Level-raising and symmetric power functoriality, III

Abstract

The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers Symn of a cuspidal representation of GL.2/ over the adèles of F , where F is a number field. In 1978, Gelbart and Jacquet proved the existence of Sym2. After this, progress was slow, eventually leading, through the work of Kim and Shahidi, to the existence of Sym3 and Sym4. 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 Sym6 and Sym8.