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

TL;DR

This paper explores the global analogue of the Fargues-Fontaine curve over function fields, establishing foundational results on moduli of G-bundles, and reformulating the global Langlands conjecture in categorical terms, with verification in the commutative case.

Contribution

It introduces the global Fargues-Fontaine curve over function fields, studies its G-bundles, and reformulates the global Langlands conjecture categorically, including a new GAGA theorem for sousperfectoid spaces.

Findings

Established foundational results on moduli of G-bundles over the global Fargues-Fontaine curve.

Reformulated the global Langlands conjecture in terms of categorical local Langlands.

Proved the conjecture for the case when G is commutative.

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.