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Functorial products for $$\text{GL}_2\times \text{GL}_3$$ and the symmetric cube for $$\text{GL}_2$$. (English) Zbl 1040.11036
Langlands’ functoriality is one of the central questions in the theory of automorphic forms. In this paper, the authors prove two cases of Langlands functoriality. The first is a functorial product for cusp forms on $$\text{GL}_2 \times \text{GL}_3$$ as automorphic forms on $$\text{GL}_6$$, from which they obtain the second case, the long awaited functorial symmetric cube map for cusp forms on $$\text{GL}_2$$. Their approach is based on the Converse Theorem of Cogdell and Piatetski-Shapiro and the Langlands-Shahidi method, which studies automorphic $$L$$-functions via the constant and non-constant terms of the Eisenstein series. This is a powerful machinery which was later used for proving a number of cases of functoriality in papers by the authors and also in joint works with Cogdell and Piatetski-Shapiro.
The paper concludes with a large number of applications. The first result establishes the bound 5/34 for Hecke eigenvalues of Maass forms over any number field at every place, finite or infinite, breaking the bound 1/6 towards the Ramanujan-Petersson and Selberg conjectures for $$\text{GL}_2$$. Among other applications, we mention new examples of Artin’s conjecture and global Langlands correspondence.

##### MSC:
 11F70 Representation-theoretic methods; automorphic representations over local and global fields 22E50 Representations of Lie and linear algebraic groups over local fields
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