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Functorial products for $GL_2\times GL_3$ and the symmetric cube for $GL_2$

Henry H. Kim·Freydoon Shahidi·Colin J. Bushnell·Guy Henniart
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Abstract

In this paper we prove two new cases of Langlands functoriality. The first is a functorial product for cusp forms on $GL_2\times GL_3$ as automorphic forms on $GL_6$, from which we obtain our second case, the long awaited functorial symmetric cube map for cusp forms on $GL_2$. 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 $GL_2$. Many other applications are obtained.

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Henry H. Kim, Freydoon Shahidi, Colin J. Bushnell, Guy Henniart. 2004-09-30. Functorial products for $GL_2\times GL_3$ and the symmetric cube for $GL_2$. https://arxiv.org/abs/math/0409607

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