  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.
