SIYAN DANIEL LI-HUERTA

We prove the plectic conjecture of Nekovář–Scholl over global function fields Q. For example, when the cocharacter is defined over Q and the structure group is a Weil restriction from a geometric degree d separable extension F/Q, consider the complex computing ℓ-adic intersection cohomology with compact support of the associated moduli space of shtukas over Q_I. We endow this with the structure of a complex of (Weil(F)^d S_d)^I-modules, which extends its structure as a complex of Weil(Q)^I-modules constructed by Arinkin–Gaitsgory–Kazhdan–Raskin–Rozenblyum–Varshavsky. We show that the action of (Weil(F)^d S_d)^I commutes with the Hecke action, and we give a moduli-theoretic description of the action of Frobenius elements in Weil(F)^d×I.

Let F be a local field of characteristic p > 0. By adapting methods of Scholze, 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→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).