Triple product $L$-functions and the Ramanujan conjecture

Abstract

We prove that the Ramanujan conjecture is true under the assumption that the expected analytic properties of triple product $L$-functions hold. Further, we explain how these analytic properties imply certain reduction steps in the construction of functorial transfers in the sense of Langlands. Roughly, at the level of stably automorphic representations, they allow one to reduce any functorial transfer from a given reductive group $G$ to a general linear group to a finite family of transfers depending on $G.$