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

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

2004-09-30

arxiv.org/abs/math/0409607v1

Ann. of Math. (2), Vol. 155 (2002), no. 3, 837--893

Mathematics

Number Theory

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

Scientific paper

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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