Modular functoriality in the Local Langlands Correspondence

TL;DR

This paper advances the understanding of the Local Langlands Correspondence by developing new geometric tools and applying them to prove conjectures and compute parameters related to automorphic representations.

Contribution

It introduces a theory of Smith-Treumann localization and relative parity sheaves, applying it to prove conjectures and calculate parameters in the mod-$ ext{ell}$ setting.

Findings

Proved conjectures of Treumann-Venkatesh on mod-$ ext{ell}$ Local Langlands functoriality.

Calculated Fargues-Scholze parameters for mod-$ ext{ell}$ Howe-unramified toral representations.

Developed a new geometric framework for the Local Langlands Correspondence.

Abstract

We develop a theory of Smith-Treumann localization and relative parity sheaves in the context of Fargues-Scholze's Geometrization of the Local Langlands Correspondence. We then apply this theory to prove some conjectures of Treumann-Venkatesh concerning mod-$\ell$ Local Langlands functoriality between a reductive group $G$ and its fixed subgroup under an order $\ell$ automorphism. As another application, we calculate the Fargues-Scholze parameters of mod-$\ell$ Howe-unramified toral representations.