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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 , then, writing for the Frobenius, the morphism of varieties : is surjective. Note that the kernel of this map (i.e., ) is precisely . The theorem implies that vanishes,〔This is "unwinding definition". Here, is Galois cohomology; cf. Milne, Class field theory.〕 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. 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 proof (given below) actually goes through for any that induces a nilpotent operator on the Lie algebra of ''G''. == The Lang–Steinberg theorem == 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Lang's theorem」の詳細全文を読む スポンサード リンク
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