Level-raising and symmetric power functoriality. I (Q2877500)

Let \(\mathbb{A}\) be the ring of adeles of a number field \(F\), and \(\pi\) be an automorphic representation of \(\mathrm{GL}(2,\mathbb{A})\). This paper is concerned with the Langlands functoriality for symmetric powers, i.e., the existence of the symmetric power lifting \(S^n(\pi)\) as an automorphic representation of \(\mathrm{GL}_{n+1}(\mathbb{A})\), which is known for \(n=2,3,4\) due to the work of Gelbart-Jacquet, Kim-Shahidi, and Kim. In this paper, assuming the level raising conjecture (LR\(_n\)) and the tensor product conjecture TP\(_n\), the authors are able to deduce the general case. In particular, consider a regular algebraic automorphic representation \(\pi\) of \(\mathrm{GL}_2(\mathbb{A}_\mathbb{Q})\) which is not automorphically induced from a quadratic extension. Then as an application the authors' result implies thatNEWLINENEWLINE{\parindent=5mm \begin{itemize}\item[(1)] \(S^n(\pi)\) exists for each odd integer \(1\leq n\leq 25\), assuming LR\(_n\); \item[(2)] \(S^n(\pi)\) exists for each integer \(n\geq 1\), assuming both LR\(_n\) and TP\(_n\). NEWLINENEWLINE\end{itemize}} In the proof, a recent result of the second named author on deformation theory for `Schur representations' is used.