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The local Langlands correspondence for $\mathrm{GL}_n$ over function fields

### Siyan Daniel Li-Huerta

Vol. 16 (2022), No. 9, 2119–2214
##### 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 ${\mathrm{GL}}_{n}$ 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 $\pi ↦\mathrm{rec}\left(\pi \right)$ from isomorphism classes of irreducible smooth representations of ${\mathrm{GL}}_{n}\left(F\right)$ to isomorphism classes of $n$-dimensional semisimple continuous representations of ${W}_{F}$. Our map $\mathrm{rec}$ is characterized in terms of a local compatibility condition on traces of a certain test function ${f}_{\tau ,h}$, and we prove that $\mathrm{rec}$ equals the usual local Langlands correspondence (after forgetting the monodromy operator).

##### Keywords
local Langlands correspondence, function fields, D-elliptic sheaves
##### Mathematical Subject Classification
Primary: 11F70, 11S37
Secondary: 11G09