Additive and Multiplicative Lifting Properties of the Igusa Modular Form

  • Bernhard HeimEmail author
  • Atsushi Murase
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 115)


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.

Mathematics Subject Classification (2010)

11F11 11F25 11F41 



The authors thank the referee for many helpful comments.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.German University of Technology in OmanNorth GhubrahSultanate of Oman
  2. 2.Department of Mathematics, Faculty of ScienceKyoto Sangyo UniversityKita-kuJapan

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