I want to understand the proof of the fact
Genus $1$ curve over $\Bbb{F}_q$ always have rational points
The result was originally proven by F.K. Schmidt and is currently presented as Exercise $10.6$ in Silverman's 'The arithmetic of elliptic curves'.
Let $C/\Bbb{F}_q$ be an arbitrary genus $1$ curve. $C/\overline{\Bbb{F}_q}$ can be regarded as an elliptic curve because it has rational point.
Let $\phi:C\to C$ be the $q$-th power Frobenius map on $C$.
I want to prove that
There are an endomorphism $f\in End(C)$ and a point $P_0\in C(\overline{\Bbb{F}_q})$ satisfying $\phi(P)=f(P)+P_0$ and there exits a point $P_1 \in C(\overline{\Bbb{F}_q})$ such that $(1-f)(P_1)=P_0$.
If we could prove this, $\phi(P_1)=P_1$ and hence that $P_1\in C(\Bbb{F}_q)$. Thus $C$ turns out to be an elliptic curve over $\Bbb{F}_q$ and we are done ■
But how can we take $f, P_1 and P_0$ ?
P.S. Thanks to comments, $P_0$ is base point and $f(P)=\phi(P)-\phi(P_0)$.
$P_0$ is the only problem.