Additive and Multiplicative Lifting Properties of the Igusa Modular Form

Abstract

The first cusp form χ 10 for the Siegel modular group of genus 2 is the Igusa modular form. It has been known by Gritsenko and Nikulin based on work of Borcherds that χ 10 is a Borcherds lift (multiplicative lift) and by Maass that it is a Saito–Kurokawa lift (additive lift). In this paper we show that these two properties characterize the Igusa modular form. By Bruinier, Siegel modular forms of genus 2 with Heegner divisor are Borcherds products. Hence every Saito–Kurokawa lift has a divisor different from a Heegner divisor except the lift is equal to the Igusa modular form. This implies that Siegel-type Eisenstein series do not have a Heegner divisor. Since in string theory Siegel modular forms, which are additive and multiplicative lifts play a prominent role, our uniqueness result may have some applications in this theory.