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

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. Dale Husemoller, Elliptic Curves.
10. Richelot, Essai sur une méthode générale pour déterminer les valeurs des intégrales ultra-elliptiques, fondée sur des transformations remarquables de ces transcendantes, C. R. Acad. Sci. Paris. 2 (1836), 622-627; De transformatione integralium Abelianorum primi ordinis commentatio, J. Reine Angew. Math. 16 (1837), 221-341.
11. Gopel biography
12. http://www.gap-system.org/~history/Biographies/Rosenhain.html
13. Theorie der Abel'schen Funktionen, J. Reine Angew. Math. 54 (1857), 115-180
14. http://www.gap-system.org/~history/Biographies/Thomae.html
15. Robert Langlands, Some Contemporary Problems with Origins in the Jugendtraum
16. Über die Reduction einer bestimmten Klasse Abel'scher Integrale Ranges auf elliptische Integrale, Acta Math. 4, 392–414 (1884).
17. PDF, p. 168.
18. Ruggiero Torelli, Sulle varietà di Jacobi, Rend. della R. Acc. Nazionale dei Lincei , (5), 22, 1913, 98–103.
19. G. Scorza, Intorno alla teoria generale delle matrici di Riemann e ad alcune sue applicazioni,Rend. del Circolo Mat. di Palermo 41 (1916)
20. C. L. Siegel, Einführung in die Theorie der Modulfunktionen n-ten Grades, Math. Ann. 116 (1939), 617–657
21. Jean-Pierre Serre and John Tate, Good Reduction of Abelian Varieties, The Annals of Mathematics, Second Series, Vol. 88, No. 3 (Nov., 1968), pp. 492–517.
22. Daniel Huybrechts, Fourier–Mukai transforms in algebraic geometry (2006), Ch. 9.
23. J.-M. Fontaine, Il n'y a pas de variété abélienne sur Z, Invent. Math. (1985) no. 3, 515–538.