We give a new proof of the well-known classification of finite subgroups of \({\text {SL}}(2,\mathbb C)\), that generalizes to the three dimensional case of \({\text {SL}}(3,\mathbb C)\); recall the geometric proof, based on study of the motions of the Platonic solids, that does not seem generalizable to higher \({\text {SL}}(n)\) nor to other fields, but gives geometric intuition; use the classification to give a more algebraic proof that two dimensional representations of the Weil group of tetrahedral and octahedral type are automorphic; and use this approach to construct an automorphic representation \(\pi \) of \({\text {GL}}(3,\mathbb A_F)\) that is the unique candidate to be the automorphic representation \(\pi (\rho )\) corresponding to a certain three dimensional representation \(\rho \) of the Weil group of a number field F, to initiate study of the global Galois-automorphic correspondence in dimension \(>2\) for number fields.

我们给出了\（{\ text {SL}}（2，\ mathbb C）\）的有限子组的著名分类的新证明，它归纳为\（{\ text {SL} }（3，\ mathbb C）\）；回顾基于对柏拉图式固体运动的研究的几何证明，该证明似乎不能推广到更高的\（{\ text {SL}}（n）\）或其他领域，但给出了几何直觉；使用分类来提供更多的代数证明，即四面体和八面体类型的Weil组的二维表示是自同构的；和使用这种方法来构建自守表示\（\ PI \）的\（{\文本{GL}}（3，\ mathbb A_F）\）即是唯一候选者，它是对应于数域F的Weil组的某个三维表示\（\ rho \）的自守形态表示\（\ pi（\ rho）\），以开始对整体Galois的研究数字字段的尺寸\（> 2 \）中的-自构对应。