The Gauss problem for central leaves

Abstract

We solve two related Gauss problems. In the arithmetic setting, we consider genera of maximal $O$-lattices, where $O$ is the maximal order in a definite quaternion $\mathbb{Q}$-algebra, and we list all cases where they have class number $1$. We also prove a unique orthogonal decomposition result for more general $O$-lattices. In the geometric setting, we study the Siegel modular variety $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ of genus $g$, and we list all $x$ in $\mathcal{A}_g \otimes \overline{\mathbb{F}}_p$ for which the corresponding central leaf $\mathcal{C}(x)$ consists of one point, that is, such that $x$ is the unique point in the locus consisting of the points whose associated polarised $p$-divisible groups are isomorphic to that of $x$. The solution to the second Gauss problem involves mass formulae, computations of automorphism groups, and a careful analysis of Ekedahl-Oort strata in genus $g = 4$.