Abelian Variety - History and Motivation

History and Motivation

For more details on this topic, see History of manifolds and varieties.

In the early nineteenth century, the theory of elliptic functions succeeded in giving a basis for the theory of elliptic integrals, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials. When those were replaced by polynomials of higher degree, say quintics, what would happen?

In the work of Niels Abel and Carl Jacobi, the answer was formulated: this would involve functions of two complex variables, having four independent periods (i.e. period vectors). This gave the first glimpse of an abelian variety of dimension 2 (an abelian surface): what would now be called the Jacobian of a hyperelliptic curve of genus 2.

After Abel and Jacobi, some of the most important contributors to the theory of abelian functions were Riemann, Weierstrass, Frobenius, Poincaré and Picard. The subject was very popular at the time, already having a large literature.

By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions. Eventually, in the 1920s, Lefschetz laid the basis for the study of abelian functions in terms of complex tori. He also appears to be the first to use the name "abelian variety". It was André Weil in the 1940s who gave the subject its modern foundations in the language of algebraic geometry.

Today, abelian varieties form an important tool in number theory, in dynamical systems (more specifically in the study of Hamiltonian systems), and in algebraic geometry (especially Picard varieties and Albanese varieties).