On close fields and the local Langlands correspondence

2024/07/09 by Siyan Daniel Li-Huerta, Li-Huerta, Siyan Daniel
Mathematics · Physics and Astronomy · #11S37 (Primary) 11F77 #14G45 (Secondary) #Advanced Algebra and Geometry #Advanced Differential Geometry Research #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Representation Theory (math.RT)

paper · pdf · doi:10.48550/arxiv.2407.07063

openalex publication_date 2024/07/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove that Fargues-Scholze's semisimplified local Langlands correspondence (for quasisplit groups) with \mathbbF_ℓ-coefficients is compatible with Deligne and Kazhdan's philosophy of close fields. From this, we deduce that the same holds with ℚ_ℓ-coefficients after restricting to wild inertia, addressing questions of Gan-Harris-Sawin and Scholze. The proof involves constructing a moduli space of nonarchimedean local fields and then extending Fargues-Scholze's work to this context.

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