Abelian variety

An abelian variety is a complete algebraic variety with a compatible commutative group law. It is also a projective algebraic group, so its geometry and group structure interact.

Algebraic variety: A geometric space defined by polynomial equations over a field. An abelian variety is a special algebraic variety whose geometry supports a group law.
Group law: A binary operation satisfying the axioms of a group. Its algebraic compatibility lets points on the variety be added and translated geometrically.
Mordell–Weil theorem: The theorem that the rational points of an abelian variety over a number field form a finitely generated abelian group. It makes rational points on abelian varieties accessible through finite generation.
Bernhard Riemann: A German mathematician whose work shaped complex analysis, geometry, and the theory of algebraic curves. His theory of Riemann surfaces supplied foundational ideas behind Jacobians and abelian functions.
Affine algebraic group: An algebraic group whose underlying variety is affine. Affine algebraic groups contrast with abelian varieties, which are complete and projective.
Algebraic group: An algebraic variety equipped with a group operation that is compatible with its algebraic structure. This supplies the group-law framework, with commutativity and completeness distinguishing abelian varieties.
Translation (algebraic geometry): The automorphism of an algebraic group obtained by adding a fixed group element to every point. Translations preserve the structure and reveal the uniformity of abelian varieties.
Faltings's theorem: A theorem proving that a curve of genus greater than one over a number field has only finitely many rational points. Its proof uses abelian varieties, especially Jacobians, to control rational points on curves.
Alfred Clebsch: A German mathematician who contributed to algebraic geometry and the theory of algebraic curves. His work helped develop the nineteenth-century geometric setting in which abelian functions were studied.
Linear algebraic group: An algebraic group that admits a faithful representation as a group of invertible matrices. Linear algebraic groups are affine, unlike the complete geometry of abelian varieties.

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