The local Langlands correspondence for GLn over function fields

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 GLn 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 pi F> rec(pi) from isomorphism classes of irreducible smooth representations of GLn(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 f tau,h, and we prove that rec equals the usual local Langlands correspondence (after forgetting the monodromy operator).