Estimating Siegel modular forms of genus 2 using Jacobi forms

Abstract

We give a new elementary proof of Igusa’s theorem on the structure of Siegel modular forms of genus 2. The key point of the proof is the estimation of the dimension of Jacobi forms appearing in the Fourier-Jacobi development of Siegel modular forms. This proves not only Igusa’s theorem, but also gives the canonical lifting from Jacobi forms to Siegel modular forms of genus 2.