Antisymmetric Paramodular Forms of Weights 2 and 3

TL;DR

This paper constructs and analyzes algebraic sets related to antisymmetric paramodular forms of weights 2 and 3, providing new examples of modular forms and evidence for the Paramodular Conjecture.

Contribution

It defines a 23-dimensional algebraic set linking rational points to Borcherds products, constructing explicit modular forms and differential forms, and supporting the modularity of certain abelian surfaces.

Findings

Explicit examples of holomorphic Borcherds products

Construction of antisymmetric differential forms on Siegel threefolds

Evidence supporting the Paramodular Conjecture for rank one abelian surfaces

Abstract

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 $\mathbb Q$.