Cotype zeta functions enumerating subalgebras of $R$-algebras

Abstract

We introduce and study subalgebra cotype zeta functions, multivariate zeta functions enumerating fixed-index subalgebras of $R$-algebras of a given cotype. This generalizes and unifies previous works on subalgebra zeta functions and cotype zeta functions of $R$-algebras. We prove the local functional equations for the generic Euler factors of these zeta functions, and give an explicit formula for the subalgebra cotype zeta function of a general $\mathbb{Z}$-Lie algebra $L$ of rank 3. We also give an asymptotic formula for the number of subalgebras $\Lambda$ of $L$ of index at most $X$ for which $L/\Lambda$ has rank at most, answering a question of Chinta, Kaplan, and Koplewitz. In particular, we show that unlike ${\mathbb{Z}}^3$, $\mathbb{Z}$-Lie algebras of rank 3 with additional multiplication structure exhibit different distribution of cocyclic subalgebras.