Skip to content
DraconiticResearch
Search Sources API
194,161 studies31 sourcesUpdated 52 min ago
PreprintarXiv licencearXiv

Depth-zero local Langlands via shtukas in mixed characteristic

  • Chenji Fu
Open at sourcePDFDOICite

Abstract

Assuming a smoothness property for mixed-characteristic shtuka spaces and hyperbolic localization for twisted sheaves, we determine the inertial restriction of the Fargues-Scholze L-parameter for irreducible depth-zero representations. Let E be a non-archimedean local field of residue characteristic p, let G be an unramified connected reductive group over E, and let ฯ€ be an irreducible smooth depth-zero representation of G(E) over ๐”ฝฬ…โ„“ where โ„“ is a prime number which does not divide p|ฯ€0Z(G)|. Let ฯ†ฯ€FS be the Fargues-Scholze L-parameter associated to ฯ€. We describe the restriction of ฯ†ฯ€FS to the inertia subgroup in terms of Deligne-Lusztig theory. We prove this by computing the inertia action on the cohomology of shtuka spaces with depth-zero level structure. As an application, we prove that the Fargues--Scholze L-parameters agree with DeBacker--Reeder L-parameters over โ„šฬ…โ„“.

Related

  1. Finite subgroups of outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields
    Yu Nishio
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  2. Hartl's dictionary and a transcendence characterization of equicharacteristic local shtukas with complex multiplication
    Minhua Cheng, Sean Howe
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  3. Endoscopic description of the local Langlands correspondence for G2
    Yugo Takanashi
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  4. Local Langlands functoriality for Yu's supercuspidals II: Fargues--Scholze's parametrization
    Sean Cotner, Tony Feng
    arXivPreprintarXiv licenceFull textAlgebra & number theoryarXiv
  5. A LOCAL PROOF OF LANGLANDS'S SECOND MAIN LEMMA FOR LOCAL EPSILON FACTORS OVER NONARCHIMEDEAN LOCAL FIELDS
    Fukuhiro Ueda
    HALPreprintPublisher termsFull textAlgebra & number theoryHALarXiv

Cite

BibTeXRIS
BibTeX
@misc{fu2026depthzero,
  title = {{Depth-zero local Langlands via shtukas in mixed characteristic}},
  author = {Chenji Fu},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.16445},
  eprint = {2610.16445},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.16445},
}
RIS
TY  - UNPB
TI  - Depth-zero local Langlands via shtukas in mixed characteristic
AU  - Chenji Fu
PY  - 2026
DA  - 2026/10/12
PB  - arXiv
DO  - 10.48550/arxiv.2610.16445
UR  - https://arxiv.org/abs/2610.16445
AB  - Assuming a smoothness property for mixed-characteristic shtuka spaces and hyperbolic localization for twisted sheaves, we determine the inertial restriction of the Fargues-Scholze L-parameter for irreducible depth-zero representations. Let E be a non-archimedean local field of residue characteristic p, let G be an unramified connected reductive group over E, and let ฯ€ be an irreducible smooth depth-zero representation of G(E) over ๐”ฝฬ…_โ„“ where โ„“ is a prime number which does not divide p|ฯ€_0Z(G)|. Let ฯ†_ฯ€^FS be the Fargues-Scholze L-parameter associated to ฯ€. We describe the restriction of ฯ†_ฯ€^FS to the inertia subgroup in terms of Deligne-Lusztig theory. We prove this by computing the inertia action on the cohomology of shtuka spaces with depth-zero level structure. As an application, we prove that the Fargues--Scholze L-parameters agree with DeBacker--Reeder L-parameters over โ„šฬ…_โ„“.
ER  -