The local Langlands correspondence for GLn over function fields

Abstract

Let $F$ be a local field of characteristic $p>0$. By adapting methods of Scholze (2013), we give a new proof of the local Langlands correspondence for ${\text{GL}}_n$ over $F$. More specifically, we construct $\ell$-adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map $\pi \mapsto \text{rec}(\pi )$ from isomorphism classes of irreducible smooth representations of ${\text{GL}}_n(F)$ to isomorphism classes of $n$-dimensional semisimple continuous representations of $W_F$. Our map $\text{rec}$ is characterized in terms of a local compatibility condition on traces of a certain test function $f_{\tau ,h}$, and we prove that $\text{rec}$ equals the usual local Langlands correspondence (after forgetting the monodromy operator).