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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

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 -adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map πrec (π) from isomorphism classes of irreducible smooth representations of GL n(F) to isomorphism classes of n-dimensional semisimple continuous representations of WF. Our map rec is characterized in terms of a local compatibility condition on traces of a certain test function fτ,h, and we prove that rec equals the usual local Langlands correspondence (after forgetting the monodromy operator).

local Langlands correspondence, function fields, D-elliptic sheaves
Mathematical Subject Classification
Primary: 11F70, 11S37
Secondary: 11G09
Received: 12 March 2021
Revised: 7 November 2021
Accepted: 3 January 2022
Published: 19 December 2022
Siyan Daniel Li-Huerta
Harvard University
United States