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**Thetanulls and differential equations.**
*(English.
Russian original)*
Zbl 1021.11011

Sb. Math. 191, No. 12, 1827-1871 (2000); translation from Mat. Sb. 191, No. 12, 77-122 (2000).

Summary: 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 nonlinear 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.