Pointwise Properties of Fourier-Jacobi-Forms

TL;DR

This paper investigates the properties of Fourier-Jacobi-forms, showing how their coefficient modular forms relate to Siegel cusp forms and establishing conditions for full rank in the Satake boundary, with implications for automorphic embeddings.

Contribution

It demonstrates the full rank of coefficient modular forms from Siegel cusp forms at boundary points under certain conditions and compares the abundance of Jacobi and Fourier-Jacobi cusp forms.

Findings

Full rank of coefficient modular forms at boundary points for large weights.

Existence of more Jacobi cusp forms than Fourier-Jacobi cusp forms at high weights.

Local automorphic embedding of Siegel modular varieties.

Abstract

Jacobi-Forms can be decomposed as a linear combination of Thetafunctions with modular forms as coefficients. It is shown that the space of these coefficient modular forms of Fourier-Jacobi-Forms, which come from Siegel cusp forms, has full rank in every point of the Satake boundary, if the index is 1, the weight is sufficiently large and the Satake boundary point has trivial stabilizer in $\Gamma_{n-1}$. This yields a local automorphic embedding of the Siegel modular variety. Klingen-Poincare series are the main tool. Despite of this richness it is proved that there are more Jacobi index 1 cusp forms than Fourier-Jacobi index 1 cusp forms for all sufficiently large weights extending a result of Dulinski.