Timeline of abelian varieties

This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.

Early history

 * 3rd century AD Diophantus of Alexandria studies rational points on elliptic curves


 * c. 1000 Al-Karaji writes on congruent numbers

Seventeenth century

 * Fermat studies descent for elliptic curves
 * 1643 Fermat poses an elliptic curve Diophantine equation
 * 1670 Fermat's son published his Diophantus with notes

Eighteenth century

 * 1718 Giulio Carlo Fagnano dei Toschi, studies the rectification of the lemniscate, addition results for elliptic integrals.
 * 1736 Leonhard Euler writes on the pendulum equation without the small-angle approximation.
 * 1738 Euler writes on curves of genus 1 considered by Fermat and Frenicle
 * 1750 Euler writes on elliptic integrals
 * 23 December 1751 – 27 January 1752: Birth of the theory of elliptic functions, according to later remarks of Jacobi, as Euler writes on Fagnano's work.
 * 1775 John Landen publishes Landen's transformation, an isogeny formula.
 * 1786 Adrien-Marie Legendre begins to write on elliptic integrals
 * 1797 Carl Friedrich Gauss discovers double periodicity of the lemniscate function
 * 1799 Gauss finds the connection of the length of a lemniscate and a case of the arithmetic-geometric mean, giving a numerical method for a complete elliptic integral.

Nineteenth century

 * 1826 Niels Henrik Abel, Abel-Jacobi map
 * 1827 Inversion of elliptic integrals independently by Abel and Carl Gustav Jacob Jacobi
 * 1829 Jacobi, Fundamenta nova theoriae functionum ellipticarum, introduces four theta functions of one variable
 * 1835 Jacobi points out the use of the group law for diophantine geometry, in De usu Theoriae Integralium Ellipticorum et Integralium Abelianorum in Analysi Diophantea
 * 1836-7 Friedrich Julius Richelot, the Richelot isogeny.
 * 1847 Adolph Göpel gives the equation of the Kummer surface
 * 1851 Johann Georg Rosenhain writes a prize essay on the inversion problem in genus 2.
 * c. 1850 Thomas Weddle - Weddle surface
 * 1856 Weierstrass elliptic functions
 * 1857 Bernhard Riemann lays the foundations for further work on abelian varieties in dimension > 1, introducing the Riemann bilinear relations and Riemann theta function.
 * 1865 Carl Johannes Thomae, Theorie der ultraelliptischen Funktionen und Integrale erster und zweiter Ordnung
 * 1866 Alfred Clebsch and Paul Gordan, Theorie der Abel'schen Functionen
 * 1869 Karl Weierstrass proves an abelian function satisfies an algebraic addition theorem
 * 1879, Charles Auguste Briot, Théorie des fonctions abéliennes
 * 1880 In a letter to Richard Dedekind, Leopold Kronecker describes his Jugendtraum, to use complex multiplication theory to generate abelian extensions of imaginary quadratic fields
 * 1884 Sofia Kovalevskaya writes on the reduction of abelian functions to elliptic functions
 * 1888 Friedrich Schottky finds a non-trivial condition on the theta constants for curves of genus $$g = 4$$, launching the Schottky problem.
 * 1891 Appell–Humbert theorem of Paul Émile Appell and Georges Humbert, classifies the holomorphic line bundles on an abelian surface by cocycle data.
 * 1894 Die Entwicklung der Theorie der algebräischen Functionen in älterer und neuerer Zeit, report by Alexander von Brill and Max Noether
 * 1895 Wilhelm Wirtinger, Untersuchungen über Thetafunktionen, studies Prym varieties
 * 1897 H. F. Baker, Abelian Functions: Abel's Theorem and the Allied Theory of Theta Functions

Twentieth century

 * c.1910 The theory of Poincaré normal functions implies that the Picard variety and Albanese variety are isogenous.
 * 1913 Torelli's theorem
 * 1916 Gaetano Scorza applies the term "abelian variety" to complex tori.
 * 1921 Solomon Lefschetz shows that any complex torus with Riemann matrix satisfying the necessary conditions can be embedded in some complex projective space using theta-functions
 * 1922 Louis Mordell proves Mordell's theorem: the rational points on an elliptic curve over the rational numbers form a finitely-generated abelian group
 * 1929 Arthur B. Coble, Algebraic Geometry and Theta Functions
 * 1939 Siegel modular forms
 * c. 1940 André Weil defines "abelian variety"
 * 1952 Weil defines an intermediate Jacobian
 * Theorem of the cube
 * Selmer group
 * Michael Atiyah classifies holomorphic vector bundles on an elliptic curve
 * 1961 Goro Shimura and Yutaka Taniyama, Complex Multiplication of Abelian Varieties and its Applications to Number Theory
 * Néron model
 * Birch–Swinnerton–Dyer conjecture
 * Moduli space for abelian varieties
 * Duality of abelian varieties
 * c.1967 David Mumford develops a new theory of the equations defining abelian varieties
 * 1968 Serre–Tate theorem on good reduction extends the results of Max Deuring on elliptic curves to the abelian variety case.
 * c. 1980 Mukai–Fourier transform: the Poincaré line bundle as Mukai–Fourier kernel induces an equivalence of the derived categories of coherent sheaves for an abelian variety and its dual.
 * 1983 Takahiro Shiota proves Novikov's conjecture on the Schottky problem
 * 1985 Jean-Marc Fontaine shows that any positive-dimensional abelian variety over the rationals has bad reduction somewhere.

Twenty-first century

 * 2001 Proof of the modularity theorem for elliptic curves is completed.