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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.

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.

Reviewer: Dubravka Ban (Carbondale)