Geometrization of the local Langlands correspondence, motivically

TL;DR

This paper extends the geometric approach to the local Langlands correspondence using motivic sheaves, establishing independence of the choice of prime $\

Contribution

It introduces a motivic framework for the local Langlands correspondence, generalizing previous $\

Findings

Proves independence of $\

Extends the formalism of rigid-analytic motives to the local Langlands setting

Bridges $\

Abstract

Based on the formalism of rigid-analytic motives of Ayoub--Gallauer--Vezzani, we extend our previous work with Fargues from $\ell$-adic sheaves to motivic sheaves. In particular, we prove independence of $\ell$ of the $L$-parameters constructed there.