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, studies the 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 Takahiro Shiota proves Novikov's conjecture on the Schottky problem
- 1985 Jean-Marc 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
- ^ Miscellaneous Diophantine Equations at MathPages[unreliable source?]
- ^ Fagnano_Giulio biography
- ^ E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies (fourth edition 1937), p. 72.
- ^ André Weil, Number Theory: An approach through history (1984), p. 1.
- ^ Landen biography
- ^ Chronology of the Life of Carl F. Gauss
- ^ Semen Grigorʹevich Gindikin, Tales of Physicists and Mathematicians (1988 translation), p. 143.
- ^ Dale Husemoller, Elliptic Curves.
- ^ 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.
- ^ Gopel biography
- ^ http://www.gap-system.org/~history/Biographies/Rosenhain.html
- ^ Theorie der Abel'schen Funktionen, J. Reine Angew. Math. 54 (1857), 115-180
- ^ http://www.gap-system.org/~history/Biographies/Thomae.html
- ^ Robert Langlands, Some Contemporary Problems with Origins in the Jugendtraum
- ^ Über die Reduction einer bestimmten Klasse Abel'scher Integrale Ranges auf elliptische Integrale, Acta Math. 4, 392–414 (1884).
- ^ PDF, p. 168.
- ^ Ruggiero Torelli, Sulle varietà di Jacobi, Rend. della R. Acc. Nazionale dei Lincei , (5), 22, 1913, 98–103.
- ^ G. Scorza, Intorno alla teoria generale delle matrici di Riemann e ad alcune sue applicazioni,Rend. del Circolo Mat. di Palermo 41 (1916)
- ^ C. L. Siegel, Einführung in die Theorie der Modulfunktionen n-ten Grades, Mathematische Annalen 116 (1939), 617–657
- ^ 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.
- ^ Daniel Huybrechts, Fourier–Mukai transforms in algebraic geometry (2006), Ch. 9.
- ^ Jean-Marc Fontaine, Il n'y a pas de variété abélienne sur Z, Inventiones Mathematicae (1985) no. 3, 515–538.