Thetanulls and differential equations

The closedness of the system of thetanulls (and the Siegel modular forms) and their first derivatives with respect to differentiation is well known in the one-dimensional case. It is shown in the present paper that thetanulls and their various logarithmic derivatives satisfy a non-linear system of differential equations; only one- and two-dimensional versions of this result were known before. Several distinct examples of such systems are presented, and a theorem on the transcendence degree of the differential closure of the field generated by all thetanulls is established. On the basis of a study of the modular properties of logarithmic derivatives of thetanulls (previously unknown) relations between these functions and thetanulls themselves are obtained in dimensions 2 and 3.