In most books on elliptic curve like Silverman or Lang, the author only discuss reduction of elliptic curve over local field. My question is: what about number fields?
Suppose that $E$ is an elliptic curve defined over a number field $K$ and let $\mathfrak{p}$ be a prime in ring of integers of $K$ where $E$ has good reduction. For simplicity, assume $\mathfrak{p} = (\pi)$ is principal; otherwise, replace $K$ by its Hilbert class field. Let us denote $E_\pi$ the elliptic curve over residue field $\kappa(\mathfrak{p})$ at $\mathfrak{p}$ obtained by reducing the equation of $E$ modulo $\pi$ (let me called this geometric reduction). Also, define the set $$E(K)/\pi = \{P \bmod \pi \;|\; P \in E(K)\}$$ where $P \bmod \pi$ is obtained by clearing powers of $\pi$ in denominators of the coordinates of $P$ and reducing the resulting point modulo $\pi$ (I shall called this process arithmetic reduction).
If $K$ is local field, these two kinds of reduction coincide due to Hensel's lemma i.e. there is a bijection between $E(K)/\pi$ and $E_\pi(\kappa(\mathfrak{p}))$ but for global field like number field, this might not be true due to failure of local-global principle. (We still have $E(K)/\pi \subseteq E_\pi(\kappa(\mathfrak{p}))$ for obvious reason.) (I don't have an example for failure in number field case so it would be great to have a pointer to one.)
What I want to target is the situation where $K = \overline{\mathbb{Q}}$ with ring of integers $R = \{\text{algebraic integers}\}$ which can be viewed as direct limit of number fields. A prime $\mathfrak{p}$ of $K$ is now a compatible system $(\mathfrak{p}_L)$ for each number field $L$. We then have a directed system of sets $E(M)/\pi_M$ where $M$ runs over number fields where $\mathfrak{p} \cap M$ is principal so we can take $E(\overline{\mathbb{Q}})/\mathfrak{p}$ as direct limit of those sets which can be viewed as a set of points in $\mathbb{P}^2(\overline{\mathbb{F}_p})$ where $(p) = \mathfrak{p} \cap \mathbb{Z}$.
The question is: What can we say about $E(\overline{\mathbb{Q}})/\mathfrak{p}$ and $E_{\mathfrak{p}}$?