Topological representation zeta functions of unipotent groups

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.Comment: 27 page

Computing topological zeta functions of groups, algebras, and modules, II

Building on our previous work (arXiv:1405.5711), we develop the first practical algorithm for computing topological zeta functions of nilpotent groups, non-associative algebras, and modules. While we previously depended upon non-degeneracy assumptions, the theory developed here allows us to overcome these restrictions in various interesting cases.Comment: 33 pages; sequel to arXiv:1405.571