Lang's theorem

In algebraic geometry, Lang's theorem, introduced by Serge Lang, states: if G is a connected smooth algebraic group over a finite field ${\displaystyle \mathbf {F} _{q}}$, then, writing ${\displaystyle \sigma :G\to G,\,x\mapsto x^{q}}$ for the Frobenius, the morphism of varieties

${\displaystyle G\to G,\,x\mapsto x^{-1}\sigma (x)}$ 

is surjective. Note that the kernel of this map (i.e., ${\displaystyle G=G({\overline {\mathbf {F} _{q}}})\to G({\overline {\mathbf {F} _{q}}})}$) is precisely ${\displaystyle G(\mathbf {F} _{q})}$.

The theorem implies that ${\displaystyle H^{1}(\mathbf {F} _{q},G)=H_{\mathrm {{\acute {e}}t} }^{1}(\operatorname {Spec} \mathbf {F} _{q},G)}$   vanishes, and, consequently, any G-bundle on ${\displaystyle \operatorname {Spec} \mathbf {F} _{q}}$ is isomorphic to the trivial one. Also, the theorem plays a basic role in the theory of finite groups of Lie type.

It is not necessary that G is affine. Thus, the theorem also applies to abelian varieties (e.g., elliptic curves.) In fact, this application was Lang's initial motivation. If G is affine, the Frobenius ${\displaystyle \sigma }$ may be replaced by any surjective map with finitely many fixed points (see below for the precise statement.)

The proof (given below) actually goes through for any ${\displaystyle \sigma }$ that induces a nilpotent operator on the Lie algebra of G.