Finite subgroups of SL (2 , F¯) and automorphy

We give a new proof of the well-known classification of finite subgroups of SL (2 , C) , that generalizes to the three dimensional case of SL (3 , C) ; recall the geometric proof, based on study of the motions of the Platonic solids, that does not seem generalizable to higher 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 π of GL (3 , A_F) that is the unique candidate to be the automorphic representation π(ρ) corresponding to a certain three dimensional representation ρ of the Weil group of a number field F, to initiate study of the global Galois-automorphic correspondence in dimension > 2 for number fields.