Arithmetic lattices in unipotent algebraic groups

AbstractFixing an arithmetic lattice Γ in an algebraic groupG, the commensurability growth function assigns to eachnthe cardinality of the set of subgroups Δ with[Γ:Γ∩Δ][Δ:Γ∩Δ]=n{[\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 algebraicℤ{\mathbb{Z}}-group with arithmetic lattice Γ, the Dirichlet function associated to the commensurability growth function satisfies an Euler decomposition. Moreover, the local parts are rational functions inp-s{p^{-s}}, where the degrees of the numerator and denominator are independent ofp. This gives regularity results for the set of arithmetic lattices inG.