The starting point of this paper is the maximal extension of, the subgroup of Sp4(Q) which is conjugate to the paramodular group. Correspondingly we call the quotient, the minimal Siegel modular threefold. The space, and the intermediate spaces between which is the space of (1,t)-polarized abelian surfaces and have not yet been studied in any detail. Using the Torelli theorem we first prove that can be interpreted as the space of Kummer surfaces of (1,t)-polarized abelian surfaces and that a certain degree 2 quotient of, which lies over is a moduli space of lattice polarized K3 surfaces. Using the action of on the space of Jacobi forms we show that many spaces between and 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 themselves. As a first necessary step in this direction we determine the divisorial part of the ramification locus of the finite map. This is a union of Humbert surfaces which can be interpreted as Hilbert modular surfaces. © 1998, Cambridge University Press. All rights reserved.