Counting arithmetic subgroups and subgroup growth of virtually free groups

Let K be a p-adic field, and let H = PSL₂(K) endowed with the Haar measure determined by giving a maximal compact subgroup measure 1. Let AL_H(x) denote the number of conjugacy classes of arithmetic lattices in H with co-volume bounded by x. We show that under the assumption that K does not contain the element ζ + ζ^-1, where ζ denotes the p-th root of unity over ℚ_p, we have (Formula presented.) where q denotes the order of the residue field of K. The main innovation of this paper is the proof of a sharp bound on subgroup growth of lattices in H as above.