Most subrings of $\mathbb{Z}^n$ have large corank

TL;DR

This paper investigates the algebraic structure of sublattice quotients of b^n, revealing that subring quotients tend to have large rank and rarely are cyclic when n  7, contrasting with general sublattice quotients.

Contribution

It demonstrates that subring quotients of b^n typically have large rank for large n, using a combination of analytic number theory and combinatorics.

Findings

Subring quotients rarely cyclic for n  7.

Quotients generally have very large rank as n increases.

Analysis of zeta functions and Hermite normal form matrices underpins results.

Abstract

If $\Lambda \subseteq \mathbb{Z}^n$ is a sublattice of index $m$, then $\mathbb{Z}^n/\Lambda$ is a finite abelian group of order $m$ and rank at most $n$. Several authors have studied statistical properties of these groups as we range over all sublattices of index at most $X$. In this paper we investigate quotients by sublattices that have additional algebraic structure. While quotients $\mathbb{Z}^n/\Lambda$ follow the Cohen-Lenstra heuristics and are very often cyclic, we show that if $\Lambda$ is actually a subring, then once $n \ge 7$ these quotients are very rarely cyclic. More generally, we show that once $n$ is large enough the quotient typically has very large rank. In order to prove our main theorems, we combine inputs from analytic number theory and combinatorics. We study certain zeta functions associated to $\mathbb{Z}^n$ and also prove several results about matrices in Hermite normal form whose columns span a subring of $\mathbb{Z}^n$.