The plectic conjecture over function fields

Abstract

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 $\ell$-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 $\DeclareMathOperator{\Weil}{Weil}(\Weil(F)^d\rtimes\mathfrak{S}_d)^I$-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 $(\Weil(F)^d\rtimes\mathfrak{S}_d)^I$ commutes with the Hecke action, and we give a moduli-theoretic description of the action of Frobenius elements in $\Weil(F)^{d\times I}$.

Major revision: upgraded main result to level of complexes. 41 pages. Comments welcome!

References in corpus (3)

• Geometrization of the local Langlands correspondence
• Smoothness of cohomology sheaves of stacks of shtukas
• Functoriality of Moduli Spaces of Global $\mathbb{G}$-Shtukas