Zeta functions of 3-dimensional p-adic Lie algebras

TL;DR

This paper derives an explicit formula for the subalgebra zeta function of 3-dimensional p-adic Lie algebras using Igusa's local zeta functions linked to associated ternary quadratic forms.

Contribution

It provides a novel explicit formula connecting subalgebra zeta functions of 3D p-adic Lie algebras with Igusa's local zeta functions via ternary quadratic forms.

Findings

Explicit formula for subalgebra zeta functions

Connection to Igusa's local zeta functions

Application to general 3D p-adic Lie algebras

Abstract

We give an explicit formula for the subalgebra zeta function of a general 3-dimensional Lie algebra over the p-adic integers $\mathbb{Z}_p$. To this end, we associate to such a Lie algebra a ternary quadratic form over $\mathbb{Z}_p$. The formula for the zeta function is given in terms of Igusa's local zeta function associated to this form.