Timeline of abelian varieties

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

Early history

• c. 1000 Al-Karaji writes on congruent numbers^[1]

Seventeenth century

• Fermat studies descent for elliptic curves
• 1643 Fermat poses an elliptic curve Diophantine equation^[2]
• 1670 Fermat's son published his Diophantus with notes

Eighteenth century

• 1718 Giulio Carlo Fagnano dei Toschi, rectification of the lemniscate, addition results for elliptic integrals.^[3]
• 1736 Euler writes on the pendulum equation without the small-angle approximation.^[4]
• 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.^[5]
• 1775 John Landen publishes Landen's transformation,^[6] an isogeny formula.
• 1786 Adrien-Marie Legendre begins to write on elliptic integrals
• 1797 C. F. Gauss discovers double periodicity of the lemniscate function^[7]
• 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.^[8]

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 Du usu Theoriae Integralium Ellipticorum et Integralium Abelianorum in Analysi Diophantea^[9]
• 1836-7 Friedrich Julius Richelot, the Richelot isogeny.^[10]
• 1847 Adolph Göpel gives the equation of the Kummer surface^[11]
• 1851 Johann Georg Rosenhain writes a prize essay on the inversion problem in genus 2.^[12]
• c. 1850 Thomas Weddle - Weddle surface
• 1856 Weierstrass elliptic functions
• 1857 Bernhard Riemann^[13] 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^[14]
• 1866, Alfred Clebsch and Paul Gordan, Theorie der Abel'schen Functionen
• 1869 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,^[15] 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^[16]
• 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.^[17]
• 1913 Torelli's theorem^[18]
• 1916 Gaetano Scorza^[19] applies the term "abelian variety" to complex tori.
• 1921 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^[20]
• c. 1940 Weil defines "abelian variety"
• 1952 André 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 Deuring on elliptic curves to the abelian variety case.^[21]
• c. 1980 Mukai–Fourier transform: the Poincare bundle as Mukai–Fourier kernel induces an equivalence of the derived categories of coherent sheaves for an abelian variety and its dual.^[22]
• 1983 Shiota proves Novikov's conjecture on the Schottky problem
• 1985 J.-M. Fontaine shows that any positive-dimensional abelian variety over the rationals has bad reduction somewhere.^[23]

Twenty-first century

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

Notes