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Level-raising and symmetric power functoriality, I
- Part of
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- Journal:
- Compositio Mathematica / Volume 150 / Issue 5 / May 2014
- Published online by Cambridge University Press:
- 26 March 2014, pp. 729-748
- Print publication:
- May 2014
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- Article
- Export citation
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As the simplest case of Langlands functoriality, one expects the existence of the symmetric power
$S^n(\pi )$, where 
$\pi $ is an automorphic representation of 
${\rm GL}(2,{\mathbb{A}})$ and 
${\mathbb{A}}$ denotes the adeles of a number field 
$F$. This should be an automorphic representation of 
${\rm GL}(N,{\mathbb{A}})$ (
$N=n+1)$. This is known for 
$n=2,3$ and 
$4$. In this paper we show how to deduce the general case from a recent result of J.T. on deformation theory for ‘Schur representations’, combined with expected results on level-raising, as well as another case (a particular tensor product) of Langlands functoriality. Our methods assume 
$F$ totally real, and the initial representation 
$\pi $ of classical type.
