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

for the Frobenius, the morphism of varieties is surjective.

Note that the kernel of this map (i.e.,

The theorem implies that

vanishes,[1] and, consequently, any G-bundle on

is isomorphic to the trivial one.

Also, the theorem plays a basic role in the theory of finite groups of Lie type.

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

may be replaced by any surjective map with finitely many fixed points (see below for the precise statement.)

that induces a nilpotent operator on the Lie algebra of G.[2] 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.

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

is bijective since the differential of the Frobenius

[4] Let X be the closure of the image of

The smooth points of X form an open dense subset; thus, there is some b in G such that

is a smooth point of X.

Since the tangent space to X at

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

then contains an open dense subset U of G. Now, given an arbitrary element a in G, by the same reasoning, the image of

contains an open dense subset V of G. The intersection

is then nonempty but then this implies a is in the image of