**Henry Kim**

**Title**: Functoriality of symmetric powers of **GL**_{2}

**Abstract**:
One of the most important and a test case of Langlands
functoriality is the functoriality of the symmetric powers of cuspidal
representations of
**GL**_{2}. 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.