This is a timeline of the theory of
abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...
in
algebraic geometry, including elliptic curves.
Early history
* c. 1000
Al-Karaji writes on
congruent number
In number theory, a congruent number is a positive integer that is the area of a right triangle with three rational number sides. A more general definition includes all positive rational numbers with this property.
The sequence of (integer) c ...
s
Seventeenth century
* Fermat studies
descent for elliptic curves
* 1643 Fermat poses an elliptic curve
Diophantine equation
* 1670 Fermat's son published his ''Diophantus'' with notes
Eighteenth century
* 1718
Giulio Carlo Fagnano dei Toschi, studies the rectification of the
lemniscate
In algebraic geometry, a lemniscate is any of several figure-eight or -shaped curves. The word comes from the Latin "''lēmniscātus''" meaning "decorated with ribbons", from the Greek λημνίσκος meaning "ribbons",. or which alternative ...
, addition results for
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s.
* 1736
Leonhard Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
writes on the
pendulum equation without the small-angle approximation.
* 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.
* 1775
John Landen publishes
Landen's transformation, an
isogeny In mathematics, in particular, in algebraic geometry, an isogeny is a morphism of algebraic groups (also known as group varieties) that is surjective and has a finite kernel.
If the groups are abelian varieties, then any morphism of the underlyi ...
formula.
* 1786
Adrien-Marie Legendre
Adrien-Marie Legendre (; ; 18 September 1752 – 9 January 1833) was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are name ...
begins to write on
elliptic integral
In integral calculus, an elliptic integral is one of a number of related functions defined as the value of certain integrals, which were first studied by Giulio Fagnano and Leonhard Euler (). Their name originates from their originally arising in ...
s
* 1797
Carl Friedrich Gauss
Johann Carl Friedrich Gauss (; german: Gauß ; la, Carolus Fridericus Gauss; 30 April 177723 February 1855) was a German mathematician and physicist who made significant contributions to many fields in mathematics and science. Sometimes refer ...
discovers
double periodicity of the
lemniscate function
* 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.
Nineteenth century
* 1826
Niels Henrik Abel
Niels Henrik Abel ( , ; 5 August 1802 – 6 April 1829) was a Norwegian mathematician who made pioneering contributions in a variety of fields. His most famous single result is the first complete proof demonstrating the impossibility of solvin ...
,
Abel-Jacobi map
* 1827
Inversion of elliptic integrals independently by Abel and
Carl Gustav Jacob Jacobi
Carl Gustav Jacob Jacobi (; ; 10 December 1804 – 18 February 1851) was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasiona ...
* 1829 Jacobi, ''Fundamenta nova theoriae functionum ellipticarum'', introduces four
theta function
In mathematics, theta functions are special functions of several complex variables. They show up in many topics, including Abelian varieties, moduli spaces, quadratic forms, and solitons. As Grassmann algebras, they appear in quantum field ...
s of one variable
* 1835 Jacobi points out the use of the group law for
diophantine geometry
In mathematics, Diophantine geometry is the study of Diophantine equations by means of powerful methods in algebraic geometry. By the 20th century it became clear for some mathematicians that methods of algebraic geometry are ideal tools to study ...
, in ''De usu Theoriae Integralium Ellipticorum et Integralium Abelianorum in Analysi Diophantea''
* 1836-7
Friedrich Julius Richelot
Friedrich Julius Richelot (6 November 1808 – 31 March 1875) was a German mathematician, born in Königsberg. He was a student of Carl Gustav Jacob Jacobi.
He was promoted in 1831 at the Philosophical Faculty of the University of Königsberg wit ...
, the
Richelot isogeny.
* 1847
Adolph Göpel gives the equation of the
Kummer surface
In algebraic geometry, a Kummer quartic surface, first studied by , is an irreducible nodal surface of degree 4 in \mathbb^3 with the maximal possible number of 16 double points. Any such surface is the Kummer variety of the Jacobian varie ...
* 1851
Johann Georg Rosenhain writes a prize essay on the inversion problem in genus 2.
*c. 1850
Thomas Weddle
Thomas Weddle (30 November 1817 Stamfordham, Northumberland – 4 December 1853 Bagshot) was a mathematician who introduced the Weddle surface. He was mathematics professor at the Royal Military College, Sandhurst
The Royal Military College (RMC) ...
-
Weddle surface
* 1856
Weierstrass elliptic function
In mathematics, the Weierstrass elliptic functions are elliptic functions that take a particularly simple form. They are named for Karl Weierstrass. This class of functions are also referred to as ℘-functions and they are usually denoted by t ...
s
* 1857
Bernhard Riemann 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
Carl Johannes Thomae (sometimes called ''Johannes Thomae'', ''Karl Johannes Thomae'', or ''Johannes Karl Thomae''; 11 December 1840 in Laucha an der Unstrut – 1 April 1921 in Jena) was a German mathematician.
Biography
Thomae, son of Karl Au ...
, ''Theorie der ultraelliptischen Funktionen und Integrale erster und zweiter Ordnung''
* 1866
Alfred Clebsch and
Paul Gordan
__NOTOC__
Paul Albert Gordan (27 April 1837 – 21 December 1912) was a Jewish-German mathematician, a student of Carl Jacobi at the University of Königsberg before obtaining his PhD at the University of Breslau (1862),. and a professor ...
, ''Theorie der Abel'schen Functionen''
* 1869
Karl Weierstrass
Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics ...
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
Leopold Kronecker (; 7 December 1823 – 29 December 1891) was a German mathematician who worked on number theory, algebra and logic. He criticized Georg Cantor's work on set theory, and was quoted by as having said, "'" ("God made the integers, ...
describes his ''
Jugendtraum'', to use
complex multiplication
In mathematics, complex multiplication (CM) is the theory of elliptic curves ''E'' that have an endomorphism ring larger than the integers. Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible wh ...
theory to generate
abelian extension In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. Going in the other direction, a Galois extension is called solvable ...
s of
imaginary quadratic field
In algebraic number theory, a quadratic field is an algebraic number field of degree two over \mathbf, the rational numbers.
Every such quadratic field is some \mathbf(\sqrt) where d is a (uniquely defined) square-free integer different from 0 ...
s
* 1884
Sofia Kovalevskaya
Sofya Vasilyevna Kovalevskaya (russian: link=no, Софья Васильевна Ковалевская), born Korvin-Krukovskaya ( – 10 February 1891), was a Russian mathematician who made noteworthy contributions to analysis, partial differen ...
writes on the
reduction of abelian functions to elliptic functions
* 1888
Friedrich Schottky
Friedrich Hermann Schottky (24 July 1851 – 12 August 1935) was a German mathematician who worked on elliptic, abelian, and theta functions and introduced Schottky groups and Schottky's theorem. He was born in Breslau, Germany (now Wrocław, ...
finds a non-trivial condition on the
theta constant In mathematics, a theta constant or
Thetanullwert' (German for theta zero value; plural Thetanullwerte) is the restriction ''θ'm''(''τ'') = θ''m''(''τ'',''0'') of a theta function ''θ'm''(τ,''z'') with rational characteristic ...
s for curves of genus
, launching the
Schottky problem In mathematics, the Schottky problem, named after Friedrich Schottky, is a classical question of algebraic geometry, asking for a characterisation of Jacobian varieties amongst abelian varieties.
Geometric formulation
More precisely, one should co ...
.
* 1891
Appell–Humbert theorem
In mathematics, the Appell–Humbert theorem describes the line bundles on a complex torus or complex abelian variety.
It was proved for 2-dimensional tori by and , and in general by
Statement
Suppose that T is a complex torus given by V/\Lam ...
of
Paul Émile Appell
:''M. P. Appell is the same person: it stands for Monsieur Paul Appell''.
Paul Émile Appell (27 September 1855, in Strasbourg – 24 October 1930, in Paris) was a French mathematician and Rector of the University of Paris. Appell polynomials an ...
and
Georges Humbert, classifies the
holomorphic line bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
s 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
Alexander Wilhelm von Brill (20 September 1842 – 18 June 1935) was a German mathematician.
Born in Darmstadt, Hesse, Brill was educated at the University of Giessen, where he earned his doctorate under supervision of Alfred Clebsch. He held a c ...
and
Max Noether
Max Noether (24 September 1844 – 13 December 1921) was a German mathematician who worked on algebraic geometry and the theory of algebraic functions. He has been called "one of the finest mathematicians of the nineteenth century". He was the f ...
* 1895
Wilhelm Wirtinger
Wilhelm Wirtinger (19 July 1865 – 16 January 1945) was an Austrian mathematician, working in complex analysis, geometry, algebra, number theory, Lie groups and knot theory.
Biography
He was born at Ybbs on the Danube and studied at the Unive ...
, ''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
In mathematics, the Albanese variety A(V), named for Giacomo Albanese, is a generalization of the Jacobian variety of a curve.
Precise statement
The Albanese variety is the abelian variety A generated by a variety V taking a given point of V to ...
are
isogenous.
*1913
Torelli's theorem
* 1916
Gaetano Scorza
Bernardino Gaetano Scorza (29 September 1876, in Morano Calabro – 6 August 1939, in Rome) was an Italian mathematician working in algebraic geometry
Algebraic geometry is a branch of mathematics, classically studying zeros of multivariat ...
applies the term "abelian variety" to
complex tori.
* 1921
Solomon Lefschetz
Solomon Lefschetz (russian: Соломо́н Ле́фшец; 3 September 1884 – 5 October 1972) was an American mathematician who did fundamental work on algebraic topology, its applications to algebraic geometry, and the theory of non-linear o ...
shows that any complex torus with Riemann matrix satisfying the necessary conditions can be embedded in some
complex projective space
In mathematics, complex projective space is the projective space with respect to the field of complex numbers. By analogy, whereas the points of a real projective space label the lines through the origin of a real Euclidean space, the points of a ...
using theta-functions
* 1922
Louis Mordell
Louis Joel Mordell (28 January 1888 – 12 March 1972) was an American-born British mathematician, known for pioneering research in number theory. He was born in Philadelphia, United States, in a Jewish family of Lithuanian extraction.
Educati ...
proves
Mordell's theorem: the rational points on an elliptic curve over the rational numbers form a
finitely-generated abelian group
In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x_1,\dots,x_s in G such that every x in G can be written in the form x = n_1x_1 + n_2x_2 + \cdots + n_sx_s for some integers n_1,\dots, n ...
*1929
Arthur B. Coble, ''Algebraic Geometry and Theta Functions''
*1939
Siegel modular form
In mathematics, Siegel modular forms are a major type of automorphic form. These generalize conventional ''elliptic'' modular forms which are closely related to elliptic curves. The complex manifolds constructed in the theory of Siegel modular form ...
s
* c. 1940
André Weil defines "abelian variety"
* 1952 Weil defines an
intermediate Jacobian
In mathematics, the intermediate Jacobian of a compact Kähler manifold or Hodge structure is a complex torus that is a common generalization of the Jacobian variety of a curve and the Picard variety and the Albanese variety. It is obtained by ...
*
Theorem of the cube
*
Selmer group
In arithmetic geometry, the Selmer group, named in honor of the work of by , is a group constructed from an isogeny of abelian varieties.
The Selmer group of an isogeny
The Selmer group of an abelian variety ''A'' with respect to an isogeny ' ...
*
Michael Atiyah
Sir Michael Francis Atiyah (; 22 April 1929 – 11 January 2019) was a British-Lebanese mathematician specialising in geometry. His contributions include the Atiyah–Singer index theorem and co-founding topological K-theory. He was awarded th ...
classifies
holomorphic vector bundle In mathematics, a holomorphic vector bundle is a complex vector bundle over a complex manifold such that the total space is a complex manifold and the projection map is holomorphic. Fundamental examples are the holomorphic tangent bundle of a ...
s on an elliptic curve
* 1961
Goro Shimura
was a Japanese mathematician and Michael Henry Strater Professor Emeritus of Mathematics at Princeton University who worked in number theory, automorphic forms, and arithmetic geometry. He was known for developing the theory of complex multipli ...
and
Yutaka Taniyama
was a Japanese mathematician known for the Taniyama–Shimura conjecture.
Contribution
Taniyama was best known for conjecturing, in modern language, automorphic properties of L-functions of elliptic curves over any number field. A partial and r ...
, ''Complex Multiplication of Abelian Varieties and its Applications to Number Theory''
*
Néron model In algebraic geometry, the Néron model (or Néron minimal model, or minimal model)
for an abelian variety ''AK'' defined over the field of fractions ''K'' of a Dedekind domain ''R'' is the "push-forward" of ''AK'' from Spec(''K'') to Spec(''R''), ...
*
Birch–Swinnerton–Dyer conjecture
*
Moduli space for abelian varieties
*
Duality of abelian varieties
* c.1967
David Mumford
David Bryant Mumford (born 11 June 1937) is an American mathematician known for his work in algebraic geometry and then for research into vision and pattern theory. He won the Fields Medal and was a MacArthur Fellow. In 2010 he was awarded t ...
develops a new theory of the
equations defining abelian varieties
* 1968
Serre–Tate theorem on good reduction extends the results of
Max Deuring on elliptic curves to the abelian variety case.
*c. 1980
Mukai–Fourier transform: the
Poincaré line bundle as Mukai–Fourier kernel induces an equivalence of the
derived categories
In mathematics, the derived category ''D''(''A'') of an abelian category ''A'' is a construction of homological algebra introduced to refine and in a certain sense to simplify the theory of derived functors defined on ''A''. The construction proce ...
of
coherent sheaves
In mathematics, especially in algebraic geometry and the theory of complex manifolds, coherent sheaves are a class of sheaves closely linked to the geometric properties of the underlying space. The definition of coherent sheaves is made with refer ...
for an abelian variety and its dual.
*1983
Takahiro Shiota proves
Novikov's conjecture on the Schottky problem
*1985
Jean-Marc Fontaine
Jean-Marc Fontaine (13 March 1944 – 29 January 2019) was a French mathematician. He was one of the founders of p-adic Hodge theory. He was a professor at Paris-Sud 11 University from 1988 to his death.
Life
In 1962 Fontaine entered the Écol ...
shows that any positive-dimensional abelian variety over the rationals has bad reduction somewhere.
Jean-Marc Fontaine
Jean-Marc Fontaine (13 March 1944 – 29 January 2019) was a French mathematician. He was one of the founders of p-adic Hodge theory. He was a professor at Paris-Sud 11 University from 1988 to his death.
Life
In 1962 Fontaine entered the Écol ...
, ''Il n'y a pas de variété abélienne sur Z'', Inventiones Mathematicae (1985) no. 3, 515–538.
Twenty-first century
* 2001 Proof of the
modularity theorem
The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. And ...
for elliptic curves is completed.
Notes
{{Reflist
Abelian varieties
Abelian varieties
In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an abelian variety is a Algebraic variety#Projective variety, projective algebraic variety that is also an algebraic group, i.e., has a group law th ...