On the cohomological dimension of Siegel modular varieties and the modularity of formal Siegel modular forms

TL;DR

This paper establishes an upper bound on the cohomological dimension of Siegel modular varieties, leading to the automatic classicality of formal Siegel modular forms for genus at least 2, and generalizes prior modularity results.

Contribution

It proves a new bound on the cohomological dimension of Siegel modular varieties and shows formal Siegel modular forms are classical for genus ≥ 2, extending previous work.

Findings

Bound on the cohomological dimension: at most g(g+1)/2-2 for g≥2.

Formal Siegel modular forms of genus g≥2 are automatically classical.

Generalization of Bruinier and Raum's modularity results.

Abstract

We prove that the coherent cohomological dimension of the Siegel modular variety $A_{g,\Gamma}$ is at most $g(g+1)/2-2$ for $g\geq 2$. As a corollary, we show that the boundary of the compactified Siegel modular variety satisfies the Grothendieck-Lefschetz condition. This implies, in particular, that formal Siegel modular forms of genus $g\geq2$ are automatically classical Siegel modular forms. Our result generalizes the work of Bruinier and Raum on the modularity of formal Siegel modular forms.