I'm trying to study the Picard's Little Theorem, from here I noticed an interesting (also seems very simple) way of showing theorem, which is introduced by @Matt E also from here a likely proof. But there is some points which are not so clear to me.
Please allow me to cite the proof from here:
For this proof, we use the fact that there is a holomorphic covering map :$\phi: \Bbb D \to \Bbb C \backslash \{0, 1\}$
where $\Bbb D = \{z \in \Bbb C: |z| < 1\}$.
This follows from the Riemann Uniformization Theorem, but is much easier to prove.
Indeed, such a covering is given by the elliptic modular function.
Now suppose that an entire function $f: \Bbb C \to \Bbb C$ omits two different complex values.
Assume for simplicity that the omitted values are $0$ and $1$. This can always be achieved by postcomposing $f$ with a suitable complex affine map.
Because $\Bbb C$ is simply connected and $\phi$ is a covering, we can lift $f$ to a holomorphic function: :$F: \Bbb C \to \Bbb D$ with $\phi \circ F = f$.
By Liouville's Theorem, $F$ is constant. Therefore $f$ is also constant.
I tried to search it everywhere but it only makes things more complecated.
My first question is both of them (@Matt E and ProofWiki)mentioned the holomorphic covering map is given by a modular function(Without considering the Uniformization theorem, I think this might make the proof much harder), but what is this function and how it maps $\Bbb C \backslash \{0,1\}$ to unit circle? Then, at the end they mentioned "we can lift $f$ to a holomorphic function", I believe this follows Lifting lemma, but I can't find the suitable one, how can we prove this is true for holomorphic case?
Thank you very much fro your help.
