This is a **timeline of the theory of abelian varieties ** in algebraic geometry, including elliptic curves.

- c. 1000 Al-Karaji writes on congruent numbers
^{ [1] }

- Fermat studies descent for elliptic curves
- 1643 Fermat poses an elliptic curve Diophantine equation
^{ [2] }^{[ unreliable source? ]} - 1670 Fermat's son published his
*Diophantus*with notes

- 1718 Giulio Carlo Fagnano dei Toschi, studies the rectification of the lemniscate, addition results for elliptic integrals.
^{ [3] } - 1736 Leonhard 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 Carl Friedrich 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] }

- 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 , 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*

- 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 Solomon 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 André Weil defines "abelian variety"
- 1952 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 Max Deuring on elliptic curves to the abelian variety case.
^{ [21] } - c. 1980 Mukai–Fourier transform: the Poincaré line 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] }

- 2001 Proof of the modularity theorem for elliptic curves is completed.

- ↑ 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. - ↑ Robert Langlands ,
*Some Contemporary Problems with Origins in the Jugendtraum* - ↑
*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.

In mathematics, an **elliptic curve** is a smooth, projective, algebraic curve of genus one, on which there is a specified point *O*. An elliptic curve is defined over a field *K* and describes points in *K*^{2}, the Cartesian product of *K* with itself. If the field's characteristic is different from 2 and 3, then the curve can be described as a plane algebraic curve which, after a linear change of variables, consists of solutions (*x*,*y*) for:

In the mathematical field of complex analysis, **elliptic functions** are a special kind of meromorphic functions, that satisfy two periodicity conditions. They are named elliptic functions because they come from elliptic integrals. Originally those integrals occurred at the calculation of the arc length of an ellipse.

**Carl Gustav Jacob Jacobi** was a German mathematician who made fundamental contributions to elliptic functions, dynamics, differential equations, determinants, and number theory. His name is occasionally written as **Carolus Gustavus Iacobus Iacobi** in his Latin books, and his first name is sometimes given as **Karl**.

In mathematics, particularly in algebraic geometry, complex analysis and algebraic number theory, an **abelian variety** is a projective algebraic variety that is also an algebraic group, i.e., has a group law that can be defined by regular functions. Abelian varieties are at the same time among the most studied objects in algebraic geometry and indispensable tools for much research on other topics in algebraic geometry and number theory.

In mathematics, the **arithmetic of abelian varieties** is the study of the number theory of an abelian variety, or a family of abelian varieties. It goes back to the studies of Pierre de Fermat on what are now recognized as elliptic curves; and has become a very substantial area of arithmetic geometry both in terms of results and conjectures. Most of these can be posed for an abelian variety *A* over a number field *K*; or more generally.

In mathematics, the **Jacobian variety***J*(*C*) of a non-singular algebraic curve *C* of genus *g* is the moduli space of degree 0 line bundles. It is the connected component of the identity in the Picard group of *C*, hence an abelian variety.

In mathematics, an **abelian integral**, named after the Norwegian mathematician Niels Henrik Abel, is an integral in the complex plane of the form

In mathematics, **arithmetic geometry** is roughly the application of techniques from algebraic geometry to problems in number theory. Arithmetic geometry is centered around Diophantine geometry, the study of rational points of algebraic varieties.

This is a glossary of **arithmetic and diophantine geometry** in mathematics, areas growing out of the traditional study of Diophantine equations to encompass large parts of number theory and algebraic geometry. Much of the theory is in the form of proposed conjectures, which can be related at various levels of generality.

In mathematics, the **lemniscate elliptic functions** are elliptic functions related to the arc length of the lemniscate of Bernoulli. They were first studied by Giulio Fagnano in 1718 and later by Leonhard Euler and Carl Friedrich Gauss, among others.

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.

In mathematics, the concept of abelian variety is the higher-dimensional generalization of the elliptic curve. The **equations defining abelian varieties** are a topic of study because every abelian variety is a projective variety. In dimension *d* ≥ 2, however, it is no longer as straightforward to discuss such equations.

The study of manifolds combines many important areas of mathematics: it generalizes concepts such as curves and surfaces as well as ideas from linear algebra and topology. Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are called Lie Groups. Alternatively, they may be described by polynomial equations, in which case they are called algebraic varieties, and if they additionally carry a group structure, they are called algebraic groups.

In number theory, **Fermat's Last Theorem** states that no three positive integers *a*, *b*, and *c* satisfy the equation *a*^{n} + *b*^{n} = *c*^{n} for any integer value of *n* greater than 2. The cases *n* = 1 and *n* = 2 have been known since antiquity to have infinitely many solutions.

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