Geodesic
Thetanulls and differential equations
Sbornik. Mathematics, Tome 191 (2000) no. 12, pp. 1827-1871

Voir la notice de l'article provenant de la source Math-Net.Ru

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. 
@article{SM_2000_191_12_a4,
     author = {W. V. Zudilin},
     title = {Thetanulls and differential equations},
     journal = {Sbornik. Mathematics},
     pages = {1827--1871},
     publisher = {mathdoc},
     volume = {191},
     number = {12},
     year = {2000},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2000_191_12_a4/}
}
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W. V. Zudilin. Thetanulls and differential equations. Sbornik. Mathematics, Tome 191 (2000) no. 12, pp. 1827-1871. http://geodesic.mathdoc.fr/item/SM_2000_191_12_a4/