Zeta functions related to the pro-p group SL1(Delta(p))

Abstract

Let D-p be a central simple Q(p)-division algebra of index 2, with maximal Z(p)-order Delta(p). We give an explicit formula for the number of subalgebras of any given finite index in the Z(p) Lie algebra L := sl(1) (Delta(p)). From this we obtain a closed formula for the zeta function zeta(L)(s) := Sigma(M less than or equal to L) \ L: M\(-s). The results are extended to the p-power congruence subalgebras of L, and as an application we obtain the zeta functions of the corresponding congruence subgroups of the uniform pro-p group SL12 (Delta(p)).