On close fields and the local Langlands correspondence

TL;DR

This paper demonstrates the compatibility of the Fargues-Scholze local Langlands correspondence with the philosophy of close fields, extending results to different coefficient fields and addressing open questions in the field.

Contribution

It proves the compatibility of the local Langlands correspondence with close fields for quasisplit groups and extends the results to $ar{bQ}_ell$-coefficients, involving new moduli space constructions.

Findings

Compatibility with Deligne and Kazhdan's philosophy established

Extension to $ar{bQ}_ell$-coefficients after wild inertia restriction

Construction of a moduli space of nonarchimedean local fields

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.