Let be a local field of characteristic . By adapting methods of Scholze (2013), we give a new proof of the local Langlands correspondence for over . More specifically, we construct -adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map from isomorphism classes of irreducible smooth representations of to isomorphism classes of -dimensional semisimple continuous representations of . Our map is characterized in terms of a local compatibility condition on traces of a certain test function , and we prove that equals the usual local Langlands correspondence (after forgetting the monodromy operator).
Siyan Daniel Li-Huerta. "The local Langlands correspondence for over function fields." Algebra Number Theory 16 (9) 2119 - 2214, 2022. https://doi.org/10.2140/ant.2022.16.2119