Why is reduction map $\phi: End(E)\rightarrow End(\widetilde{E})$ a ring homomorphism? Let $E/L$ be an elliptic curve over a number field $L$. Let $\mathfrak{B}$ be a maximal ideal of the ring of integers, and let $E$ have good reduction at $\mathfrak{B}$. In Silverman's Advanced Topic in the Arithmetic of Elliptic Curves, Proposition II.4.4 yields that the natural reduction (wrt $\mathfrak{B}$) morphism $\phi: End(E)\rightarrow End(\widetilde{E})$ is injective. As far as I can see the proof only yields that $\phi$ is an injective morphism of sets.
However soon after in the text, e.g. in Lemma II.5.2 it is heavily insinuated that $\phi$ is a morphism of rings. I don't see a priori why this should hold, especially when $End(E)\supsetneq \mathbb{Z}$.
Question: Why is reduction map $\phi: End(E)\rightarrow End(\widetilde{E})$ a ring homomorphism? Bonus points if this can be shown in as low-tech a way as possible.
 A: The high-tech solution is as follows: since $E$ has good reduction at $\mathfrak{B}$, it has a smooth proper Néron model $\mathcal{E}$ over the valuation ring $\mathcal{O}$ of $L$ at $\mathfrak{B}$. Note that the reduction $\tilde{E}$ of $E$ mod $\mathfrak{B}$ is exactly the special fiber of the Néron model.
The Néron model property is as follows: for every separated scheme $X$ smooth over $\mathcal{O}$, the restriction to the generic fiber $\mathrm{Hom}_{\mathcal{O}}(X,\mathcal{E}) \rightarrow \mathrm{Hom}_L(X_L,E)$ is a bijection.
Here, the Néron model is easy to construct: take a good (homogeneous) Weierstrass equation for $\mathfrak{B}$ (ie with coefficients in $\mathcal{O}$) and nonsingular mod $\mathfrak{B}$; then consider the corresponding $\mathrm{Proj}$ over $\mathcal{O}$. But I’m not sure how easy the complete property is to show even in this very special case, which is why I’m calling this “high-tech”.
Anyway, the Néron model property shows that we have a unit section $e \in \mathcal{E}(\mathcal{O})$, a multiplication $m :\mathcal{E} \times_{\mathcal{O}} \mathcal{E} \rightarrow \mathcal{E}$ and an inverse map $i: \mathcal{E} \rightarrow \mathcal{E}$, all extending the “elliptic curve data” on $E$. All the expected relations hold thanks to the Néron model property, since they hold over the generic fiber and $\mathcal{E}$ is smooth over $\mathcal{O}$. Note that the special fibers of $e,m,i$ are the expected operations on $\tilde{E}$.
In particular, we can define a notion of $\mathcal{O}$-endomorphisms of $\mathcal{E}$ (with addition and composition), as those $\mathcal{O}$-maps $\mathcal{E} \rightarrow \mathcal{E}$ that commute to $m$. But by the Néron model property, this occurs iff the morphism is an endomorphism on the generic fiber.
It follows that the restriction to the generic fiber $\mathrm{End}(\mathcal{E}) \rightarrow \mathrm{End}(E)$ is a ring isomorphism, and that we also have a natural ring homomorphism of restriction to the special fiber $\mathrm{End}(\mathcal{E}) \rightarrow \mathrm{End}(\tilde{E})$. Compose the inverse of the first map and the second, and we’re done.
