Zeta functions related to the pro-$p$ group SL$_1${\bfseries (}$\Delta_{\hbox{\scriptsize{\bfseries\itshape p}}}${\bfseries )$}

Abstract

Let ${\bb D}_p$ be a central simple ${\bb Q}_p$-division algebra of index 2, with maximal ${\bb Z}_p$-order $\Delta_p$. We give an explicit formula for the number of subalgebras of any given finite index in the ${\bb Z}_p$-Lie algebra $\mathcal{L}\colone \spl_1(\Delta_p)$. From this we obtain a closed formula for the zeta function $\zeta_\mathcal{L}(s) \colone \sum_{M \leq \mathcal{L}} |\mathcal{L}:M|^{-s}$. The results are extended to the $p$-power congruence subalgebras of $\mathcal{L}$, and as an application we obtain the zeta functions of the corresponding congruence subgroups of the uniform pro-$p$ group $\SL_1^2(\Delta_p)$.