Let $G$ be a connected commutative algebraic group over $\mathbb{F}\_q$. If $\text{Fr}\_q : G \to G$ denotes the $q$-Frobenius morphism, we define the *Lang isogeny* $L\_q$ to be the endomorphism of $G$ given by $g \mapsto \text{Fr}\_q(g)g^{-1}$.


I have two questions about this important map.


1. It is not too hard to see that $L\_q$ is etale by computing its differential, but why is $L\_q$ finite?
2. Granted that $L\_q$ is a finite Galois covering with group $G(\mathbb{F}\_q)$, we get a surjection $\pi\_1(G,\overline{1}) \to G(\mathbb{F}\_q)$. What can we say about the kernel of this homomorphism?


Is there a good modern reference for basic results about $L\_q$? If not, would someone kindly explain these two points to me?



The Lang isogeny
1. Every etale morphism is finite over some nonempty open set. (For instance, locally somewhere write it as a standard etale morphism $(A[t]/f(t))\_{g(t)}$, then consider the open set where the norm of $g$, that is, the resultant of $f$ and $g$, is nonzero.) If a group homomorphism $H \to G$ is finite over some nonempty open set, it is finite over all translates of that set, so it is finite everywhere.
2. It is isomorphic to $\pi\_1(G,\overline{1})$. In general, if $X \to Y$ is finite etale and Galois, the kernel of the map $\pi\_1(Y) \to Gal(X/Y)$ is $\pi\_1(X)$.


