Varietà Jacobiane di Curve Spettrali e Funzioni Theta

David Mumford described an explicit isomorphism between an open set of the Jacobian variety of a hyperelliptic curve C and a set of triples of polynomials satisfying certain conditions. Moreover he is able to express the coefficients of these polynomials in terms of Riemann's theta function of the curve C, thus providing an explicit expression for the algebraic coordinates on Jac(C) as theta functions. A. Beauville generalized the algebraic construction of Mumford to the case of spectral curves. It turns out that an open set of the Jacobian variety of such a curve is isomorphic to the set of polynomial matrices of fixed degree whose characteristic polynomial is equal to the polynomial defining the spectral curve, modulo conjugation by PGL(n,C). In this paper we generalize the expressions of the algebraic coordinates on Jac(C) in terms of theta functions, given by Mumford in the case of a hyperelliptic curve, to the general case of spectral curves. More precisely, we give an expression for the coefficients of a matrix A(x) corresponding to a line bundle L, using the classical Riemann's theta function of the spectral curve.