Courbes et fibrés vectoriels en théorie de Hodge $z$-adique globale

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.

💡 Research Summary

The paper develops a global analogue of the Fargues–Fontaine curve for a global function field F (with constant field F_q) and studies the moduli stack of G‑bundles on this curve, denoted Bun_{G,F}. The authors begin by explaining why a naïve construction of a global Hartl–Pink curve fails: the product Spec F × Spa K is not well‑behaved and the Frobenius action ceases to be free. To bypass these issues they work with the smooth proper curve C over F_q attached to F and consider dense open subschemes U⊂C. For each U, the adic space U_an K (the analytification of U over K) is well defined, and the inverse limit over U yields a well‑behaved v‑stack Bun_{G,F} that parametrises G‑bundles together with a Frobenius descent datum.

Theorem A shows that for any such U, Bun_{G,U} is a small Artin v‑stack, ℓ‑adic smooth, and its dualizing complex with ℓ‑adic coefficients is constant. This mirrors the local results of Fargues–Scholze and Hamann–Imai, confirming that the global moduli enjoys the expected cohomological finiteness.

Theorem B identifies the open substack of semistable G‑bundles Bun^{ss}{G,F} with a disjoint union of classifying stacks */G_b(F) indexed by basic elements b in the global Kottwitz set B(F,G). Thus Bun{G,F} geometrizes B(F,G) in the same way that the local moduli geometrizes the local Kottwitz set. Theorem C gives an explicit description of the F_q‑points of Bun_{G,F} as the quotient G(F⊗{F_q}F_q)/G(F⊗{F_q}F_q) under Frobenius conjugation, establishing a bijection of isomorphism classes with B(F,G).

The paper then relates Bun_{G,F} to global shtuka stacks. For a finite set I and a representation V of the Langlands dual group, the global shtuka moduli Sh_{I,G,V} is a Deligne–Mumford stack over C^I. By partitioning I according to places z∈Z (the complement of U), one obtains local shtuka stacks LocSht_{I_z,G,V_z}. Theorem D constructs a Cartesian diagram of small v‑stacks linking Sh_{I,G,V}, the product of local shtuka stacks, and Bun_{G,F} via Beauville–Laszlo uniformization and a function‑field analogue of the Hodge–Tate period map π_{HT}. This shows that Bun_{G,F} plays the role of Igusa stacks in the function‑field setting, explaining why the point set of Bun_{G,F} need not biject with B(F,G) when non‑basic elements are present.

In the categorical Langlands direction, the authors consider Hecke operators T_{V_z} acting on the derived category D(Bun_{G,F_z},Λ) for a coefficient ring Λ (ℚ_ℓ, ℤ_ℓ, ℱ_ℓ). Theorem E (assuming a forthcoming result of Étale‑Gaitsgory‑Genestier‑Lafforgue) identifies the push‑forward of the intersection complex of the infinite‑level shtuka stack with the iterated Hecke functor applied to the object loc_Z!Λ in the product of local derived categories. This provides a categorical reformulation of the global Langlands correspondence: automorphic sheaves on Bun_{G,F} correspond to Galois‑parameter stacks LS_{bG,F}.

Conjecture F proposes a precise global Langlands statement: under the local Fargues–Scholze equivalences Lψ_z, the object loc_Z!Λ should correspond to the pull‑back of the canonical line bundle ω_{LS_{bG,U}} on the stack of continuous \hat G‑valued Weil‑group representations over U. This conjecture unifies earlier proposals of Arinkin–Gaitsgory–Kazhdan–Raskin–Rozenblyum–Varshavsky and Zhu, and predicts that automorphic forms for all inner forms G_b (b∈B(F,G)) appear simultaneously.

The conjecture is proved when G is a torus (not necessarily split) in Theorem G. The proof uses classical class‑field theory for tori (Langlands) to compute the Galois side explicitly and match it with the automorphic side.

A crucial technical tool is Theorem H (Appendix A), a GAGA theorem for sousperfectoid spaces: for a smooth proper scheme X over a sousperfectoid affinoid algebra, the analytification functor induces an equivalence of vector‑bundle categories. This result, independently obtained by Wang, underpins the comparison between algebraic and analytic descriptions of Bun_{G,F} and is used throughout the paper (e.g., in the proof of Theorem C).

Appendix B develops basic properties of over‑convergent motivic sheaves needed to compare motivic and classical ℓ‑adic sheaf theories, while Appendix C (by Peter Dillery) adapts the construction to the global Kaletha set, extending the framework to refined inner‑form parametrizations.

Overall, the work establishes a robust global geometric framework mirroring the local Fargues–Fontaine theory, connects it to Igusa stacks and shtukas, and formulates a categorical global Langlands conjecture that is verified for commutative groups. It opens a pathway for future advances in the global Langlands program over function fields, especially in the non‑abelian case.