Legitimacy of reducing mod p a complex multiplication action of an elliptic curve?

Question

I scoured Silverman's two books on arithmetic of elliptic curves to find an answer to the following question, and did not find an answer:

Given an elliptic curve E defined over H, a number field, with complex multiplication by R, and P is a prime ideal in the maximal order of H and E has good reduction at P. Is it legitimate to reduce an endomorphism of E mod P?

In the chapter "Complex Multiplication" of the advanced arithmetic topics book by Silverman, a few propositions and theorems mention reducing an endomorphism mod P.

A priori, this doesn't seem trivial to me. Sure, the endomorphism is comprised of two polynomials with coefficients in H. But I still don't see why if a point Q is in the kernel of reduction mod P, why is phi(Q) also there. When I put Q inside the two polynomials, how can I be sure that P is still in the "denominator" of phi(Q)?

(*) I looked at the curves with CM by sqrt(-1), sqrt(-2) and sqrt(-3), and it seems convincing that one can reduce the CM action mod every prime, except maybe in the case of sqrt(-2) at the ramified prime.

Answer 1

I'm not sure if there's a trivial way to see this. One answer is to use the fact that every rational map from a variety X / $\mathbb{Z}_p$ to an abelian scheme is actually defined on all of X (see for instance Milne's abelian varieties notes). Here, since the generic fiber is open in X you can apply this by viewing the map you started with as a rational map.

Answer 2

Let $\mathcal{E}$ denote the Neron model of $E$ over $R$ and $k=R/P$. Thus $\mathcal{E}$ is the unique (up to isomorphism) smooth commutative group scheme over $R$ with generic fiber $E$ such that for any smooth $X/R$ the natural map $\mathcal{E}(X)\to E(X_H)$ is an isomorphism. Then "$E$ mod $P$" is the special fiber $\mathcal{E}_k = \mathcal{E}\times_R k$. (This makes sense and is canonical even if $E$ has bad reduction at $P$.) Now take $X=\mathcal{E}$ in the definition of Neron model. Your endomorphism $\varphi$ is just an element of $E(E) = \mathcal{E}(\mathcal{E})$, so extends uniquely to a morphism $\tilde{\varphi}:\mathcal{E} \to \mathcal{E}$. Base extending $\tilde{\varphi}$ by $k$ yields the reduction $\overline{\varphi}: \mathcal{E}_k \to \mathcal{E}_k$. In particular, $\overline{\varphi}$ is defined (it is "legitimate"). It even makes sense when the reduction is bad at $P$, but of course then it is a map on special fibers of Neron models.

The above all makes sense for abelian varieties as well. If you want to understand any question about reduction of abelian varieties, learning Neron models is a very good idea. Some references are Silverman's "Advanced topics in the arithmetic of elliptic curves" which you mention above (only for elliptic curves), and Bosch-Lutkebohmert-Raynaud "Neron Models" in general.