Siegel's theorem states the following:

Let $C$ be a smooth projective curve over a number field $K$. Let $\tilde C\subset C$ be an open affine subvariety, and $i:\tilde C\hookrightarrow \mathbb{A}^m_K$ be a closed immersion. Then if $i(\tilde C)$ lies over infinitely many $\mathbb{A}^m_{\mathcal{O}_K}(\mathcal{O}_K)$-points, then the genus of $C$ is $0$, and furthermore $|C(\bar{\mathbb{Q}})\smallsetminus \tilde C(\bar{\mathbb{Q}})|\leq 2$.

In order for me to explain what I find confusing about this statement, consider the following definition:

Call a subset $S$ of $C(K)$ *Siegel* if there exists an open affine $\tilde C\subset C$ and a closed immersion $i:\tilde C\hookrightarrow \mathbb{A}^m_K$ so that the points of $S$ are exactly the $K$-points of $\tilde C$ lying above $\mathcal{O}_K$-points of $\mathbb{A}^m_{\mathcal{O}_K}$.

### Question

- Can you give a reasonably intuitive characterization of the Siegel subsets of $C(K)$? For example, can you re-phrase the definition of a Siegel set in a manner that avoids reference to affine space, and only deals with $\mathcal{O}_K$-models of $C$ or $\tilde C$?
- What can one say about the set of Siegel sets? Is it true that any two maximal Siegel sets are disjoint? Is the set of maximal Siegel sets an equivalence relation?
- Can one describe maximal Siegel sets in some nice fashion? Presumably $C(K)$ itself is not a Siegel set in general, because if that were the case then Siegel's theorem would imply Falting's theorem, which is apocryphal.