Modular functoriality in the Local Langlands Correspondence

Abstract

We develop a theory of Smith--Treumann localization 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 explicitly calculate the Fargues-Scholze parameters of certain mod-$\ell$ toral representations.

arXiv commentMajor revision, eliminating the old sections 5 and 6 (but breaking other sections so that sections 8-10 have the same numbering). Improved results to cover all primes \ell, eliminating earlier technical assumptions, and giving a cleaner definition of the Brauer functor. Several errors corrected