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AVERAGING FUNCTORS IN FARGUES’ PROGRAM FOR $\mathrm {GL}_n$

Part of: Arithmetic problems. Diophantine geometry (Co)homology theory Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  24 September 2025

Johannes Anschütz Open the ORCID record for Johannes Anschütz [Opens in a new window]  and
Arthur-César Le Bras Open the ORCID record for Arthur-César Le Bras [Opens in a new window]
Johannes Anschütz
Affiliation:
https://ror.org/03xjwb503Université Paris-Saclay , Laboratoire de Mathématiques d’Orsay, Orsay, France (janschuetz@math.cnrs.fr)
Arthur-César Le Bras*
Affiliation:
https://ror.org/02hwgty18Université de Strasbourg UFR de Mathématique et d’Informatique , France
*
lebras@math.uni-bonn.de
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Abstract

We study the so-called averaging functors from the geometric Langlands program in the setting of Fargues’ program. This makes explicit certain cases of the spectral action which was recently introduced by Fargues-Scholze in the local Langlands program for $\mathrm {GL}_n$. Using these averaging functors, we verify (without using local Langlands) that the Fargues-Scholze parameters associated to supercuspidal modular representations of $\mathrm {GL}_2$ are irreducible. We also attach to any irreducible $\ell $-adic Weil representation of degree n an Hecke eigensheaf on $\mathrm {Bun}_n$ and show, using the local Langlands correspondence and recent results of Hansen and Hansen-Kaletha-Weinstein, that it satisfies most of the requirements of Fargues’ conjecture for $\mathrm {GL}_n$.

MSC classification

Primary: 11S37: Langlands-Weil conjectures, nonabelian class field theory 14F20: Étale and other Grothendieck topologies and (co)homologies 14G22: Rigid analytic geometry

Information

Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , First View , pp. 1 - 40
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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