Graded rings of paramodular forms of levels $5$ and $7$

Abstract

We compute generators and relations for the graded rings of paramodular forms of degree two and levels 5 and 7. The generators are expressed as quotients of Gritsenko lifts and Borcherds products. The computation is made possible by a characterization of modular forms on the Humbert surfaces of discriminant 4 that arise from paramodular forms by restriction.