Courbes et fibr\'es vectoriels en th\'eorie de Hodge $z$-adique globale

Abstract

We study the global analogue of the Fargues-Fontaine curve over function fields $F$. We prove some foundational results about its moduli of $G$-bundles $\operatorname{Bun}_{G,F}$, which is a geometrization of the global Kottwitz set $B(F,G)$. For example, $\operatorname{Bun}_{G,F}$ plays the role of Igusa stacks over function fields. We use $\operatorname{Bun}_{G,F}$ to reformulate the global Langlands conjecture for $G$ over $F$ in terms of categorical local Langlands, refining conjectures of Arinkin-Gaitsgory-Kazhdan-Raskin-Rozenblyum-Varshavsky and Zhu. Finally, we verify this conjecture when $G$ is commutative. Along the way, we prove a GAGA theorem for smooth proper schemes over sousperfectoid spaces, which is of independent interest.