Minimal Siegel modular threefolds

Abstract

The starting point of this paper is the maximal extension Γ*_t of Γ_t, the subgroup of Sp₄(ℚ) which is conjugate to the paramodular group. Correspondingly we call the quotient [Ascr ]*_t=Γ*_t\ℍ₂ the minimal Siegel modular threefold. The space [Ascr ]*_t and the intermediate spaces between [Ascr ]_t=Γ_t\ℍ₂ which is the space of (1, t)-polarized abelian surfaces and [Ascr ]*_t have not yet been studied in any detail. Using the Torelli theorem we first prove that [Ascr ]*_t can be interpreted as the space of Kummer surfaces of (1, t)-polarized abelian surfaces and that a certain degree 2 quotient of [Ascr ]_t which lies over [Ascr ]*_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 [Ascr ]_t and [Ascr ]*_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 [Ascr ]*_t themselves. As a first necessary step in this direction we determine the divisorial part of the ramification locus of the finite map [Ascr ]_t→[Ascr ]*_t. This is a union of Humbert surfaces which can be interpreted as Hilbert modular surfaces.