Let $\text{char}(K) = p > 0$, and let $E/K$ be an elliptic curve with $j(E) \notin \overline{\mathbb{F}}\_p$.


Does it follow that $\text{End}(E) = \mathbb{Z}$?



$\text{char}(K) = p > 0$, $E/K$ elliptic curve with $j(E) \notin \overline{\mathbb{F}}_p$, follows that $\text{End}(E) = \mathbb{Z}$?
Yes. This seems to be due to Deuring. There are various proofs. You'll find a proof, for example, in Mumford's *Abelian Varieties*, page 217.


