abelian varieties

abelian varieties
Name	Abelian variety
Type	Projective algebraic variety, Group variety
Field	Field k
Notable	André Weil, Abel, Riemann, Serre, Tate, Grothendieck

abelian varieties Abelian varieties are complete projective algebraic group varieties that generalize elliptic curves and furnish fundamental objects in algebraic geometry, arithmetic geometry, and number theory. Originating in the work of Niels Henrik Abel, Bernhard Riemann, and André Weil, they play central roles in the theories developed by Alexander Grothendieck, John Tate, and Jean-Pierre Serre, and connect to moduli problems studied by David Mumford and Goro Shimura.

Definition and basic properties

An abelian variety over a field k is a connected projective group variety that is smooth and complete, carrying both algebraic group structure and projective geometry; foundational contributors include Niels Henrik Abel, Bernhard Riemann, André Weil, Alexander Grothendieck, Jean-Pierre Serre, John Tate, and David Mumford. Over the complex numbers, an abelian variety of dimension g is analytically isomorphic to a complex torus C^g/Λ subject to Riemann's bilinear relations studied by Riemann and later formalized by Weil and Mumford. Basic properties such as the existence of translation-invariant differential forms, the structure of the Picard variety, and the duality theory were developed by Émile Picard, Poincaré, Weil, and Grothendieck in the context of the formalism of schemes and cohomology by Alexander Grothendieck, Jean-Pierre Serre, and SGA seminars. The group law is commutative by construction, and the structure of torsion points is governed by results of Tate, Serre, and Shimura.

Examples and classification

Standard examples include elliptic curves (dimension one) studied by Niels Henrik Abel and Andrew Wiles in relation to the Taniyama–Shimura–Weil conjecture and the proof of Fermat's Last Theorem, and Jacobian varieties of algebraic curves investigated by Riemann, Abel, Jacobi, Mumford, and Igusa. Products of elliptic curves and simple abelian varieties appear in the classification theory influenced by A. A. Albert's work on endomorphism algebras, and the Poincaré reducibility theorem attributed to Poincaré and refined by Weil and Mumford decomposes abelian varieties up to isogeny into simple factors. The classification over finite fields follows from Honda–Tate theory developed by Taira Honda and John Tate, while complex multiplication varieties involve contributions of Carl Friedrich Gauss's class field ideas and Shimura's theory. Moduli of principally polarized objects include the Siegel modular variety investigated by Carl Ludwig Siegel and later by Igusa, Faltings, and Deligne.

Morphisms, isogenies, and endomorphism rings

Morphisms of abelian varieties are homomorphisms of algebraic groups that are regular maps; the theory of homs and Exts was advanced by Grothendieck and Serre and formalized in SGA and the work of Mumford. Isogenies—surjective morphisms with finite kernel—play central roles in the Torelli theorem context studied by Torelli, in the Honda–Tate classification by Taira Honda and John Tate, and in modern isogeny-based cryptography influenced by Craig Gentry-era research and Jao and De Feo. The endomorphism algebra End^0(A) = End(A) ⊗ Q was analyzed by A. A. Albert and later by Mumford, Tate, and Shimura; its structure links to division algebras, quaternion algebras examined by Hamilton, and complex multiplication studied by Hecke and Shimura. Results such as Faltings's isogeny theorem stem from techniques of Gerd Faltings and relate to conjectures by Mordell and Shafarevich.

Polarizations and dual abelian varieties

A polarization on an abelian variety generalizes an ample line bundle and yields the Rosati involution on End^0(A) studied by Rosati and formalized by Mumford and Poincaré; principally polarized abelian varieties (PPAVs) are central objects in the Schottky problem addressed by Schottky and later by Igusa and Faltings. The dual abelian variety, or Picard variety, arises from the Pic^0 functor and was developed by Picard, Weil, Mumford, and Grothendieck; the Fourier–Mukai transform relating derived categories of A and its dual was introduced by Mukai and built on ideas of Grothendieck and Mumford. Theta functions and theta divisors, with origins in the work of Riemann and Jacobi, produce polarizations and connect to classical works by Igusa and Mumford.

Complex tori, moduli, and Shimura varieties

Complex analytic descriptions identify abelian varieties with complex tori satisfying Riemann bilinear relations studied by Riemann and Poincaré; moduli spaces of principally polarized abelian varieties are Siegel modular varieties investigated by Carl Ludwig Siegel, Igusa, Deligne, and Mumford. Shimura varieties generalize these moduli spaces and were defined by Goro Shimura and Yukiyoshi Taniyama's circle of ideas; contributions by Deligne, Kottwitz, Harris, Taylor, and Langlands connect these moduli to automorphic representations and the Langlands program. Toroidal and minimal compactifications studied by Ash, Mumford, and Faltings provide tools for geometry and arithmetic on these moduli.

Arithmetic and reduction (including Néron models)

Arithmetic aspects include rational points, the Mordell–Weil theorem proven in various settings by Louis Mordell, André Weil, and generalized by Gerd Faltings's finiteness theorems; L-functions and the Birch and Swinnerton-Dyer conjecture for elliptic curves extend to higher-dimensional analogues with input from Hasse, Tate, and Langlands. Reduction theory and integral models were developed by André Néron who introduced Néron models, and further refined by Grothendieck, Raynaud, and Bosch; p-adic Hodge theory contributions by Jean-Marc Fontaine, Kazuya Kato, and Faltings illuminate good and bad reduction, monodromy, and comparison isomorphisms. The Tate conjecture and Hodge conjecture, posed by Tate and W. V. D. Hodge respectively, bear on the arithmetic of cycles on abelian varieties and connect to work by Deligne and Mumford.

Applications and connections (including Jacobians and cryptography)

Jacobians of algebraic curves furnish canonical principally polarized examples and are crucial in the proof of the Torelli theorem by Torelli and in algebraic geometry studies by Mumford and Weil; they underpin algorithms for computing rational points used by Henri Cohen and Schoof's point-counting methods developed by Rafael Schoof and extended by Pila. In cryptography, isogeny-based protocols owe conceptual foundations to number-theoretic work by Craig Gentry successors and to contemporary implementations by De Feo, Jao, and Kohel. Further connections include mirror symmetry and derived categories via Mukai and Kontsevich, arithmetic geometry via Faltings and Serre, and explicit class field theory through complex multiplication researched by Shimura, Taniyama, and Weber.

Category:Algebraic geometry