Topological representation zeta functions of unipotent groups

TL;DR

This paper introduces topological representation zeta functions for unipotent groups over number fields, providing a new invariant that generalizes p-adic zeta functions and enabling classification of low-dimensional cases.

Contribution

It defines and studies the properties of topological representation zeta functions, offering a computational method and classifying all such functions for unipotent groups up to dimension six.

Findings

Established fundamental properties of topological zeta functions.

Developed a computation method under non-degeneracy assumptions.

Classified all topological zeta functions for unipotent groups of dimension ≤6.

Abstract

Inspired by work surrounding Igusa's local zeta function, we introduce topological representation zeta functions of unipotent algebraic groups over number fields. These group-theoretic invariants capture common features of established $p$-adic representation zeta functions associated with pro-$p$ groups derived from unipotent groups. We investigate fundamental properties of the topological zeta functions considered here. We also develop a method for computing them under non-degeneracy assumptions. As an application, among other things, we obtain a complete classification of topological representation zeta functions of unipotent algebraic groups of dimension at most six.