Functorial products for $G{L}_2\times G{L}_3$ and the symmetric cube for $G{L}_2$

Abstract

In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on $G{L}_2\times G{L}_3$ as automorphic forms on $G{L}_6$, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on $G{L}_2$. We prove these by applying a recent version of converse theorems of Cogdell and Piatetski-Shapiro to analytic properties of certain $L$-functions obtained from the method of Eisenstein series (Langlands-Shahidi method). As a consequence, we prove the bound 5/34 for Hecke eigenvalues of Maass forms over any number field and at every place, finite or infinite, breaking the crucial bound 1/6 (see below and Section 7 and 8) towards Ramanujan-Petersson and Selberg conjectures for $G{L}_2$. Many other applications are obtained.

57 pages, published version. Appendix by Colin J. Bushnell and Guy Henniart