Counting subalgebras of ${\mathfrak{o}}^n$

Let $\mathfrak{o}$ be a compact discrete valuation ring and $n\ge 2$. We introduce a method to study the cotype zeta function of subalgebras of ${\mathfrak{o}}^n$. This multivariable series encodes the number of finite-index subalgebras $\mathrm{\Lambda }$ of the $\mathfrak{o}$-algebra ${\mathfrak{o}}^n$ of a given elementary divisor type. We express this zeta function as a finite sum of $\mathfrak{o}$-adic integrals and compute these integrals in many cases. As a first application, we recover known results in a natural way from our approach. For instance, we obtain a lower bound for the abscissa of convergence of the subalgebra zeta function of ${\mathfrak{o}}^n$ by exhibiting an explicit pole. We also determine the number of irreducible subrings of ${\mathfrak{o}}^n$ of small index. As a second application, we give an explicit formula for the cotype zeta function of subalgebras of ${\mathfrak{o}}^4$.