Let W(z) be an n x 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).

We review the role played by tau-functions of special type - called Bergman tau-functions in various areas: theory of isomonodromic deformations, solutions of Einstein’s equations, theory of Dubrovin–Frobenius manifolds, geometry of moduli spaces and spectral theory of Riemann surfaces. These tau-functions are natural generalizations of Dedekind’s eta-function to higher genus. Study of their properties allows to get an explicit form of Einstein’s metrics, obtain new relations in Picard groups of various moduli spaces and derive holomorphic factorization formulas of determinants of Laplacians in flat singular metrics on Riemann surfaces, among other things.