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.
On Hilbert modular threefolds of discriminant 49