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Integrable linear equations and the Riemann-Schottky problem

I. Krichever
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Abstract

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

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I. Krichever. 2005-11-30. Integrable linear equations and the Riemann-Schottky problem. https://arxiv.org/abs/math/0504192

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