Zeta functions of three-dimensional p-adic Lie algebras

Abstract

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