Theta block conjecture for Siegel paramodular forms

TL;DR

This paper proves the theta block conjecture for specific infinite series of Siegel paramodular forms, linking Borcherds products and Jacobi lifts, using classification of reflective modular forms.

Contribution

It establishes the theta block conjecture for two new infinite series of theta blocks, expanding understanding of Siegel paramodular forms.

Findings

Proved the theta block conjecture for weights 2 and 3 series.

Connected Borcherds products with additive Jacobi lifts.

Utilized Scheithauer's classification of reflective modular forms.

Abstract

The theta-block conjecture proposed by Gritsenko--Poor--Yuen in 2013 characterizes Siegel paramodular forms which are simultaneously Borcherds products and additive Jacobi lifts. In this paper, we prove this conjecture for two new infinite series of theta blocks of weights 2 and 3. The proof is based on Scheithauer's classification of reflective modular forms of singular weight.