I'm reading Ehlen's Singular moduli of higher level and special cycles (arXiv link) which uses the following definition of an elliptic curve:
Definition 1.2: Let $S$ be a scheme. A proper smooth curve $E \to S$ with geometrically connected fibers of genus one together with a section $0 : S \to E$, is called an elliptic curve.
I'm somewhat (not super) familiar/comfortable with schemes, and I know enough to see that this is a mild generalization of the definition given in say, Silverman's books. The author later considers a moduli problem and writes:
We consider the moduli problem which assigns to a base scheme $S$ over $\mathcal{O}_D$ the category $C_D^+(S)$ of pairs $(E, \iota)$, where
(i) $E$ is an elliptic curve over $S$ with complex multiplication $\iota : \mathcal{O}_D \hookrightarrow \operatorname{End}(E)$,
(ii) such that the induced map $$\operatorname{Lie}(\iota) : \mathcal{O}_D \to \operatorname{End}_{\mathcal{O}_S}(\operatorname{Lie} E) = \mathcal{O}_S$$ coincides with the structure map $S \to \operatorname{Spec} \mathcal{O}_D$.
There are two pieces of notation I'm not familiar with here: one is the use of $\operatorname{Lie}$ - I'm not sure what $\operatorname{Lie}(\iota)$ and $\operatorname{Lie} E$ mean, and the other is $\mathcal{O}_S$ - here I suspect this is the structure sheaf for $S$, but I want to clarify that point.
Neither of the algebraic geometry books I have handy include $\operatorname{Lie}$ in their symbol indices, and I haven't had any luck searching online, so would anyone be able to explain what this means, or point me to the right resource on the topic?