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ABSTRACT: Let $L$ be a finite-dimensional $\mathbb{Z}$-Lie algebra (also called a Lie ring), and for a prime $p$ let $L_p:=L\otimes \mathbb{Z}_p$, where $\mathbb{Z}_p$ is the usual ring of $p$-adic integers. We call $\zeta_{L_{p}}^{\triangleleft}(s):=\sum_{H\triangleleft L_{p}}|L_p:H|^{-s}$ the local ideal zeta functions of $L$, enumerating ideals of finite index in $L_p$. In this talk, we show how these local factors $\zeta_{L_{p}}^{\triangleleft}(s)$ can be expressed in terms of certain $p$-adic integrals, allowing us to associate $L$ with certain algebraic varieties intrinsically encoded in its structure. Then we discuss how, under certain conditions, one can control and study these varieties.