Integrable linear equations and the Riemann-Schottky problem

Abstract

We prove that an indecomposable principally polarized abelian variety $X$ is the Jacobain of a curve if and only if there exist vectors $U\ne 0,V$ such that the roots $x_i(y)$ of the theta-functional equation $\theta (Ux+Vy+Z)=0$ satisfy the equations of motion of the {\it formal infinite-dimensional Calogero-Moser system}

20 pages, Latex, minor erros are corrected, missing argumants are clarified