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.

- It is not too hard to see that $L_q$ is etale by computing its differential, but why is $L_q$ finite?
- 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?

anyhomomorphism $f:G \rightarrow H$ between finite type group schemes over a field (with no smoothness or commutativity hypotheses) is finite if its kernel has finitely many geometric points. When $G$ and $H$ are moreover smooth then you should show to yourself that such an $f$ is flat. One needs such facts to have a robust theory in positive characteristic, and they are good exercises in understanding quotients. $\endgroup$ – user30379 Jan 28 '13 at 0:03