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arXiv · 1807.11258

Algebraic spectral curves over $\mathbb Q$ and their tau-functions

Boris Dubrovin
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Abstract

Let $W(z)$ be a $n\times n$ matrix polynomial with rational coefficients. Denote $C$ the spectral curve $\det \left( w\cdot{\bf 1}-W(z)\right) =0$. Under some natural assumptions about the structure of $W(z)$ we prove that certain combinations of logarithmic derivatives of the Riemann theta-function of $C$ of an arbitrary order starting from the third one all take rational values at the point of the Jacobi variety $J(C)$ specified by the line bundle of eigenvectors of $W(z)$.

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Boris Dubrovin. 2018-07-30. Algebraic spectral curves over $\mathbb Q$ and their tau-functions. https://arxiv.org/abs/1807.11258

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