Vector addition theorems and Baker-Akhiezer functions

Abstract

Functional equations that arise naturally in various problems of modern mathematical physics are discussed. We introduce the concepts of anN-dimensional addition theorem for functions of a scalar argument and Cauchy equations of rankN for a function of ag-dimensional argument that generalize the classical functional Cauchy equation. It is shown that forN=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 rankN for functions of ag-dimensional argument withN≤2 g in the case of a generalg-dimensional Abelian variety andN≤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 ag-dimensional argument is characteristic for θ functions of a Jacobian variety of an algebraic curve of genusg, i.e., solves the Riemann—Schottky problem.