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 functions give solutions of a Cauchy equation of rank $N$ for functions of a $g$-dimensional argument with $N\le 2^{g}$ in the case of a general $g$-dimensional Abelian variety and $N\le g$ in the case of a Jacobian variety of an algebra curve of genusg. It is conjectured that a functional Cauchy equation of rankg for a function of a $g$-dimensional argument is characteristic for functions of a Jacobian variety of an algebraic curve of genusg, i. e., solves the Riemann–Schottky problem.