Subgroup growth and congruence subgroups

Let k be a global field, O its ring of integers, G an almost simple, simply connected, connected algebraic subgroups of GL_m, defined over k and Γ=G(O) which is assumed to be infinite. Let σ_n(G{cyrillic}) (resp. γ_n(G{cyrillic}) be the number of all (resp. congruence) subgroups of index at most n in Γ. We show:(a) If char(k)=0 then: (i) {Mathematical expression} for suitable constants C₁ and C₂. Under some mild assumptions we also have: (ii) Γ has the congruence subgroup property if and only if log σ_n(G{cyrillic})=o(log²n). (iii) If Γ is boundedly generated the Γ has the congruence subgroup property. (This confirms a conjecture of Rapinchuk [R1] which was also proved by Platonov and Rapinchuk [PR3].) (b) If char(k)>0 (and under somewhat stronger conditions on G) then for suitable constants C₃ and C₄, C₃ log²n≦log γn(Γ)≦C₄log³n.