Reduction mod $p$ map is injective?

Question

Let $E$ be an elliptic curve over $\mathbb{Q}$ with good or multiplicative reduction at $p$. How do I see that the reduction mod $p$ map from the endomorphisms of $E$ over $\overline{\mathbb{Q}}$ to the endomorphisms of the reduction of $E$ mod $p$ over $\overline{\mathbb{F}}$ is injective?

Answer 1

[The following is a vast simplification of an answer I gave a bit earlier, so I deleted most of the earlier answer. For anyone who cares about the sanctity of upvoting points it is all OK since I am certain anyone who upvoted the earlier answer would upvote this one too.]

Any specific endomorphism is defined over an actual number field $K$ inside $\overline{\mathbf{Q}}$, and in case of semistable reduction the formation of the identity component of the special fiber of the Neron model commutes with any local extension of the discrete valuation ring. Thus, we can work relative to the discrete valuation ring $O_{K, \mathfrak{p}}$ (localization at a prime over $p$). It is sufficient to prove more generally that if $A$ and $B$ are abelian varieties (or just elliptic curves) over the fraction field $F$ of a discrete valuation ring $R$ with residue field $k$ and their Neron models $\mathcal{A}$ and $\mathcal{B}$ over $R$ have $k$-fibers with semistable reduction then the natural map $${\rm{Hom}}_F(A,B) = {\rm{Hom}}_R(\mathcal{A}, \mathcal{B}) \rightarrow {\rm{Hom}}_k(\mathcal{A}^0_k, \mathcal{B}^0_k)$$ is injective. In what follows one can work throughout with elliptic curves if that is preferred, and the main idea can be carried out in more elementary terms for elliptic curves, but I give the general case because it is more attractive.

The first thing to observe is that if $n$ is a nonzero integer then its effect on the semi-abelian identity components of the special fibers of the Neron models is an isogeny (by viewing such identity components as extensions of abelian varieties by tori, and noting that a nonzero integer acts as an isogeny on any torus and on any abelian variety).

The next thing to observe is since an isogeny of abelian varieties admits a map in the opposite direction such that the composition in both directions is multiplication by a common nonzero integer, the preceding observation about the effect of nonzero integers on the identity components of the special fibers implies that an isogeny between abelian varieties over $F$ with semistable reduction induces an isogeny between identity components of special fibers of the Neron model. (This already settles the case of elliptic curves in the question posed above.)

Consequently, for our purpose we can replace either of $A$ or $B$ with an $F$-isogenous abelian variety (which always has semistable reduction too: this is well-known for elliptic curves, and follows from Grothendieck's inertial semistable reduction criterion in general). Thus, in case either of $A$ or $B$ are not $F$-simple (though they are $F$-simple for the case of elliptic curves) we can apply the Poincare reducibility theorem to reduce to the case when both are $F$-simple. But then any nonzero $F$-homomorphism between them is an isogeny and hence as we have seen above induces an isogeny between identity components of special fibers of the Neron models. In particular, the induced map between such special fibers is nonzero.