Commutator coverings of Siegel threefolds

Abstract

We investigate the existence and non-existence of modular forms of low weight with a character with respect to the paramodular group $\Gamma_t$ and discuss the resulting geometric consequences. Using an advanced version of Maa\ss\ lifting one can construct many examples of such modular forms and in particular examples of weight 3 cusp forms. Consequently we find many abelian coverings of low degree of the moduli space ${\Cal A}_t$ of (1,t)-polarized abelian surfaces which are not unirational. We also determine the commutator subgroups of the paramodular group $\Gamma_t$ and its degree 2 extension $\Gamma^+_t$. This has applications for the Picard group of the moduli stack ${\underline{\Cal A}}_t$. Finally we prove non-existence theorems for low weight modular forms. As one of our main results we obtain the theorem that the maximal abelian cover ${\Cal A}_t^{com}$ of ${\Cal A}_t$ has geometric genus 0 if and only if t=1, 2, 4 or 5. We also prove that ${\Cal A}_t^{com}$ has geometric genus 1 for t=3 and 7.