An approximation principle for congruence subgroups (Q1751612)

Summary: 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 \(\mathrm{Ker}(G(\mathbb{Z}_p)\to G(\mathbb{Z}/p^n\mathbb{Z}))\)? More precisely, we will show that for \(G_{\mathbb{Q}_p}\) simply connected there exist constants \(J\geq1\) 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 \(\leq J\) 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 \textit{M. V. Nori}'s results on the structure of finite subgroups of \(\mathrm{GL}(N_0,\mathbb{F}_p)\) for large \(p\) [Invent. Math. 88, 257--275 (1987; Zbl 0632.20030)]. As an application we give a bound for the volume of the intersection of a conjugacy class in the group \(G(\hat{\mathbb{Z}})=\prod_pG(\mathbb{Z}_p)\), 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 \(G(\mathbb{Z})\).