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

Zudilin, V. V.

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.

MSC:

11F27	Theta series; Weil representation; theta correspondences
11J72	Irrationality; linear independence over a field
11F46	Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms

Keywords:

theta functions; thetanullwerte; Siegel modular forms; logarithmic derivatives