Abelian variety

In mathematics, a abelian variety A is a algebraic Groupe whose subjacent algebraic variety is projective and geometrically just (on a body of null characteristic, one can replace the condition " geometrically intègre" by " connexe" , weaker condition). An abelian variety is always not-singular, and the law of group on A is commutative.

If A is an abelian variety on ℂ, then A (ℂ) is naturally an analytical variety complexes, and even a group of Dregs. It is the quotient (within the meaning of the analytical geometry complexes) of a ℂⁿ by a network which admits a plunging in a projective Espace.

The abelian varieties of dimension 1 are the elliptic curved .

The jacobienne of a algebraic Courbe projective not-singular is an abelian variety.

If A is an abelian variety of size g definite on a body k and if n is a natural entirety, then the unit $A\; (\backslash \; bar\; \{K\})$ of the elements of A with coordinates in an algebraic fence $\backslash \; bar\; \{K\}$ of k and who are of a nature dividing n is a finished group, isomorphous with ℤ/n^2gℤ.

Reference

• David Mumford: Abelian varieties , 1974 (republished in 1985)

• Andre Weil: Curved algebraic and abelian varieties , 1948

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