Arithmetic lattices in unipotent algebraic groups

TL;DR

This paper studies the growth of subgroups in arithmetic lattices within unipotent algebraic groups, revealing regularity and rationality properties of associated Dirichlet functions and their local factors.

Contribution

It introduces a new approach to analyze the subgroup growth in arithmetic lattices using methods from nilpotent group theory, establishing rationality of local factors.

Findings

Dirichlet functions decompose into Euler products

Local factors are rational functions in p^{-s}

Degrees of numerator and denominator are p-independent

Abstract

Fixing an arithmetic lattice $\Gamma$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $\Delta$ 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 $\Gamma$, 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 $G$.