Timeline of abelian varieties

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

Contents 1 Early history 2 Seventeenth century 3 Eighteenth century 4 Nineteenth century 5 Twentieth century 6 Twenty-first century 7 Notes

[edit] Early history

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

[edit] 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

[edit] 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].

[edit] 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^[9]
• 1851 Johann Georg Rosenhain writes a prize essay on the inversion problem in genus 2.^[10]
• c. 1850 Thomas Weddle - Weddle surface
• 1856 Weierstrass elliptic functions
• 1857 Bernhard Riemann^[11] lays the foundations for further work on abelian varieties in dimension > 1.
• Riemann bilinear relations
• Riemann theta function
• 1865 Carl Johannes Thomae, Theorie der ultraelliptischen Funktionen und Integrale erster und zweiter Ordnung^[12]
• 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^[13], 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^[14]
• 1888 Schottky finds a non-trivial condition on the theta constants for curves of genus g = 4, launching the Schottky problem
• 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

[edit] Twentieth century

• c.1910 The theory of Poincaré normal functions implies that the Picard variety and Albanese variety are isogenous^[15].
• 1913 Torelli's theorem^[16]
• 1916 Gaetano Scorza^[17] 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^[18]
• 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 Schottky problem

[edit] Twenty-first century

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

[edit] Notes

1. ^ PDF
2. ^ Miscellaneous Diophantine Equations at MathPages
3. ^ Fagnano_Giulio biography
4. ^ E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (fourth edition 1937), p. 72.
5. ^ André Weil, Number Theory: An approach through history (1984), p. 1.
6. ^ Landen biography
7. ^ Chronology of the Life of Carl F. Gauss
8. ^ Semen Grigorʹevich Gindikin, Tales of Physicists and Mathematicians (1988 translation), p. 143.
9. ^ Gopel biography
10. ^ http://www.gap-system.org/~history/Biographies/Rosenhain.html
11. ^ Theorie der Abel'schen Funktionen, J. Reine Angew. Math. 54 (1857), 115-180
12. ^ http://www.gap-system.org/~history/Biographies/Thomae.html
13. ^ Robert Langlands, Some Contemporary Problems with Origins in the Jugendtraum
14. ^ Über die Reduction einer bestimmten Klasse Abel'scher Integrale Ranges auf elliptische Integrale, Acta Math. 4, 392–414 (1884).
15. ^ PDF, p. 168.
16. ^ Ruggiero Torelli, Sulle varietà di Jacobi, Rend. della R. Acc. Nazionale dei Lincei , (5), 22, 1913, 98-103.
17. ^ G. Scorza, Intorno alla teoria generale delle matrici di Riemann e ad alcune sue applicazioni,Rend. del Circolo Mat. di Palermo 41 (1916)
18. ^ C. L. Siegel, Einführung in die Theorie der Modulfunktionen n-ten Grades, Math. Ann. 116 (1939), 617–657