Level-raising and symmetric power functoriality, III

Abstract

The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers ${Sym}^n$ of a cuspidal representation of $GL\left(2\right)$ over the adèles of $F$, where $F$ is a number field. In 1978, Gelbart and Jacquet proved the existence of ${Sym}^2$. After this, progress was slow, eventually leading, through the work of Kim and Shahidi, to the existence of ${Sym}^3$ and ${Sym}^4$. In this series of articles we revisit this problem using recent progress in the deformation theory of modular Galois representations. As a consequence, our methods apply only to classical modular forms on a totally real number field; the present article proves the existence, in this “classical” case, of ${Sym}^6$ and ${Sym}^8$.