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Level raising and symmetric power functoriality. II. (English) Zbl 1339.11060
This paper is a sequel to the authors’ [Compos. Math. 150, No. 5, 729–748 (2014; Zbl 1304.11040)]. In that first paper, they had outlined a strategy to reduce a certain symmetric power lifting conjecture to two other conjectures about automorphic forms (concerning the existence of automorphic tensor products and the construction of level raising congruences between certain automorphic representations).
In the present paper, they carry out this strategy in certain cases. This allows them to prove that if $$\ell \geq 5$$ is a prime number, then the implication $$\mathrm{SP}_{\ell-1}(K(\zeta_\ell)) \implies \mathrm{SP}_{\ell+1}(K(\zeta_\ell))$$ holds. Here $$K$$ is a finite Galois extension of $$\mathbb{Q}$$, and $$\mathrm{SP}_{n+1}(K)$$ is a conjecture (conjecture 1.1 of the paper) asserting the existence of the $$n$$th symmetric power lifting of certain automorphic representations of $$\mathrm{GL}_2(\mathbb{A}_F)$$ with $$F$$ a totally real number field linearly disjoint from $$K$$.
Thanks to some previous work of H. H. Kim and F. Shahidi [Ann. Math. (2) 155, No. 3, 837–893 (2002; Zbl 1040.11036)], establishing $$\mathrm{SP}_4(\mathbb{Q})$$, this allows the authors to prove $$\mathrm{SP}_6(\mathbb{Q}(\zeta_5))$$ and $$\mathrm{SP}_8(\mathbb{Q}(\zeta_{35}))$$. The proof of the paper’s main result relies on C. P. Mok’s study [Mem. Am. Math. Soc. 1108, iii-v, 248 p. (2015; Zbl 1316.22018)] of the space of automorphic forms on certain groups, and uses automorphy lifting methods.

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E55 Representations of Lie and linear algebraic groups over global fields and adèle rings 11F41 Automorphic forms on $$\mbox{GL}(2)$$; Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces 11F80 Galois representations
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