Abstract
Let f be a cuspidal Hecke eigenform without complex multiplication. We prove the automorphy of the symmetric power lifting operatorname{Sym}^{n} f for every n geq 1.
Highlights
Let F be a number field, and let π be a cuspidal automorphic representation of GL2(AF)
Langlands’s functoriality principle predicts the existence, for any n ≥ 1, of an automorphic representation Symn π of GLn+1(AF), characterized by the requirement that for any place v of F, the Langlands parameter of (Symn π )v is the image of the Langlands parameter of πv under the nth symmetric power Symn : GL2 → GLn+1 of the standard representation of GL2
For a more detailed discussion of the context surrounding this problem, including known results by other authors, we refer the reader to the introduction of [NT21], of which this paper is a continuation
Summary
Let F be a number field, and let π be a cuspidal automorphic representation of GL2(AF). For each integer n ≥ 1, Symn π exists, as a regular algebraic, cuspidal automorphic representation of GLn+1(AQ). Regular algebraic, and polarizable, in the sense of [BLGGT14], for any isomorphism ι : Qp → C there exists a continuous, semisimple representation rπ,ι : GF → GLn(Qp) such that for each finite place v of F, WD(rπ,ι|GFv )F−ss ∼= recTFv (ι−1πv) If F is a number field, G is a reductive group over F, v is a finite place of F, and Uv is an open compact subgroup of G(Fv), we write H(G(Fv), Uv) for the convolution algebra of compactly supported Uv-biinvariant functions f : G(Fv) → Z (convolution defined with respect to the Haar measure on G(Fv) which gives Uv volume 1).
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