Lang's theorem (original) (raw)

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

G → G , x ↦ x − 1 σ ( x ) {\displaystyle G\to G,\,x\mapsto x^{-1}\sigma (x)} {\displaystyle G\to G,\,x\mapsto x^{-1}\sigma (x)}

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

The theorem implies that H 1 ( F q , G ) = H e ´ t 1 ( Spec ⁡ F q , G ) {\displaystyle H^{1}(\mathbf {F} _{q},G)=H_{\mathrm {{\acute {e}}t} }^{1}(\operatorname {Spec} \mathbf {F} _{q},G)} {\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 Spec ⁡ F q {\displaystyle \operatorname {Spec} \mathbf {F} _{q}} {\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 σ {\displaystyle \sigma } {\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 σ {\displaystyle \sigma } {\displaystyle \sigma } that induces a nilpotent operator on the Lie algebra of G.[2]

The Lang–Steinberg theorem

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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_−1_F(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

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Define:

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

Then, by identifying the tangent space at a with the tangent space at the identity element, we have:

( d f a ) e = d ( h ∘ ( x ↦ ( x − 1 , a , σ ( x ) ) ) ) e = d h ( e , a , e ) ∘ ( − 1 , 0 , d σ e ) = − 1 + d σ e {\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}} {\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 h ( x , y , z ) = x y z {\displaystyle h(x,y,z)=xyz} {\displaystyle h(x,y,z)=xyz}. It follows ( d f a ) e {\displaystyle (df_{a})_{e}} {\displaystyle (df_{a})_{e}} is bijective since the differential of the Frobenius σ {\displaystyle \sigma } {\displaystyle \sigma } vanishes. Since f a ( b x ) = f f a ( b ) ( x ) {\displaystyle f_{a}(bx)=f_{f_{a}(b)}(x)} {\displaystyle f_{a}(bx)=f_{f_{a}(b)}(x)}, we also see that ( d f a ) b {\displaystyle (df_{a})_{b}} {\displaystyle (df_{a})_{b}} is bijective for any b.[4] Let X be the closure of the image of f 1 {\displaystyle f_{1}} {\displaystyle f_{1}}. The smooth points of X form an open dense subset; thus, there is some b in G such that f 1 ( b ) {\displaystyle f_{1}(b)} {\displaystyle f_{1}(b)} is a smooth point of X. Since the tangent space to X at f 1 ( b ) {\displaystyle f_{1}(b)} {\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 f 1 {\displaystyle f_{1}} {\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 f a {\displaystyle f_{a}} {\displaystyle f_{a}} contains an open dense subset V of G. The intersection U ∩ V {\displaystyle U\cap V} {\displaystyle U\cap V} is then nonempty but then this implies a is in the image of f 1 {\displaystyle f_{1}} {\displaystyle f_{1}}.