Integrable equations, addition theorems, and the RiemannSchottky problem
Abstract
The classical Weierstrass theorem claims that, among the analytic functions, the only functions admitting an algebraic addition theorem are the elliptic functions and their degenerations. This survey is devoted to farreaching generalizations of this result that are motivated by the theory of integrable systems. The authors discovered a strong form of the addition theorem for theta functions of Jacobian varieties, and this form led to new approaches to known problems in the geometry of Abelian varieties. It is shown that strong forms of addition theorems arise naturally in the theory of the socalled trilinear functional equations. Diverse aspects of the approaches suggested here are discussed, and some important open problems are formulated.
 Publication:

Russian Mathematical Surveys
 Pub Date:
 February 2006
 DOI:
 10.1070/RM2006v061n01ABEH004298
 Bibcode:
 2006RuMaS..61...19B