Minimal Siegel modular threefolds

The starting point of this paper is the maximal extension Gamma(t)* of Gamma(t), the subgroup of Sp(4)(Q) which is conjugate to the paramodular group. Correspondingly we call the quotient A(t)* = Gamma(t)*\H2 the minimal Siegel modular threefold. The space A(t)* and the intermediate spaces between A(t) = Gamma(t)\H-2 which is the space of(1, t)-polarized abelian surfaces and A(t)* have not Set been studied in any detail. Using the Torelli theorem we first prove that 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 dl which lies over A(t)* is a moduli space of lattice polarized K3 surfaces. Using the action of Gamma(t)* on the space of Jacobi forms we show that many spaces between A(t) and A(t)* possess a nontrivial 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 A(t)* themselves. As a first necessary step in this direction we determine the divisorial part of the ramification locus of the finite map A(t) --> A(t)*. This is a union of Humbert surfaces which can be interpreted as Hilbert modular surfaces.