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 [ [http://www.cms.math.ca/Events/summer05/abs/pdf/hm.pdf PDF] ]

eventeenth century

* Fermat studies descent for elliptic curves
* 1643 Fermat poses an elliptic curve Diophantine equation [ [http://www.mathpages.com/home/kmath022/kmath022.htm Miscellaneous Diophantine Equations] at MathPages]
* 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. [ [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Fagnano_Giulio.html Fagnano_Giulio biography ] ]
* 1736 Euler writes on the pendulum equation without the small-angle approximation [E. T. Whittaker, "A Treatise on the Analytical Dynamics of Particles and Rigid Bodies" (fourth edition 1937), p. 72.] .
* 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 [André Weil, "Number Theory: An approach through history" (1984), p. 1.] .
* 1775 John Landen publishes Landen's transformation [ [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Landen.html Landen biography ] ] , an isogeny formula.
* 1786 Adrien-Marie Legendre begins to write on elliptic integrals
* 1797 C. F. Gauss discovers double periodicity of the lemniscate function [ [http://www.geocities.com/RainForest/Vines/2977/gauss/appendix/chrono.html Chronology of the Life of Carl F. Gauss ] ]
* 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 [Semen GrigorʹevichGindikin, "Tales of Physicists and Mathematicians" (1988 translation), p. 143.] .

Nineteenth century

* 1826 N. H. Abel, Abel's theorem
* 1827 inversion of elliptic integrals independently by Abel and C. G. J. Jacobi
* 1829 Jacobi, "Fundamenta nova theoriae functionum ellipticarum", introduces four theta functions of one variable

* 1847 Adolph Göpel gives the equation of the Kummer surface [ [http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Gopel.html Gopel biography ] ]
* Rosenhain
* Thomae
*c. 1850 Thomas Weddle - Weddle surface
* 1856 Weierstrass elliptic functions
* 1857 Bernhard Riemann ["Theorie der Abel'schen Funktionen, J. Reine Angew. Math. 54 (1857), 115-180"] lays the foundations for further work on abelian varieties in dimension > 1.
* Riemann bilinear relations
* Riemann theta function
* 1866, Clebsch and 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 Dedekind, Leopold Kronecker describes his "Jugendtraum" [ [http://www.sunsite.ubc.ca/DigitalMathArchive/Langlands/pdf/jugend-ps.pdf Robert Langlands, "Some Contemporary Problems with Origins in the 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 ["Über die Reduction einer bestimmten Klasse Abel'scher Integrale Ranges auf elliptische Integrale," Acta Math. 4, 392–414 (1884).]
* 1888 Schottky finds a non-trivial condition on the theta constants for curves of genus "g" = 4, launching the Schottky problem
* 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 [ [http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183548682 PDF] , p. 168.] .
*1913 Torelli's theorem [Ruggiero Torelli, "Sulle varietà di Jacobi", Rend. della R. Acc. Nazionale dei Lincei , (5), 22, 1913, 98-103.]
* 1916 Gaetano Scorza [G. Scorza, Intorno alla teoria generale delle matrici di Riemann e ad alcune sue applicazioni,Rend. del Circolo Mat. di Palermo 41 (1916)] 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. L. Siegel, "Einführung in die Theorie der Modulfunktionen n-ten Grades", Math. Ann. 116 (1939), 617–657]
*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
*1960s David Mumford develops a new theory of the equations defining abelian varieties
*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
* 1968 Serre-Tate theorem on good reduction
*Fontaine and bad reduction
*1983 Shiota proves Novikov's conjecture on the Schottly problem

Twenty-first century

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

Notes