Antisymmetric Paramodular Forms of Weights 2 and 3

We define an algebraic set in $23$~dimensional projective space whose $\mathbb Q$-rational points correspond to meromorphic, antisymmetric, paramodular Borcherds products. We know two lines inside this algebraic set. Some rational points on these lines give holomorphic Borcherds products and thus construct examples of Siegel modular forms on degree two paramodular groups. Weight $3$ examples provide antisymmetric canonical differential forms on Siegel modular threefolds. Weight $2$ is the minimal weight and these examples, via the Paramodular Conjecture, give evidence for the modularity of some rank one abelian surfaces defined over $\mathbb Q$.