## Duke Mathematical Journal

### Level-raising and symmetric power functoriality, III

#### Abstract

The simplest case of the Langlands functoriality principle asserts the existence of the symmetric powers $\operatorname{Sym}^{n}$ of a cuspidal representation of $\operatorname{GL}(2)$ over the adèles of $F$, where $F$ is a number field. In 1978, Gelbart and Jacquet proved the existence of $\operatorname{Sym}^{2}$. After this, progress was slow, eventually leading, through the work of Kim and Shahidi, to the existence of $\operatorname{Sym}^{3}$ and $\operatorname{Sym}^{4}$. 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 $\operatorname{Sym}^{6}$ and $\operatorname{Sym}^{8}$.

#### Article information

Source
Duke Math. J., Volume 166, Number 2 (2017), 325-402.

Dates
Revised: 10 December 2015
First available in Project Euclid: 9 December 2016

https://projecteuclid.org/euclid.dmj/1481252669

Digital Object Identifier
doi:10.1215/00127094-3714971

Mathematical Reviews number (MathSciNet)
MR3600753

Zentralblatt MATH identifier
1339.11060

#### Citation

Clozel, Laurent; Thorne, Jack A. Level-raising and symmetric power functoriality, III. Duke Math. J. 166 (2017), no. 2, 325--402. doi:10.1215/00127094-3714971. https://projecteuclid.org/euclid.dmj/1481252669

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