Context: Let $\mathbb{F}$ be the algebraic closure of $\mathbb{F}_q$ for $q$ prime. We know that $\mathbb{F}(t)$ for $t$ transcendental is the function field of the projective line $\mathbb{P}^{1}(\mathbb{F})$. Assume that $K$ is a normal extension of $\mathbb{F}(t)$. Denote by $C$ the corresponding projective curve of $K$.
What is meant by the group of automorphisms of $C/\mathbb{P}^{1}(\mathbb{F})$?
Can you give me literature reference for the topic Galois Theory on Curves and covering maps (some more key words: Galois covering, groups which acts on the fiber of a covering map, ramified etc.)? As simple as possible (I am a non-expert!)
I read the subchapter about Riemann hypothesis for curves over finite fields in C. Moreno's book (google book link, page 60-61 C. Moreno, Curve over finite fields)
Some implicit indication of the author: In some other place, the author evaluates the Frobenius automorphism ($\phi(x):=x^q$ for $x \in K$) at points of $C$. RigorouslyIs this is impossible, is it in some way possible, if do some identification of $K$ and $C$?
In the amazing (as Mumford calls it) correspondence between algebra and geometry in the question, a finite extension $\mathbb F(t)\subset K$ (not necessary normal nor separable) corresponds to a smooth projective curve plus a finite morphism $\pi:C\to \mathbb P^1_\mathbb F$.
The group of automorphisms of $C/\mathbb P^1_\mathbb F$ is then simply the group of automorphisms $f:C\to C$ commuting with $f$ i.e. such that $\pi \circ f=\pi$.
As Qiaochu comments, this group (of very geometric nature) is isomorphic to the group (of very algebraic nature) of automorphisms $Aut(K/\mathbb F(t))$ .
(Do not write that last group $Gal(K/\mathbb F(t))$ because the extension $\mathbb F(t)\subset K$ needn't be Galois)
And to answer your last question: this has nothing to do with Frobenius.