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

摘要

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 grosencharacters. 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.