Minimal Siegel modular threefolds

Details Minimal Siegel modular threefolds Minimal Siegel modular threefolds

1995-06-23

arxiv.org/abs/alg-geom/9506017v2

Mathematics

Algebraic Geometry

We have included a theorem explaining the connection of the minimal Siegel modular threefold with the space of Kummer surfaces

Scientific paper

In this paper we study the maximal extension $\Gamma_t^*$ of the subgroup $\Gamma_t$ of $\operatorname{Sp}_4 (\bq)$ which is conjugate to the paramodular group. The index of this extension is $2^{\nu(t)}$ where $\nu(t)$ is the number of prime divisors of $t$. The group $\Gamma_t^*$ defines the minimal modular threefold ${\Cal A}_t^*$ which is a finite quotient of the moduli space ${\Cal A}_t$ of $(1,t)$-polarized abelian surfaces. A certain degree 2 quotient of ${\Cal A}_t$ is a moduli space of lattice polarized $K3$ surfaces. The space ${\Cal A}_t^*$ can be interpreted as the space of Kummer surfaces associated to $(1,t)$-polarized abelian surfaces. Using the action of $\Gamma_t^*$ on the space of Jacobi forms we show that many spaces between ${\Cal A}_t$ and ${\Cal A}_t^*$ posess a non-trivial 3-form, i.e. the Kodaira dimension of these spaces is non-negative. Finally we determine the divisorial part of the ramification locus of the finite map ${\Cal A}_t\rightarrow {\Cal A}_t^*$ which is a union of Humbert surfaces. We interprete the corresponding Humbert surfaces as Hilbert modular surfaces.

Affiliated with

Also associated with

No associations

Rating

Minimal Siegel modular threefolds does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Minimal Siegel modular threefolds, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Minimal Siegel modular threefolds will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-531337