Mathematical Proceedings of the Cambridge Philosophical Society

Minimal Siegel modular threefolds

a1 St. Petersburg Department Steklov Mathematical Institute, Fontanka 27, 191011 St Petersburg, Russia; e-mail:;
a2 Institut für Mathematik, Universität Hannover, Postfach 6009, D-30060 Hannover, Germany; e-mail:


The starting point of this paper is the maximal extension Γ*t of Γt, the subgroup of Sp4([open face Q]) which is conjugate to the paramodular group. Correspondingly we call the quotient [script A]*t=Γ*t\[open face H]2 the minimal Siegel modular threefold. The space [script A]*t and the intermediate spaces between [script A]tt\[open face H]2 which is the space of (1, t)-polarized abelian surfaces and [script A]*t have not yet been studied in any detail. Using the Torelli theorem we first prove that [script A]*t can be interpreted as the space of Kummer surfaces of (1, t)-polarized abelian surfaces and that a certain degree 2 quotient of [script A]t which lies over [script A]*t is a moduli space of lattice polarized K3 surfaces. Using the action of Γ*t on the space of Jacobi forms we show that many spaces between [script A]t and [script A]*t possess a non-trivial 3-form, i.e. the Kodaira dimension of these spaces is non-negative. It seems a difficult problem to compute the Kodaira dimension of the spaces [script A]*t themselves. As a first necessary step in this direction we determine the divisorial part of the ramification locus of the finite map [script A]t[rightward arrow][script A]*t. This is a union of Humbert surfaces which can be interpreted as Hilbert modular surfaces.

(Received May 7 1996)
(Revised September 9 1996)


1 Supported by Institute Fourier, Grenoble, the Schwerpunktprogramm ‘Komplexe Mannigfaltigkeiten’ grant Hu 337/2-4 and DFG grant 436 Rus 17/108/95.