# Ideal zeta functions associated to a family of class-2-nilpotent Lie rings

## TL;DR

This paper derives explicit formulas for ideal zeta functions of a family of class-2-nilpotent Lie rings, revealing their analytic properties and connections to Igusa functions, with implications for related zeta functions and group schemes.

## Contribution

It provides the first explicit formulas for these ideal zeta functions in terms of Igusa functions, advancing understanding of their analytic and algebraic properties.

## Key findings

- Explicit formulas for ideal zeta functions in terms of Igusa functions
- Results on analytic properties and functional equations of global zeta functions
- Connections established between ideal zeta functions and representation zeta functions

## Abstract

We produce explicit formulae for various ideal zeta functions associated to the members of an infinite family of class-$2$-nilpotent Lie rings, introduced in [1], in terms of Igusa functions. As corollaries we obtain information about analytic properties of global ideal zeta functions, local functional equations, topological, reduced, and graded ideal zeta functions, as well as representation zeta functions for the unipotent group schemes associated to the Lie rings in question.