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

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 $\mathbb{Z}^n$.