Timeline of abelian varieties

Updated on Dec 13, 2024

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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

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 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 C. F. 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 Du 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 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 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 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.

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.

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.

Twenty-first century

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

References

Timeline of abelian varieties Wikipedia

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