From Wikipedia, the free encyclopedia
This is a timeline of the theory of
abelian varieties in
algebraic geometry , including elliptic curves.
Early history
Seventeenth century
Eighteenth century
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 De 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
Karl 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
{\displaystyle 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
Twenty-first century
Notes
^
PDF
^
Miscellaneous Diophantine Equations at MathPages
^
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
^
"Rosenhain biography" . www.gap-system.org . Archived from
the original on 2008-09-07.
^ Theorie der Abel'schen Funktionen,
J. Reine Angew. Math. 54 (1857), 115-180
^
"Thomae biography" . www.gap-system.org . Archived from
the original on 2006-09-28.
^
Some Contemporary Problems with Origins in the Jugendtraum ,
Robert Langlands
^ Über die Reduction einer bestimmten Klasse Abel'scher Integrale Ranges auf elliptische Integrale,
Acta Mathematica 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.
^
Gaetano Scorza , Intorno alla teoria generale delle matrici di Riemann e ad alcune sue applicazioni, Rend. del Circolo Mat. di Palermo 41 (1916)
^
Carl Ludwig 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 ,
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.