Abelian variety

Abelian variety
Original: Jakob.scholbach Vector: Pbroks13 · CC BY-SA 3.0 · source
Name	Abelian variety
Field	Algebraic geometry
Introduced	19th century
Notable	Jacobian, elliptic curve, complex torus

Abelian variety An Abelian variety is a complete, projective algebraic variety equipped with a group law that is commutative and defined by regular maps, arising classically in the work of Riemann, Weierstrass, and Abel. It generalizes elliptic curves and Jacobian varieties of algebraic curves and features centrally in theories developed by Poincaré, Torelli, and Grothendieck. Abelian varieties connect arithmetic problems studied by Mordell, Faltings, Tate and Shimura to analytic constructions originating in Riemann, Siegel, and Hilbert.

Definition and basic examples

An Abelian variety is a projective variety over a field with the structure of an algebraic group; standard examples include elliptic curves such as those described by Weierstrass and modular curves studied by Klein and Hecke. The Jacobian variety associated to a smooth projective curve appears in Torelli-type statements developed by Torelli and Mumford, while complex tori constructed from lattices in C^g lead to principally polarized Jacobians investigated by Riemann and Siegel. Other classical examples arise from products of elliptic curves as in Legendre families and from Prym varieties considered by Donagi and Mumford.

Algebraic and complex structure

Over the complex numbers an Abelian variety admits the structure of a complex torus C^g/Λ, a viewpoint elaborated by Riemann and Siegel and used in the formulation of the Riemann bilinear relations by Poincaré and Lefschetz. In the scheme-theoretic approach developed by Grothendieck and Serre the functorial properties of Abelian varieties relate to Néron models of Néron and Ogg and to deformation theory studied by Kodaira and Spencer. Over finite fields the Honda–Tate theory of Tate and Honda classifies isogeny classes using Frobenius endomorphisms and Weil conjectures proved by Deligne play a central role in counting points, connecting to applications in Langlands program as pursued by Langlands and Kottwitz.

Polarizations and projectivity

A polarization on an Abelian variety is an equivalence class of ample line bundles whose theory was systematized by Lefschetz and Mumford and which yields principally polarized Jacobians central to the Schottky problem studied by Schottky and Igusa. The existence of a polarization ensures projectivity via the embedding theorems of Kodaira and Nakai while theta functions of Riemann, Siegel, and Weil realize projective embeddings and link to moduli constructions by Deligne and Mumford. Analytic theta constants studied by Igusa and Thomae give explicit descriptions used in computational approaches by Gaudry and Satoh.

Endomorphisms and isogenies

The ring of endomorphisms of an Abelian variety is a finitely generated Z-module whose structure was elucidated in work of Poincaré, Albert, and Shimura; this ring can be noncommutative in higher dimensions as in Albert classification and interacts with complex multiplication theory developed by Kronecker, Hilbert, and Shimura. Isogenies, finite surjective morphisms studied by Tate and Faltings, underlie results such as Faltings's proof of the Mordell conjecture and Tate's isogeny theorem for Abelian varieties over finite fields. Explicit isogeny constructions appear in algorithms influenced by Schoof, Atkin, and Elkies and have found cryptographic applications in proposals by Couveignes and Jao–De Feo.

Moduli and classification

Moduli spaces of polarized Abelian varieties, denoted Ag in classical literature, were constructed by Siegel, Mumford, and Deligne and compactified using methods of Satake, Baily–Borel, and Faltings–Chai; these spaces connect to locally symmetric spaces studied by Borel and Harish-Chandra. Classification problems employ Shimura varieties introduced by Shimura and Deligne, which relate to automorphic forms studied by Langlands and Arthur and to arithmetic models examined by Kisin and Scholze. Geometric invariant theory of Mumford and Kirwan provides tools for constructing coarse moduli, while Torelli-type theorems of Torelli and Welters link curves and their Jacobians.

Applications and connections

Abelian varieties appear in number theory via the Birch and Swinnerton-Dyer conjecture for elliptic curves framed by Birch, Swinnerton-Dyer, and later refined by Coates and Wiles; they enter the proof of Fermat-related results by Frey, Ribet, and Wiles. In arithmetic geometry they connect to Galois representations studied by Serre and Fontaine, to L-functions in the Langlands program via Clozel and Harris, and to p-adic Hodge theory developed by Fontaine and Colmez. In complex geometry and mathematical physics theta functions and integrable systems studied by Krichever, Novikov, and Hitchin exploit Abelian varieties, while computational aspects and cryptographic protocols draw on work by Galbraith, Menezes, and Silverman.

Category:Algebraic geometry