Vector addition theorems and Baker-Akhiezer functions

Functional equations that arise naturally in various problems of modern mathematical physics are discussed. We introduce the concepts of an N-dimensional addition theorem for functions of a scalar argument and Cauchy equations of rank N for a function of a g-dimensional argument that generalize the classical functional Cauchy equation. It is shown that for N=2 the general analytic solution of these equations is determined by the Baker—Akhiezer function of an algebraic curve of genus 2. It is also shown that θ functions give solutions of a Cauchy equation of rank N for functions of a g-dimensional argument with N≤2 ^g in the case of a general g-dimensional Abelian variety and N≤ g in the case of a Jacobian variety of an algebra curve of genus g. It is conjectured that a functional Cauchy equation of rank g for a function of a g-dimensional argument is characteristic for θ functions of a Jacobian variety of an algebraic curve of genus g, i.e., solves the Riemann—Schottky problem.