The local Langlands correspondence for $\DeclareMathOperator{\GL}{GL}\GL_n$ over function fields

TL;DR

This paper provides a new proof of the local Langlands correspondence for  over function fields by constructing -adic Galois representations associated with automorphic representations, confirming the correspondence's compatibility and equivalence to the classical form.

Contribution

It introduces a novel proof of the local Langlands correspondence for  over function fields using Scholze's methods and constructs Galois representations to establish the correspondence.

Findings

Constructed -adic Galois representations for automorphic representations.

Proved the map     the local Langlands correspondence.

Confirmed the correspondence matches the classical version after ignoring monodromy.

Abstract

Let $F$ be a local field of characteristic $p>0$. By adapting methods of Scholze, we give a new proof of the local Langlands correspondence for $\GL_n$ 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 $\DeclareMathOperator{\rec}{rec}\pi\mapsto\rec(\pi)$ from isomorphism classes of irreducible smooth representations of $\GL_n(F)$ to isomorphism classes of $n$-dimensional semisimple continuous representations of $W_F$. Our map $\rec$ is characterized in terms of a local compatibility condition on traces of a certain test function $f_{\tau,h}$, and we prove that $\rec$ equals the usual local Langlands correspondence (after forgetting the monodromy operator).