Sturm's operator for scalar weight in arbitrary genus

TL;DR

This paper demonstrates that Sturm's operator fails to serve as a holomorphic projection operator for scalar weights in arbitrary genus Siegel modular forms, extending previous genus two results.

Contribution

It proves the failure of Sturm's operator as a holomorphic projection in arbitrary genus, generalizing earlier findings for genus two.

Findings

Sturm's operator does not realize holomorphic projection for scalar weight m+1 in genus m≥2

The failure extends the known genus two case to all higher genera

Provides theoretical proof of the operator's limitations in this context

Abstract

In contrast to the wellknown cases of large weights, Sturm's operator does not realize the holomorphic projection operator for lower weights. We prove its failure for arbitrary Siegel genus $m\geq 2$ and scalar weight $\kappa=m+1$. This generalizes a result for genus two in [4].