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) .