An approximation principle for congruence subgroups

Abstract

The motivating question of this paper is roughly the following: given a flat group scheme $G$ over ${\mathbb{Z}}_p$, $p$ prime, with semisimple generic fiber $G_{{\mathbb{Q}}_p}$, how far are open subgroups of $G({\mathbb{Z}}_p)$ from subgroups of the form $X({\mathbb{Z}}_p){\mathbf{K}}_p(p^n)$, where $X$ is a subgroup scheme of $G$ and ${\mathbf{K}}_p(p^n)$ is the principal congruence subgroup Ker$(G({\mathbb{Z}}_p)\to G(\mathbb{Z}/p^n\mathbb{Z}))$? More precisely, we will show that for $G_{{\mathbb{Q}}_p}$ simply connected there exist constants $J\ge 1$ and $\epsilon >0$, depending only on $G$, such that any open subgroup of $G({\mathbb{Z}}_p)$ of level $p^n$ admits an open subgroup of index $\le J$ which is contained in $X({\mathbb{Z}}_p){\mathbf{K}}_p(p^{\lceil \epsilon n\rceil })$ for some proper, connected algebraic subgroup $X$ of $G$ defined over ${\mathbb{Q}}_p$. Moreover, if $G$ is defined over $\mathbb{Z}$, then $\epsilon$ and $J$ can be taken independently of $p$.

We also give a correspondence between natural classes of ${\mathbb{Z}}_p$-Lie subalgebras of ${\mathfrak{g}}_{{\mathbb{Z}}_p}$ and of closed subgroups of $G({\mathbb{Z}}_p)$ that can be regarded as a variant over ${\mathbb{Z}}_p$ of Nori's results on the structure of finite subgroups of GL$(N_0,{\mathbb{F}}_p)$ for large $p$ [Nor 87].

As an application we give a bound for the volume of the intersection of a conjugacy class in the group $G(\hat{\mathbb{Z}})={\prod }_{p}G({\mathbb{Z}}_p)$, for $G$ defined over $\mathbb{Z}$, with an arbitrary open subgroup. In a companion paper, we apply this result to the limit multiplicity problem for arbitrary congruence subgroups of the arithmetic lattice $G(\mathbb{Z})$.