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Level-raising and symmetric power functoriality. I. (English) Zbl 1304.11040

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 that
(1)
\(S^n(\pi)\) exists for each odd integer \(1\leq n\leq 25\), assuming LR\(_n\);
(2)
\(S^n(\pi)\) exists for each integer \(n\geq 1\), assuming both LR\(_n\) and TP\(_n\).
In the proof, a recent result of the second named author on deformation theory for ‘Schur representations’ is used.

MSC:

11F80 Galois representations
11F03 Modular and automorphic functions
11F66 Langlands \(L\)-functions; one variable Dirichlet series and functional equations
22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings
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References:

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