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Abstract: Let O be a compact discrete valuation ring of characteristic 0. Given a module M of matrices over O, we study the generating function encoding the average sizes of the kernels of the elements of M over finite quotients of O. We prove rationality and establish fundamental properties of these generating functions and determine them explicitly for various natural families of modules M. Using p‐adic Lie theory, we then show that special cases of these generating functions enumerate orbits and conjugacy classes of suitable linear pro‐ p groups. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Building on our previous work, we develop the first practical algorithm for computing topological zeta functions of nilpotent groups, possibly 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. [ABSTRACT FROM AUTHOR]