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 $J\ge1$ and $\varepsilon>0$, 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 $X(\mathbb Z_p)\mathbf K_p(p^{\lceil \varepsilon n\rceil})$ for some proper, connected algebraic subgroup $X$ of $G$ defined over $\mathbb Q_p$. Moreover, if $G$ is defined over $\mathbb Z$, then $\varepsilon$ 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 $G (\mathbb Z)$.