Hecke Operators for Curves Over Non-Archimedean Local Fields and Related Finite Rings

Alexander Braverman; David Kazhdan; Alexander Polishchuk; Ka Fai Wong

doi:10.1093/imrn/rnaf075

Hecke Operators for Curves Over Non-Archimedean Local Fields and Related Finite Rings

Alexander Braverman^*
, David Kazhdan
, Alexander Polishchuk
, Ka Fai Wong

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We study Hecke operators associated with curves over a non-archimedean local field K and over the rings O/m^N, where O ⊂ K is the ring of integers. Our main result is commutativity of a certain “small” local Hecke algebra over O/m^N, associated with a connected split reductive group G such that [G, G] is simply connected. The proof uses a Hecke algebra associated with G(K_((t))) and a global argument involving G-bundles on curves.

Original language	English
Article number	rnaf075
Journal	International Mathematics Research Notices
Volume	2025
Issue number	7
DOIs	https://doi.org/10.1093/imrn/rnaf075
State	Published - 1 Apr 2025

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© The Author(s) 2025. Published by Oxford University Press. All rights reserved.

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abstract = "We study Hecke operators associated with curves over a non-archimedean local field K and over the rings O/mN, where O ⊂ K is the ring of integers. Our main result is commutativity of a certain “small” local Hecke algebra over O/mN, associated with a connected split reductive group G such that [G, G] is simply connected. The proof uses a Hecke algebra associated with G(K((t))) and a global argument involving G-bundles on curves.",

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N2 - We study Hecke operators associated with curves over a non-archimedean local field K and over the rings O/mN, where O ⊂ K is the ring of integers. Our main result is commutativity of a certain “small” local Hecke algebra over O/mN, associated with a connected split reductive group G such that [G, G] is simply connected. The proof uses a Hecke algebra associated with G(K((t))) and a global argument involving G-bundles on curves.

AB - We study Hecke operators associated with curves over a non-archimedean local field K and over the rings O/mN, where O ⊂ K is the ring of integers. Our main result is commutativity of a certain “small” local Hecke algebra over O/mN, associated with a connected split reductive group G such that [G, G] is simply connected. The proof uses a Hecke algebra associated with G(K((t))) and a global argument involving G-bundles on curves.

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Hecke Operators for Curves Over Non-Archimedean Local Fields and Related Finite Rings

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