in the theory of automorphic functions. To each
complete algebraic variety, Abelian varieties (cf.
[[Albanese variety|Albanese variety]];
[[Picard variety|Picard variety]];
[[Intermediate Jacobian|Intermediate Jacobian]]) can be functorially
assigned. These constructions are powerful tools in studying the
geometric structures of algebraic varieties. E.g., they were used to
obtain one of the solutions of the
[[Lüroth problem|Lüroth problem]]. Another application is the proof of
the Riemann hypothesis for algebraic curves over a finite field — the
problem for which the abstract theory of Abelian varieties was
originally developed. It was also one of the sources of
[[L-adic-cohomology|$l$-adic cohomology]]. The simplest example of
such a cohomology is the
[[Tate module|Tate module]] of an Abelian variety. It is the
projective limit, as $n\to\infty$, of the groups $X[l^n]$ of points of order
$l^n$. The determination of the structure of such groups was one of the
principal achievements of the theory of Weil. In fact, if $m$ is
coprime with the characteristic $p$ of the field $k$ and if $k$ is
algebraically closed, then the group $X[m]$ is isomorphic to $(\Z/mZ)^{2\dim X}$. If $m=p$,
the situation is more complicated, which resulted in the appearance of
concepts such as finite group schemes, formal groups and $p$-divisible
groups (cf.
[[Finite group scheme|Finite group scheme]];
[[Formal group|Formal group]];
[[P-divisible group|$p$-divisible group]]). The study of the action of
endomorphisms of Abelian varieties, in particular of the
[[Frobenius endomorphism|Frobenius endomorphism]] on its Tate module,
makes it possible to give a proof of the Riemann hypothesis (for
algebraic curves over finite fields, cf.
[[Riemann hypotheses|Riemann hypotheses]]) and is also the principal
instrument in the theory of complex multiplication of Abelian
varieties. Another circle of problems connected with the Tate module
consists of a study of the action of the Galois group of the closure
of the ground field on this module. There resulted the
[[Tate conjectures|Tate conjectures]] and the theory of Tate–Honda,
which describes Abelian varieties over finite fields in terms of the
Tate module
{{Cite|Mu}}.

The study of Abelian varieties over local fields, including $p$-adic
fields, is proceeding at a fast rate. An analogue of the
above-mentioned representation of Abelian varieties as a quotient
space $\C^n/\Gamma$, usually known as uniformization, over such fields, was
constructed by D. Mumford and M. Raynaud. Unlike the complex case, not
all Abelian varieties, but only those having a reduction to a
multiplicative group modulo $p$, are uniformizable
{{Cite|Ma}}. The theory of Abelian varieties over global
(number and function) fields plays an important role in
[[Diophantine geometry|Diophantine geometry]]. Its principal result is
the Mordell–Weil theorem: The group of rational points of an Abelian
variety, defined over a finite extension of the field of rational
numbers, is finitely generated.

For recent information on the Tate conjectures see
{{Cite|Fa}}. For the theory of Tate–Honda see also
{{Cite|Ta}}. Mumford's theory of uniformization is developed
in
{{Cite|Mu2}},
{{Cite|Mu3}}.

====References====
{|
|-
|valign="top"|{{Ref|Fa}}||valign="top"| G. Faltings, "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" ''Invent. Math.'', '''73''' (1983) pp. 349–366 ((Errratum: Invent. Math. 75 (1984), p. 381))  {{MR|0718935}} {{MR|0732554}}  {{ZBL|0588.14026}}    
|-
|valign="top"|{{Ref|La}}||valign="top"| S. Lang, "Abelian varieties", Springer (1983)  {{MR|0713430}}  {{ZBL|0516.14031}}    
|-
|valign="top"|{{Ref|Ma}}||valign="top"| Yu.I. Manin, "p-Adic automorphic functions" ''J. Soviet Math.'', '''5''' : 3 (1976) pp. 279–333 ''Itogi Nauk. i Tekhn. Sovrem. Problemy'', '''3''' (1974) pp. 5–93     {{ZBL|0375.14007}} 
|-
|valign="top"|{{Ref|Mu}}||valign="top"| D. Mumford, "Abelian varieties", Oxford Univ. Press (1974) {{MR|0282985}}  {{ZBL|0326.14012}}    
|-
|valign="top"|{{Ref|Mu2}}||valign="top"| D. Mumford, "An analytic construction of degenerating curves over complete local rings" ''Compos. Math.'', '''24''' (1972) pp. 129–174  {{MR|0352105}}  {{ZBL|0243.14010}} {{ZBL|0228.14011}}    
|-
|valign="top"|{{Ref|Mu3}}||valign="top"| D. Mumford, "An analytic construction of degenerating Abelian varieties over complete local rings" ''Compos. Math.'', '''24''' (1972) pp. 239–272      
|-
|valign="top"|{{Ref|Se}}||valign="top"| J.-P. Serre, "Groupes algébrique et corps des classes", Hermann (1959)  {{MR|0103191}}     
|-
|valign="top"|{{Ref|Si}}||valign="top"| C.L. Siegel, "Automorphe Funktionen in mehrerer Variablen", Math. Inst. Göttingen (1955)      
|-
|valign="top"|{{Ref|Ta}}||valign="top"| J.T. Tate, "Classes d'isogénie des variétés abéliennes sur un corps fini (d' après T. Honda)", ''Sem. Bourbaki Exp. 352'', ''Lect. notes in math.'', '''179''', Springer (1971)      
|-
|valign="top"|{{Ref|We}}||valign="top"| A. Weil, "Courbes algébriques et variétés abéliennes. Variétés abéliennes et courbes algébriques", Hermann (1971)      
|-
|valign="top"|{{Ref|We2}}||valign="top"| A. Weil, "Courbes algébriques et variétés abéliennes. Sur les courbes algébriques et les varietés qui s'en deduisent", Hermann (1948)      {{MR|0029522}}
|-
|valign="top"|{{Ref|We3}}||valign="top"| A. Weil, "Introduction à l'étude des variétés kahlériennes", Hermann (1958)  {{MR|0111056}}  {{ZBL|0137.41103}}    
|-
|}