Enumerating submodules invariant under an endomorphism

TL;DR

This paper investigates zeta functions counting submodules invariant under endomorphisms of modules over number rings, providing explicit formulas, meromorphic continuation, and applications to nilpotent Lie algebra ideal zeta functions.

Contribution

It offers explicit formulas involving Dedekind zeta functions and proves meromorphic continuation for these zeta functions, advancing understanding of their analytic properties.

Findings

Explicit formulas involving Dedekind zeta functions

Meromorphic continuation of the zeta functions

Ideal zeta functions for nilpotent Lie algebras have abscissa of convergence 2

Abstract

We study zeta functions enumerating submodules invariant under a given endomorphism of a finitely generated module over the ring of ($S$-)integers of a number field. In particular, we compute explicit formulae involving Dedekind zeta functions and establish meromorphic continuation of these zeta functions to the complex plane. As an application, we show that ideal zeta functions associated with nilpotent Lie algebras of maximal class have abscissa of convergence $2$.