Local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms

TL;DR

This paper establishes a criterion for local functional equations of submodule zeta functions linked to nilpotent algebras of endomorphisms, with implications for ideal zeta functions of nilpotent Lie lattices and subgroup enumeration in nilpotent groups.

Contribution

It provides a sufficient condition for functional equations in submodule zeta functions and proves several conjectures for ideal zeta functions of nilpotent Lie lattices.

Findings

Proved functional equations for ideal zeta functions of nilpotent Lie lattices.

Confirmed conjectures on local functional equations for submodule zeta functions.

Applied results to enumerate normal subgroups in finitely generated nilpotent groups.

Abstract

We give a sufficient criterion for generic local functional equations for submodule zeta functions associated to nilpotent algebras of endomorphisms defined over number fields. This allows us, in particular, to prove various conjectures on such functional equations for ideal zeta functions of nilpotent Lie lattices. Via the Mal'cev correspondence, these results have corollaries pertaining to zeta functions enumerating normal subgroups of finite index in finitely generated nilpotent groups, most notably finitely generated free nilpotent groups of any given class.