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

TL;DR

This paper investigates the properties of spectral curves derived from matrix polynomials over rationals, showing that specific derivatives of associated theta-functions are rational at certain points, revealing deep algebraic structures.

Contribution

It establishes that higher-order derivatives of Riemann theta-functions related to spectral curves over  are rational at points defined by eigenvector line bundles, under natural assumptions.

Findings

Certain derivatives of theta-functions are rational at specific points.

The spectral curve's structure influences the rationality of theta-function derivatives.

Results connect algebraic geometry with spectral theory over .

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 $W(z)$.