Remarkable Siegel paramodular forms of small weights

Paramodular forms are Siegel modular forms with respect to the paramodular groups of polarisation $(1,t)$ and genus 2. The construction of paramodular forms of small weights is quite a difficult problem, equivalent to the proof of the non-triviality of the third cohomology groups or proof of general type of Siegel modular 3-folds. In this talk, we first give a brief overview of the subject and some necessary preliminary materials will be given, such as the notions of paramodular forms, Jacobi forms, theta-blocks, additive lifting and Borcherds products. Then we construct an infinite family of paramodular forms of weight 2 which are simultaneously Borcherds products and Gritsenko lifting (i.e. identities of type ‘infinite product=infinite sum’). This proves the conjecture of Gritsenko-Poor-Yuen (2013) for the known infinite series of theta blocks of weight 2. We also construct infinite families of antisymmetric paramodular forms of weights 3 and 4. These paramodular forms are obtained as quasi pull-backs of certain remarkable reflective modular forms of singular weights for the root lattices $A_4$ and $A_6$. The talk is based on a joint work with Valery Gritsenko.