Zero Cells of the Siegel–Gottschling Fundamental Domain of Degree 2

Abstract

Let $\mathcal{F}_n$ be a fundamental domain of the Siegel upper half-space of degree $n$ with respect to the Siegel modular group $\operatorname{Sp}(n, \mathbb{Z})$. According to Siegel himself, $\mathcal{F}_n$ is determined by only finitely many polynomial inequalities. In case of degree $n = 2$, Gottschling determined the minimal set of inequalities. The boundary of $\mathcal{F}_2$ is of great concern in the literature not only from a homological point of view but also from the geometry of numbers. In this paper we compute the vertices of $\mathcal{F}_2$ under the condition that the defining ideal is zero-dimensional (“0-cells”). We also discuss an equivalence relation among 0-cells.