Zeta functions of 3-dimensional 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.