2008/11/07 by Laurent Clozel, Michael Harris, Richard Taylor · 2 citations
Mathematics · #Algebraic Geometry and Number Theory #Advanced Algebra and Geometry #Homotopy and Cohomology in Algebraic Topology #Galois module #Mathematics #Automorphic form #Langlands–Shahidi method #Pure mathematics #Galois group #Unitary state #Conjecture #Rank (graph theory) #Automorphic L-function #Dimension (graph theory) #Unitary group #Mod #Galois cohomology #Group (periodic table) #Lemma (botany) #Algebra over a field #Discrete mathematics #Combinatorics
paper · doi:10.1007/s10240-008-0016-1
openalex publication_date 2008/11/07 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
We extend the methods of Wiles and of Taylor and Wiles from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>GL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> to higher rank unitary groups and establish the automorphy of suitable conjugate self-dual, regular (de Rham with distinct Hodge-Tate numbers), minimally ramified, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>l</mml:mi> </mml:math> -adic lifts of certain automorphic mod <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>l</mml:mi> </mml:math> Galois representations of any dimension. We also make a conjecture about the structure of mod <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>l</mml:mi> </mml:math> automorphic forms on definite unitary groups, which would generalise a lemma of Ihara for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>GL</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> . Following Wiles' method we show that this conjecture implies that our automorphy lifting theorem could be extended to cover lifts that are not minimally ramified.