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Article

Antisymmetric Paramodular Forms of Weights 2 and 3

International Mathematics Research Notices. 2019. Vol.  rnz011. P. 1-21.
Gritsenko V., Poor C., Yuen D. S.

We define an algebraic set in 23-dimensional projective space whose 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 2 paramodular groups. Weight 3examples provide antisymmetric canonical differential forms on Siegel modular three-folds. Weight 2 is the minimal weight and these examples, via the paramodular conjecture, give evidence for the modularity of some rank 1 abelian surfaces defined over ℚ⁠.