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 formula_1, then, writing formula_2 for the Frobenius, the morphism of varieties is surjective. Note that the kernel of this map (i.e., formula_4) is precisely formula_5. The theorem implies that formula_6   vanishes, and, consequently, any "G"-bundle on formula_8 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 formula_9 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 formula_9 that induces a nilpotent operator on the Lie algebra of "G".   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 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 (identifying the tangent space at "a" with the tangent space at the identity element) we have: where formula_13. It follows formula_14 is bijective since the differential of the Frobenius formula_9 vanishes. Since formula_16, we also see that formula_17 is bijective for any "b". Let "X" be the closure of the image of formula_19. The smooth points of "X" form an open dense subset; thus, there is some "b" in "G" such that formula_20 is a smooth point of "X". Since the tangent space to "X" at formula_20 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.