Timeline of Abelian Varieties - Twentieth Century

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.

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Famous quotes related to twentieth century:

    “Advertising is the greatest art form of the twentieth century.”
    —Marshall McLuhan (1911–1980)

    “The real passion of the twentieth century is servitude.”
    —Albert Camus (1913–1960)

    “The nineteenth century planted the words which the twentieth ripened into the atrocities of Stalin and Hitler. There is hardly an atrocity committed in the twentieth century that was not foreshadowed or even advocated by some noble man of words in the nineteenth.”
    —Eric Hoffer (1902–1983)

    “Doubt, it seems to me, is the central condition of a human being in the twentieth century.”
    —Salman Rushdie (b. 1947)

    “In the twentieth century, death terrifies men less than the absence of real life. All these dead, mechanized, specialized actions, stealing a little bit of life a thousand times a day until the mind and body are exhausted, until that death which is not the end of life but the final saturation with absence.”
    —Raoul Vaneigem (b. 1934)