An example of non-normal quintic automorphic induction and modularity of symmetric powers of cusp forms of icosahedral type

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Due to the work of Arthur and Clozel [AC], the automorphic induction for cyclic Galois extensions of prime degree is well understood in great generality. It is not the case for non-normal extensions, even for monomial representations, i.e., the ones induced from grossencharacters. The only examples we have at the moment are: first, non-normal cubic automorphic induction due to Jacquet, Piatetski-Shapiro and Shalika [JPSS]. They obtained the automorphic induction as a consequence of the converse theorem on GL3. The second example is that of Harris [H]. He constructed automorphic induction for special class of algebraic Hecke characters of (suitable) non-normal extensions with solvable Galois closure. In this note, we give an example of automorphic induction for nonnormal quintic extension whose Galois closure is not solvable (Theorem 6.3). In fact, the Galois group is A5, the alternating group on 5 letters. The key observation due to Ramakrishnan is that symmetric fourth of the two dimensional icosahedral representation is equivalent to a suitable twist (by a character) of the five dimensional monomial representation of A5 . Our result follows immediately by combining results of K. Buzzard, M. Dickinson, N. Shepherd-Barron and R. Taylor [BDST] or R. Taylor’s result [Ta2] on modularity of certain icosahedral representations and our result on the functoriality of symmetric fourth [Ki]. We also prove the modularity of all symmetric powers of cuspidal representations of icosahedral type (Theorem 6.4).