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,^[1] 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.^[2]

The Lang–Steinberg theorem

Steinberg (1968) gave a useful improvement to the theorem.

Suppose that F is an endomorphism of an algebraic group G. The Lang map is the map from G to G taking g to g⁻¹F(g).

The Lang–Steinberg theorem states^[3] that if F is surjective and has a finite number of fixed points, and G is a connected affine algebraic group over an algebraically closed field, then the Lang map is surjective.

Proof of Lang's theorem

Define:

${\displaystyle f_{a}:G\to G,\quad f_{a}(x)=x^{-1}a\sigma (x).}$ 

Then (identifying the tangent space at a with the tangent space at the identity element) we have:

${\displaystyle (df_{a})_{e}=d(h\circ (x\mapsto (x^{-1},a,\sigma (x))))_{e}=dh_{(e,a,e)}\circ (-1,0,d\sigma _{e})=-1+d\sigma _{e}}$  

where ${\displaystyle h(x,y,z)=xyz}$ . It follows ${\displaystyle (df_{a})_{e}}$  is bijective since the differential of the Frobenius ${\displaystyle \sigma }$  vanishes. Since ${\displaystyle f_{a}(bx)=f_{f_{a}(b)}(x)}$ , we also see that ${\displaystyle (df_{a})_{b}}$  is bijective for any b.^[4] Let X be the closure of the image of ${\displaystyle f_{1}}$ . The smooth points of X form an open dense subset; thus, there is some b in G such that ${\displaystyle f_{1}(b)}$  is a smooth point of X. Since the tangent space to X at ${\displaystyle f_{1}(b)}$  and the tangent space to G at b have the same dimension, it follows that X and G have the same dimension, since G is smooth. Since G is connected, the image of ${\displaystyle f_{1}}$  then contains an open dense subset U of G. Now, given an arbitrary element a in G, by the same reasoning, the image of ${\displaystyle f_{a}}$  contains an open dense subset V of G. The intersection ${\displaystyle U\cap V}$  is then nonempty but then this implies a is in the image of ${\displaystyle f_{1}}$ .

Notes

1. This is "unwinding definition". Here, ${\displaystyle H^{1}(\mathbf {F} _{q},G)=H^{1}(\operatorname {Gal} ({\overline {\mathbf {F} _{q}}}/\mathbf {F} _{q}),G({\overline {\mathbf {F} _{q}}}))}$  is Galois cohomology; cf. Milne, Class field theory.
2. Springer 1998, Exercise 4.4.18.
3. Steinberg 1968, Theorem 10.1
4. This implies that ${\displaystyle f_{a}}$  is étale.

References

• Springer, T. A. (1998). Linear algebraic groups (2nd ed.). Birkhäuser. ISBN 0-8176-4021-5. OCLC 38179868.
• Lang, Serge (1956), "Algebraic groups over finite fields", American Journal of Mathematics, 78: 555–563, doi:10.2307/2372673, ISSN 0002-9327, JSTOR 2372673, MR 0086367
• Steinberg, Robert (1968), Endomorphisms of linear algebraic groups, Memoirs of the American Mathematical Society, No. 80, Providence, R.I.: American Mathematical Society, MR 0230728