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Cyclic base change of cuspidal automorphic representations over function fields

Part of: Classical Topics in Algebraic Topology Algebraic number theory: global fields Discontinuous groups and automorphic forms Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  11 September 2024

Gebhard Böckle Open the ORCID record for Gebhard Böckle [Opens in a new window] ,
Tony Feng Open the ORCID record for Tony Feng [Opens in a new window] ,
Michael Harris Open the ORCID record for Michael Harris [Opens in a new window] ,
Chandrashekhar B. Khare Open the ORCID record for Chandrashekhar B. Khare [Opens in a new window]  and
Jack A. Thorne Open the ORCID record for Jack A. Thorne [Opens in a new window]
Gebhard Böckle
Affiliation:
Interdisciplinary Center for Scientific Computing, Universität Heidelberg, 69120 Heidelberg, Germany gebhard.boeckle@iwr.uni-heidelberg.de
Tony Feng
Affiliation:
Department of Mathematics, University of California at Berkeley, Berkeley, CA 94720, USA fengt@berkeley.edu
Michael Harris
Affiliation:
Department of Mathematics, Columbia University, New York, NY 10027, USA harris@math.columbia.edu
Chandrashekhar B. Khare
Affiliation:
Department of Mathematics, UCLA, Los Angeles, USA shekhar@math.ucla.edu
Jack A. Thorne
Affiliation:
Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge, UK thorne@dpmms.cam.ac.uk
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Abstract

Let $G$ be a split semisimple group over a global function field $K$. Given a cuspidal automorphic representation $\Pi$ of $G$ satisfying a technical hypothesis, we prove that for almost all primes $\ell$, there is a cyclic base change lifting of $\Pi$ along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $K$. 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 $G$ over a local function field $F$, and almost all primes $\ell$, any irreducible admissible representation of $G(F)$ admits a base change along any $\mathbb {Z}/\ell \mathbb {Z}$-extension of $F$. Finally, we characterize local base change more explicitly for a class of toral representations considered in work of Chan and Oi.

Keywords

base changeSmith theorymodularity liftingfunction fields

MSC classification

Primary: 55M35: Finite groups of transformations (including Smith theory)
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11R39: Langlands-Weil conjectures, nonabelian class field theory 11S37: Langlands-Weil conjectures, nonabelian class field theory
Type
Research Article
Information
Compositio Mathematica , Volume 160 , Issue 9 , September 2024 , pp. 1959 - 2004
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Copyright
© The Author(s), 2024. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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