The cotype zeta function of $\mathbb{Z}^d$

TL;DR

This paper derives an asymptotic formula for counting sublattices of $\

Contribution

It introduces a new asymptotic formula for sublattice enumeration using Petrogradsky's cotype zeta function, connecting subgroup growth and Cohen-Lenstra heuristics.

Findings

Provides an asymptotic count of sublattices with bounded index and rank conditions.

Links subgroup growth to Cohen-Lenstra heuristics.

Utilizes Petrogradsky's formulas for the cotype zeta function.

Abstract

We give an asymptotic formula for the number of sublattices $\Lambda \subseteq \mathbb{Z}^d$ of index at most $X$ for which $\mathbb{Z}^d/\Lambda$ has rank at most $m$, answering a question of Nguyen and Shparlinski. We compare this result to recent work of Stanley and Wang on Smith Normal Forms of random integral matrices and discuss connections to the Cohen-Lenstra heuristics. Our arguments are based on Petrogradsky's formulas for the cotype zeta function of $\mathbb{Z}^d$, a multivariable generalization of the subgroup growth zeta function of $\mathbb{Z}^d$.