Let $E$ be a non-supersingular elliptic curve over $\mathbb{F}\_p$, and $E^1$ be some elliptic curve, then $Hom\_{\bar{\mathbb{F}\_p}}(E, E^1)$ is free $\mathbb{Z}$ module of rank 2.


Can someone explain this please? I can't see how it is of rank 2. 



For a supersingular elliptic curve, $Hom_{\bar{\mathbb{F}_p}}(E, E^1)$ is free module of rank 2
Assume there is a non-zero isogeny $\phi:E\to E'$. Let $q$ be a power of $p$ so that both $E'$ and $\phi$ are defined over $\mathbb F\_q$. Let $F\_q:E'\to E'$ be the $q$-power Frobenius map. Then $\text{Hom}(E,E')$ at least contains $\{m\phi+n F\_q\circ\phi: m,n\in\mathbb Z\}$, so the rank is at least 2. I'll let you figure out why the rank can't be greater than 2. 


