An approximation principle for congruence subgroups

TL;DR

This paper establishes an approximation principle for open subgroups of $G(Z_p)$, showing they are close to subgroups associated with algebraic subgroups and principal congruence subgroups, with uniform bounds when $G$ is defined over $Z$.

Contribution

It introduces a new approximation framework for congruence subgroups of $p$-adic groups, extending Nori's results to a $Z_p$-setting and providing uniform bounds over all primes.

Findings

Open subgroups are contained in algebraic subgroup-related subgroups with bounded index.

The bounds are independent of $p$ if $G$ is defined over $Z$.

A volume bound for conjugacy class intersections with open subgroups is established.

Abstract

The motivating question of this paper is roughly the following: given a 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 $\operatorname{Ker} (G(\mathbb{Z}_p)\rightarrow G(\mathbb{Z}/p^n\mathbb{Z}))$? 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 $\operatorname{GL}(N_0,\mathbb{F}_p)$ for large $p$. 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 future paper, this result will be applied to the limit multiplicity problem for arbitrary congruence subgroups of the arithmetic lattice $G (\mathbb{Z})$.