Functoriality for the exterior square of GL₄ and the symmetric fourth of GL₂

Source abstract

In this paper we prove the functoriality of the exterior square of cusp forms on G L 4 GL_{4} as automorphic forms on G L 6 GL_{6} and the symmetric fourth of cusp forms on G L 2 GL_{2} as automorphic forms on G L 5 GL_{5} . We prove these by applying a converse theorem of Cogdell and Piatetski-Shapiro to analytic properties of certain L L -functions obtained by the Langlands-Shahidi method. We give several applications: First, we prove the weak Ramanujan property of cuspidal representations of G L 4 GL_{4} and the absolute convergence of the exterior square L L -functions of G L 4 GL_{4} . Second, we prove that the fourth symmetric power L L -functions of cuspidal representations of G L 2 GL_{2} are entire, except for those of dihedral and tetrahedral type. Third, we prove the bound 3 26 \frac {3}{26} for Hecke eigenvalues of Maass forms over any number field.