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On the Ramanujan conjecture for automorphic forms over function fields I. Geometry


Authors: Will Sawin and Nicolas Templier
Journal: J. Amer. Math. Soc. 34 (2021), 653-746
MSC (2020): Primary 14D24, 11F70, 14F20, 22E57, 20G30
DOI: https://doi.org/10.1090/jams/968
Published electronically: April 16, 2021
MathSciNet review: 4334190
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Abstract: Let $G$ be a split semisimple group over a function field. We prove the temperedness at unramified places of automorphic representations of $G$, subject to a local assumption at one place, stronger than supercuspidality, and assuming the existence of cyclic base change with good properties. Our method relies on the geometry of $\operatorname {Bun}_G$. It is independent of the work of Lafforgue on the global Langlands correspondence.


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Additional Information

Will Sawin
Affiliation: ETH Institute for Theoretical Studies, ETH Zurich, 8092 Zürich, Switzerland
MR Author ID: 1022068

Nicolas Templier
Affiliation: Department of Mathematics, Cornell University, Ithaca, New York 14853
MR Author ID: 853743

Received by editor(s): July 4, 2018
Received by editor(s) in revised form: September 5, 2020, and September 27, 2020
Published electronically: April 16, 2021
Additional Notes: This article begun while both the authors were in residence at the MSRI, supported by the NSF under Grant No. DMS-1440140. The authors received funding from the European Research Council under the European Community’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant agreement no. 290766 (AAMOT) to visit IHES. The first author was supported by Dr. Max Rössler, the Walter Haefner Foundation and the ETH Zürich Foundation. The second author was supported by the NSF-CAREER under agreement No. DMS-1454893, and by a Simons Fellowship under agreement 500294.
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