Paramodular Group Research Articles

Minimal Siegel modular threefolds

The starting point of this paper is the maximal extension Γ*t of Γt, 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=Γt\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.

AUTOMORPHIC FORMS AND LORENTZIAN KAC–MOODY ALGEBRAS PART II

We give variants of lifting construction, which define new classes of modular forms on the Siegel upper half-space of complex dimension 3 with respect to the full paramodular groups (defining moduli of Abelian surfaces with arbitrary polarization). The data for these liftings are Jacobi forms of integral and half-integral indices. In particular, we get modular forms which are generalizations of the Dedekind eta-function. Some of these forms define automorphic corrections of Lorentzian Kac–Moody algebras with hyperbolic generalized Cartan matrices of rank three classified in Part I of this paper. We also construct many automorphic forms which give discriminants of moduli of K3 surfaces with conditions on Picard lattice.

On the graded ring of modular forms of the siegel paramodular group of level 2

On the graded ring of modular forms of the siegel paramodular group of level 2