Estimating Siegel modular forms of genus 2 using Jacobi forms

Estimating Siegel Modular Forms of Genus 2 Using Jacobi Forms

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 FourierJacobi 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.

Some vector valued Siegel modular forms of genus 2

is a module over the ring of all modular forms with respect to the group Γ2[4, 8]. We are interested in its structure. By Igusa, the ring of modular forms is generated by the ten classical theta constants θ[m]. The module M contains a submodule N which is generated by 45 Cohen-Rankin brackets {θ[m], θ[n]}. We determine defining relations for this submodule and compute its Hilbert function (Theo...

Siegel Modular Forms

These are the lecture notes of the lectures on Siegel modular forms at the Nordfjordeid Summer School on Modular Forms and their Applications. We give a survey of Siegel modular forms and explain the joint work with Carel Faber on vector-valued Siegel modular forms of genus 2 and present evidence for a conjecture of Harder on congruences between Siegel modular forms of genus 1 and 2.

Elliptic Genus of Calabi–yau Manifolds and Jacobi and Siegel Modular Forms

In the paper we study two types of relations: a one is between the elliptic genus of Calabi–Yau manifolds and Jacobi modular forms, another one is between the second quantized elliptic genus, Siegel modular forms and Lorentzian Kac–Moody Lie algebras. We also determine the structure of the graded ring of the weak Jacobi forms with integral Fourier coefficients. It gives us a number of applicati...

Some vector valued Siegel modular forms of genus 3 Eberhard