Henry Kim

Title: Functoriality of symmetric powers of GL₂

Abstract: One of the most important and a test case of Langlands functoriality is the functoriality of the symmetric powers of cuspidal representations of GL₂. Namely, given a cuspidal representation $\pi=\otimes_v \pi_v$ of $GL_2(\Bbb A)$, where $\Bbb A$ is the ring of adeles of a number field $F$, one can form an irreducible admissible representation $Sym^m(\pi)=\otimes_v Sym^m(\pi_v)$ by the local Langlands correspondence. Then Langlands functoriality conjecture is that $Sym^m(\pi)$ is an automorphic representation of $GL_{m+1}(\Bbb A)$. In this talk I will explain the cases $m=3,4$, and the applications to the $m$th symmetric power $L$-functions up to $m=9$, and to the Ramanujan and Sato-Tate conjectures.