arithmetic geometry

On categorical local langlands program for

arXiv:2309.16505

summary

The paper investigates moduli spaces of local shtukas for GL_n in the framework of Fargues' program, providing explicit descriptions of the spectral action, constructing Hecke eigensheaves for certain ℓ‑adic Weil representations, and proving new cases of the Harris‑Viehmann conjecture and parts of Fargues' conjecture.

Abstract

We study various moduli spaces of local Shtukas in the setting of Fargues' program for . In certain cases, this gives us an explicit description of the spectral action which was recently introduced by Fargues and Scholze. This description sheds light to the categorical local Langlands program for and allows us to construct Hecke eigensheaves associated to certain -adic Weil representations of rank and to prove some parts of Fargues' conjecture. Moreover, by using this description, we can prove new cases of the Harris-Viehmann conjecture for non-basic Rapoport-Zink spaces and compute some parts of the cohomology of the Igusa varieties associated to .

Corrected some errors about shifts and twists. Simplified some proofs

Topics & keywords

#local langlands#shtukas#spectral action#hecke eigensheaves#raptorport-zhink spaces#igusa varietiesGL_nFargues-Scholzeℓ-adic Weil representationscategorical local LanglandsHarris-Viehmann conjecturelocal shtukas
On categorical local langlands program for $GL_n$ · wovepaper