The moduli space of bilevel-6 abelian surfaces

Abstract

We show that the moduli space of abelian surfaces with polarisation of type $(1, 6)$ and a bilevel structure has positive Kodaira dimension and indeed $p_{g} \geq 3$. To do this we show that three of the Siegel cusp forms with character for the paramodular symplectic group constructed by Gritsenko and Nikulin are cusp forms without character for the modular group associated to this moduli problem. We then calculate the divisors of the corresponding differential forms, using information about the fixed loci of elements of the paramodular group previously obtained by Brasch.