Vector addition theorems and BakerAkhiezer functions
Abstract
Functional equations that arise naturally in various problems of modern mathematical physics are discussed. We introduce the concepts of an Ndimensional addition theorem for functions of a scalar argument and Cauchy equations of rank N for a function of a gdimensional 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 gdimensional argument with N≤2^{ g } in the case of a general gdimensional 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 gdimensional argument is characteristic for θ functions of a Jacobian variety of an algebraic curve of genus g, i.e., solves the Riemann—Schottky problem.
 Publication:

Theoretical and Mathematical Physics
 Pub Date:
 February 1993
 DOI:
 10.1007/BF01019326
 Bibcode:
 1993TMP....94..142B