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 , then, writing σ : G → G , x ↦ x q for the Frobenius, the morphism of varieties
G → G , x ↦ x − 1 σ ( x ) is surjective. Note that the kernel of this map (i.e., G = G ( F q ¯ ) → G ( F q ¯ ) ) is precisely G ( F q ) .
The theorem implies that H 1 ( F q , G ) = H e ´ t 1 ( Spec F q , G ) vanishes, and, consequently, any G-bundle on Spec 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 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 that induces a nilpotent operator on the Lie algebra of G.
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 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.
Define:
f a : G → G , f a ( x ) = x − 1 a σ ( x ) . Then (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 where h ( x , y , z ) = x y z . It follows ( d f a ) e is bijective since the differential of the Frobenius σ vanishes. Since f a ( b x ) = f f a ( b ) ( x ) , we also see that ( d f a ) b is bijective for any b. Let X be the closure of the image of f 1 . The smooth points of X form an open dense subset; thus, there is some b in G such that f 1 ( b ) is a smooth point of X. Since the tangent space to X at 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 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 contains an open dense subset V of G. The intersection U ∩ V is then nonempty but then this implies a is in the image of f 1 .
