Computing local zeta functions of groups, algebras, and modules

TL;DR

This paper introduces a practical method for computing local zeta functions of algebraic structures, enabling classification and analysis of specific cases like unipotent groups and subalgebras.

Contribution

The paper presents a new computational approach for local zeta functions, providing complete classifications for certain algebraic groups and subalgebras, and exploring graded subobject zeta functions.

Findings

Complete classification of generic local representation zeta functions for unipotent groups up to dimension six.

Determination of local subalgebra zeta functions for rak{gl}_2(\u211d).

Introduction and computation of graded subobject zeta functions.

Abstract

We develop a practical method for computing local zeta functions of groups, algebras, and modules in fortunate cases. Using our method, we obtain a complete classification of generic local representation zeta functions associated with unipotent algebraic groups of dimension at most six. We also determine the generic local subalgebra zeta functions associated with $\mathfrak{gl}_2(\mathbf{Q})$. Finally, we introduce and compute examples of graded subobject zeta functions.