The following result can be found in Lang's Elliptic functions book (Chapter 13, Theorem 14):
Let $A_0$ be an elliptic curve in characteristic $p$, with an endomorphism $\alpha_0$ which is not trivial. Then there exists an elliptic curve $A$ defined over a number field, an endomorphism $\alpha$ of $A$, and a non-degenerate reduction of A at a place $\mathfrak{p}$ lying above $p$, such that $A_0$ is isomorphic to $\overline{A}$, and $\alpha_0$ corresponds to $\overline{\alpha}$ under the isomorphism.
In the proof, I don't understand how we get from a curve $A$ with transcendental invariant $j$ over $\mathbb{Q}$ to a curve over a number field. Is that because the curve $A(j_1)$ is defined over a number field? (I don't think it is true.)
On the other hand, I wonder if there is a simpler proof for this theorem. I think one can obtain an "easier" proof by simply consider the order $L$ in $\text{End}(A_0)$ containing $\alpha_0$ and take the curve $\mathbb{C}/L$. The problem is of course to prove that it is indeed defined over a number field i.e. $g_2(L)$ and $g_3(L)$ are algebraic integers. That seems to be hard.
Another question is about the statement of the theorem: It does not mention that $A_0$ being defined over finite field but the proof does assume that. Certainly, I do not think we can lift curves over transcendental extension of $\mathbb{F}_p$ such as $\mathbb{F}(t)$ to number fields. Is there an analogue for general field of characteristic $p$?