Symmetric power functoriality for holomorphic modular forms

Abstract

Let <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>f</mml:mi> </mml:math> be a cuspidal Hecke eigenform of level 1. We prove the automorphy of the symmetric power lifting <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mo form="prefix">Sym</mml:mo> <mml:mi>n</mml:mi> </mml:msup> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> for every <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> . We establish the same result for a more general class of cuspidal Hecke eigenforms, including all those associated to semistable elliptic curves over <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝐐</mml:mi> </mml:math> .

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