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Meromorphic vector bundles on the Fargues--Fontaine curve

Ian Gleason, Alexander B. Ivanov, Felix Zillinger

TL;DR

The paper constructs and analyzes the stack of meromorphic G-bundles on the Fargues--Fontaine curve, forging a bridge between the schematic Kottwitz stack ${\mathfrak B}(G)$ and the analytic stack ${\rm Bun}_G$ to illuminate the geometric local Langlands program. It introduces ${\rm Bun}_G^{mer}$ and proves two central results: (i) a generic Newton strata identification with Fargues--Scholze charts ${\mathcal M}$, and (ii) a meromorphic comparison theorem extending Fargues' theorem to families, which underpins the full faithfulness of analytification. The work also provides new proofs of a schematic comparison ${\rm Bun}_G^{red}\simeq {\mathfrak B}(G)$ and a topological comparison connecting the topologies of the two stacks. A rich framework of v-stacks, isocrystals, Dieudonné modules, shtukas, and semi-stable filtrations is developed, enabling a precise understanding of Newton strata and their compatibility across meromorphic, analytic, and schematic perspectives. Overall, the results push toward a unified categorical Langlands picture by relating analytic and schematic Langlands categories through a meromorphic intermediary and explicit comparison theorems.

Abstract

We introduce and study the stack of \textit{meromorphic} $G$-bundles on the Fargues--Fontaine curve. This object defines a correspondence between the Kottwitz stack $\mathfrak{B}(G)$ and $\operatorname{Bun}_G$. We expect it to play a crucial role in comparing the schematic and analytic versions of the geometric local Langlands categories. Our first main result is the identification of the generic Newton strata of ${\operatorname{Bun}}_G^{\operatorname{mer}}$ with the Fargues--Scholze charts $\mathcal{M}$. Our second main result is a generalization of Fargues' theorem in families. We call this the \textit{meromorphic comparison theorem}. It plays a key role in proving that the analytification functor is fully faithful. Along the way, we give new proofs to what we call the \textit{topological and schematic comparison theorems}. These say that the topologies of $\operatorname{Bun}_G$ and $\mathfrak{B}(G)$ are reversed and that the two stacks take the same values when evaluated on schemes.

Meromorphic vector bundles on the Fargues--Fontaine curve

TL;DR

The paper constructs and analyzes the stack of meromorphic G-bundles on the Fargues--Fontaine curve, forging a bridge between the schematic Kottwitz stack and the analytic stack to illuminate the geometric local Langlands program. It introduces and proves two central results: (i) a generic Newton strata identification with Fargues--Scholze charts , and (ii) a meromorphic comparison theorem extending Fargues' theorem to families, which underpins the full faithfulness of analytification. The work also provides new proofs of a schematic comparison and a topological comparison connecting the topologies of the two stacks. A rich framework of v-stacks, isocrystals, Dieudonné modules, shtukas, and semi-stable filtrations is developed, enabling a precise understanding of Newton strata and their compatibility across meromorphic, analytic, and schematic perspectives. Overall, the results push toward a unified categorical Langlands picture by relating analytic and schematic Langlands categories through a meromorphic intermediary and explicit comparison theorems.

Abstract

We introduce and study the stack of \textit{meromorphic} -bundles on the Fargues--Fontaine curve. This object defines a correspondence between the Kottwitz stack and . We expect it to play a crucial role in comparing the schematic and analytic versions of the geometric local Langlands categories. Our first main result is the identification of the generic Newton strata of with the Fargues--Scholze charts . Our second main result is a generalization of Fargues' theorem in families. We call this the \textit{meromorphic comparison theorem}. It plays a key role in proving that the analytification functor is fully faithful. Along the way, we give new proofs to what we call the \textit{topological and schematic comparison theorems}. These say that the topologies of and are reversed and that the two stacks take the same values when evaluated on schemes.
Paper Structure (30 sections, 94 theorems, 150 equations)

This paper contains 30 sections, 94 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Motivation and context
  3. Main results
  4. New proofs of two established results
  5. The schematic comparison
  6. The topological comparison
  7. Acknowledgements
  8. Notation, terminology and generalities
  9. Grothendieck topologies
  10. Combs and product of points
  11. Categories of vector bundles
  12. $E_\infty$-sous-perfectoid spaces
  13. On ${\mathcal{V}}_Y^{\operatorname{mer}}$ and ${\mathcal{V}}_Y^{\operatorname{An}}$
  14. Meromorphic vector bundles on the Fargues--Fontaine curve
  15. Dieudonné modules and Isocrystals
  16. ...and 15 more sections

Key Result

Theorem 1.1

We have a commutative diagram with Cartesian square

Theorems & Definitions (249)

  • Theorem 1.1: \ref{['maintheorem']}
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4: \ref{['meromorphic-comparison-theorem']}
  • Remark 1.5
  • Corollary 1.6
  • Remark 1.7
  • Proposition 1.8: \ref{['lemma-extending-at-infty']}
  • Corollary 1.9
  • Remark 1.10
  • ...and 239 more