On close fields and the local Langlands correspondence

Abstract

We prove that Fargues-Scholze's semisimplified local Langlands correspondence (for quasisplit groups) with $\overline{\mathbb{F}}_\ell$-coefficients is compatible with Deligne and Kazhdan's philosophy of close fields. From this, we deduce that the same holds with $\overline{\mathbb{Q}}_\ell$-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.