Arithmetic lattices in unipotent algebraic groups

Fixing an arithmetic lattice $\mathrm{\Gamma }$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\mathrm{\Delta }$ with $[\Gamma : \Gamma \cap \Delta] [\Delta: \Gamma \cap \Delta] = n$. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smith’s “Subgroups of finite index in nilpotent groups” apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic $\mathbb{Z}$-group with arithmetic lattice $\mathrm{\Gamma }$, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions in $p^{-s}$, where the degrees of the numerator and denominator are independent of $p$. This gives regularity results for the set of arithmetic lattices in $G$.