# The Lang isogeny

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 $g \mapsto \text{Fr}_q(g)g^{-1}$.

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?

• So $G$ is assumed to be smooth? You should step away from the specificity of the Lang isogeny and commutativity and prove to yourself that any homomorphism $f:G \rightarrow H$ 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. – user30379 Jan 28 '13 at 0:03

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 $\pi_1(X)$.