is surjective. Note that the kernel of this map (i.e., ) is precisely .
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 may be replaced by any surjective map with finitely many fixed points (see below for the precise statement.)
The Lang–Steinberg 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.
Proof of Lang's theorem
Then (identifying the tangent space at a with the tangent space at the identity element) we have:
where . It follows is bijective since the differential of the Frobenius vanishes. Since , we also see that is bijective for any b. 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 .
- T.A. Springer, "Linear algebraic groups", 2nd ed. 1998.
- Lang, Serge (1956), "Algebraic groups over finite fields", American Journal of Mathematics 78: 555–563, doi:10.2307/2372673, ISSN 0002-9327, 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