Ideal growth in amalgamated powers of nilpotent rings of class two and zeta functions of quiver representations

TL;DR

This paper establishes explicit rational formulas for ideal zeta functions of amalgamated powers of class two nilpotent rings and quiver representations, revealing uniform rationality over integers.

Contribution

It extends rationality results to amalgamated powers of nilpotent rings and quiver representations, providing explicit formulas and uniformity over integers.

Findings

Explicit rational functions for zeta functions of amalgamated powers.

Rationality results hold uniformly over integers.

Applicable to graded submodules of graded modules.

Abstract

Let $L$ be a nilpotent algebra of class two over a compact discrete valuation ring $A$ of characteristic zero or of sufficiently large positive characteristic. Let $q$ be the residue cardinality of $A$. The ideal zeta function of $L$ is a Dirichlet series enumerating finite-index ideals of $L$. We prove that there is a rational function in $q$, $q^m$, $q^{-s}$, and $q^{-ms}$ giving the ideal zeta function of the amalgamation of $m$ copies of $L$ over the derived subring, for every $m \geq 1$, up to an explicit factor. More generally, we prove this for the zeta functions of nilpotent quiver representations of class two defined by Lee and Voll, and in particular for Dirichlet series counting graded submodules of a graded $A$-module. If the algebra $L$, or the quiver representation, is defined over $\mathbb{Z}$, then we obtain a uniform rationality result.