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Cyclic base change of cuspidal automorphic representations over function fields

Let GG be a split semisimple group over a global function field KK. Given a cuspidal automorphic representation Π\Pi of GG satisfying a technical hypothesis, we prove that for almost all primes ℓ\ell, there is a cyclic base change lifting of Π\Pi along any Z/ℓZ\mathbb {Z}/\ell \mathbb {Z}-extension...

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Bibliographic Details
Main Authors: Böckle, Gebhard (Author) , Feng, Tony (Author) , Harris, Michael (Author) , Khare, Chandrashekhar B. (Author) , Thorne, Jack A. (Author)
Format: Article (Journal)
Language:English
Published: 11 September 2024
In: Compositio mathematica
Year: 2024, Volume: 160, Issue: 9, Pages: 1959-2004
ISSN:1570-5846
DOI:10.1112/S0010437X24007243
Online Access:Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1112/S0010437X24007243
Verlag, lizenzpflichtig, Volltext: https://www.cambridge.org/core/journals/compositio-mathematica/article/cyclic-base-change-of-cuspidal-automorphic-representations-over-function-fields/3405D309A46E5D9B90FEF92FF0C142AF
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Author Notes:Gebhard Böckle, Tony Feng, Michael Harris, Chandrashekhar B. Khare and Jack A. Thorne
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Summary:Let GG be a split semisimple group over a global function field KK. Given a cuspidal automorphic representation Π\Pi of GG satisfying a technical hypothesis, we prove that for almost all primes ℓ\ell, there is a cyclic base change lifting of Π\Pi along any Z/ℓZ\mathbb {Z}/\ell \mathbb {Z}-extension of KK. Our proof does not rely on any trace formulas; instead it is based on using modularity lifting theorems, together with a Smith theory argument, to obtain base change for residual representations. As an application, we also prove that for any split semisimple group GG over a local function field FF, and almost all primes ℓ\ell, any irreducible admissible representation of G(F)G(F) admits a base change along any Z/ℓZ\mathbb {Z}/\ell \mathbb {Z}-extension of FF. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.
Item Description:Gesehen am 24.02.2025
Physical Description:Online Resource
ISSN:1570-5846
DOI:10.1112/S0010437X24007243

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