Let $C/\mathbb{Z}_p$ be a non-singular, projective, curve, with good reduction. Now, let $P\in C(\mathbb{F}_p)$ be a point of its special fibre, and denote by $\mathcal{D}$ the residue disk of $C$, which consists of points $Q\in C(\mathbb{Z}_p)$, which reduce to $P$.
It is said that the residue disk $\mathcal{D}$ may be parameterized by the ideal $\langle p\rangle \in \text{Spec}\mathbb{Z}_p$. How is this parameterization constructed? This strikes me as some type of a generalization of Hensel's Lemma, but I am not sure how to approach this in most generality.