The local Langlands correspondence for GLn over function fields

Li-Huerta, Siyan Daniel

doi:10.2140/ant.2022.16.2119

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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

DOI: 10.2140/ant.2022.16.2119

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 ${GL}_{n}$ over $F$ . More specifically, we construct $ℓ$ -adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map $π \mapsto rec (π)$ 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 $f_{τ, h}$ , and we prove that $rec$ 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

Milestones

Received: 12 March 2021

Revised: 7 November 2021

Accepted: 3 January 2022

Published: 19 December 2022

Authors

Siyan Daniel Li-Huerta
Harvard University Cambridge MA United States

Vol. 16, No. 9, 2022