Duke Mathematical Journal

Level-raising and symmetric power functoriality, III

Laurent Clozel and Jack A. Thorne

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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.

Article information

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

Dates
Received: 17 July 2014
Revised: 10 December 2015
First available in Project Euclid: 9 December 2016

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1481252669

Digital Object Identifier
doi:10.1215/00127094-3714971

Mathematical Reviews number (MathSciNet)
MR3600753

Zentralblatt MATH identifier
1339.11060

Subjects
Primary: 11F03: Modular and automorphic functions 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations
Secondary: 11F80: Galois representations

Keywords
Galois representations Hilbert modular forms Langlands functoriality

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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