Normal forms for ordinary differential operators. I

In this paper, we develop the generalized Schur theory offered in the recent paper by the second author in the dimension one, and apply it to obtain a new explicit parametrization of torsion free rank one sheaves on projective irreducible curves with vanishing cohomology groups. This parametrization is obtained with the help of normal forms - the notion we introduce in this paper. Namely, considering the ring of ordinary differential operators $D_1=K[[x]][\partial]$ as a subring of a certain complete non-commutative ring $\widehat{D}_1^{\mathrm{sym}}$, the normal forms of differential operators mentioned here are obtained after conjugation by some invertible operator ("Schur operator"), calculated using one of the operators in a ring. Normal forms of commuting operators are polynomials with constant coefficients in the differentiation, integration, and shift operators, which have a restricted finite order in each variable, and can be effectively calculated for each given commuting operators.