An approximation principle for congruence subgroups

  • Tobias Finis

    Universität Leipzig, Germany
  • Erez Lapid

    Weizmann Institute of Science, Rehovot, Israel
An approximation principle for congruence subgroups cover
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The motivating question of this paper is roughly the following: given a flat group scheme GG over Zp\mathbb Z_p, pp prime, with semisimple generic fiber GQpG_{\mathbb Q_p}, how far are open subgroups of G(Zp)G(\mathbb Z_p) from subgroups of the form X(Zp)Kp(pn)X(\mathbb Z_p)\mathbf K_p(p^n), where XX is a subgroup scheme of GG and Kp(pn)\mathbf K_p(p^n) is the principal congruence subgroup Ker(G(Zp)G(Z/pnZ))(G(\mathbb Z_p)\to G(\mathbb Z/p^n\mathbb Z))? More precisely, we will show that for GQpG_{\mathbb Q_p} simply connected there exist constants J1J\ge1 and ε>0\varepsilon>0, depending only on GG, such that any open subgroup of G(Zp)G (\mathbb Z_p) of level pnp^n admits an open subgroup of index J\le J which is contained in X(Zp)Kp(pεn)X(\mathbb Z_p)\mathbf K_p(p^{\lceil \varepsilon n\rceil}) for some proper, connected algebraic subgroup XX of GG defined over Qp\mathbb Q_p. Moreover, if GG is defined over Z\mathbb Z, then ε\varepsilon and JJ can be taken independently of pp.

We also give a correspondence between natural classes of Zp\mathbb Z_p-Lie subalgebras of gZp\mathfrak {g}_{\mathbb Z_p} and of closed subgroups of G(Zp)G(\mathbb Z_p) that can be regarded as a variant over Zp\mathbb Z_p of Nori's results on the structure of finite subgroups of GL(N0,Fp)(N_0,\mathbb F_p) for large pp [Nor 87].

As an application we give a bound for the volume of the intersection of a conjugacy class in the group G(Z^)=pG(Zp)G (\hat{\mathbb Z}) = \prod_p G (\mathbb Z_p), for GG defined over Z\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(Z)G (\mathbb Z).

Cite this article

Tobias Finis, Erez Lapid, An approximation principle for congruence subgroups. J. Eur. Math. Soc. 20 (2018), no. 5, pp. 1075–1138

DOI 10.4171/JEMS/783