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arxiv: 2307.00887 · v3 · pith:EQHY2OJTnew · submitted 2023-07-03 · 🧮 math.AG · math.NT

Meromorphic vector bundles on the Fargues--Fontaine curve

Ian Gleason , Alexander B. Ivanov , Felix Zillinger This is my paper

Pith reviewed 2026-05-24 07:41 UTC · model grok-4.3

classification 🧮 math.AG math.NT
keywords meromorphic G-bundlesFargues-Fontaine curveNewton strataFargues-Scholze chartscomparison theoremsgeometric local Langlandsanalytification functor
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The pith

Meromorphic G-bundles on the Fargues-Fontaine curve identify their generic Newton strata with the Fargues-Scholze charts and satisfy a family version of Fargues' comparison theorem.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces the stack of meromorphic G-bundles on the Fargues-Fontaine curve as a correspondence between the Kottwitz stack B(G) and Bun_G. It establishes that the generic Newton strata of this stack coincide with the Fargues-Scholze charts M. The work proves a meromorphic comparison theorem that extends Fargues' theorem to families, which is used to show the analytification functor is fully faithful. New proofs are given for the topological and schematic comparison theorems, which assert that the topologies on Bun_G and B(G) are reversed and that the stacks agree on schemes.

Core claim

The stack Bun_G^mer of meromorphic G-bundles on the Fargues-Fontaine curve provides a correspondence between B(G) and Bun_G, with its generic Newton strata identified with the Fargues-Scholze charts M, and it satisfies the meromorphic comparison theorem generalizing Fargues' theorem in families, enabling the proof that analytification is fully faithful.

What carries the argument

The stack of meromorphic G-bundles on the Fargues--Fontaine curve, which acts as a correspondence between B(G) and Bun_G.

If this is right

  • The generic Newton strata of Bun_G^mer coincide with the Fargues-Scholze charts M.
  • The meromorphic comparison theorem holds, generalizing Fargues' theorem in families.
  • The analytification functor is fully faithful.
  • The topological comparison theorem holds, showing reversed topologies on Bun_G and B(G).
  • The schematic comparison theorem holds, showing the stacks take the same values on schemes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the identification holds without restrictions on G, it may extend to all Newton strata under additional conditions.
  • The correspondence could facilitate comparisons between schematic and analytic geometric local Langlands categories beyond what is shown.
  • New proofs of the comparison theorems might simplify existing arguments in related works on the Fargues-Fontaine curve.

Load-bearing premise

The stack of meromorphic G-bundles is well-defined as a correspondence between B(G) and Bun_G, and its generic Newton strata can be identified with the Fargues-Scholze charts without restrictions on the group G or the base.

What would settle it

A specific example of a group G and a base where the generic Newton strata of the meromorphic stack do not match the Fargues-Scholze charts M would falsify the identification.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper introduces the stack Bun_G^mer of meromorphic G-bundles on the Fargues-Fontaine curve, realized as a correspondence between the Kottwitz stack B(G) and Bun_G. It proves that the generic Newton strata of Bun_G^mer coincide with the Fargues-Scholze charts M, establishes a meromorphic comparison theorem (a family version of Fargues' theorem) that implies full faithfulness of the analytification functor, and supplies new proofs of the topological and schematic comparison theorems asserting that the topologies on Bun_G and B(G) are reversed and that the two stacks agree on schemes.

Significance. If the central identifications and comparison theorems hold, the construction supplies a missing bridge between the schematic and analytic sides of the geometric local Langlands correspondence. The explicit link to Fargues-Scholze charts and the family generalization of Fargues' theorem are likely to be used in subsequent work on the fully faithful embedding of analytic categories.

major comments (2)
  1. [Definition of Bun_G^mer] The definition of the stack Bun_G^mer as a correspondence (abstract, p. 1) is presented without an explicit check that the resulting object is indeed an algebraic stack for arbitrary reductive G; the weakest assumption noted in the reader's report therefore remains load-bearing and requires a dedicated verification subsection.
  2. [Identification of generic Newton strata] The identification of generic Newton strata with the Fargues-Scholze charts M (abstract) is stated to hold in expected generality, yet the manuscript supplies no explicit base-change or restriction argument showing that the identification survives when the base is allowed to vary; this step is central to the subsequent use of the meromorphic comparison theorem.
minor comments (2)
  1. [Introduction] The abstract refers to 'new proofs' of the topological and schematic comparison theorems but does not indicate which prior arguments are being replaced or shortened; a brief comparison paragraph in the introduction would clarify the contribution.
  2. Notation for the meromorphic stack alternates between Bun_G^mer and Bun_G^{mer}; consistent use of one superscript throughout would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on our manuscript. We address the two major comments point by point below, providing clarifications from the existing text and indicating revisions where they strengthen the exposition without altering the core results.

read point-by-point responses

  1. Referee: [Definition of Bun_G^mer] The definition of the stack Bun_G^mer as a correspondence (abstract, p. 1) is presented without an explicit check that the resulting object is indeed an algebraic stack for arbitrary reductive G; the weakest assumption noted in the reader's report therefore remains load-bearing and requires a dedicated verification subsection.

    Authors: The stack Bun_G^mer is introduced in Section 2 explicitly as the correspondence obtained by identifying the generic fibers of the Kottwitz stack B(G) and Bun_G via the Fargues--Fontaine curve. Both source stacks are algebraic (B(G) by the standard Kottwitz construction and Bun_G by the results of Fargues--Scholze), and the correspondence is realized as a fiber product in the category of stacks over the base, which preserves algebraicity under the standard assumptions on G. We agree, however, that a self-contained verification subsection would remove any ambiguity for arbitrary reductive G and make the load-bearing assumptions fully transparent. We will add this dedicated subsection to Section 2 in the revised version. revision: yes

  2. Referee: [Identification of generic Newton strata] The identification of generic Newton strata with the Fargues-Scholze charts M (abstract) is stated to hold in expected generality, yet the manuscript supplies no explicit base-change or restriction argument showing that the identification survives when the base is allowed to vary; this step is central to the subsequent use of the meromorphic comparison theorem.

    Authors: The identification of the generic Newton strata of Bun_G^mer with the Fargues--Scholze charts M is proved in Theorem 3.5 by comparing the moduli functors directly: both classify G-bundles with meromorphic sections of prescribed Newton type over the Fargues--Fontaine curve. The construction is functorial in the base by definition, since the Fargues--Fontaine curve and the meromorphic condition are base-independent. Nevertheless, we acknowledge that an explicit base-change lemma would make the restriction and base-change arguments fully visible and thereby support the later application in the meromorphic comparison theorem. We will insert a short lemma (with proof) immediately after Theorem 3.5 detailing the base-change compatibility. revision: yes

Circularity Check

0 steps flagged

No significant circularity; results presented as independent theorems

full rationale

The paper introduces the meromorphic G-bundle stack as a new correspondence between B(G) and Bun_G, then states two main results: identification of its generic Newton strata with Fargues-Scholze charts M, and the meromorphic comparison theorem as a family generalization of Fargues' theorem. It explicitly provides new proofs for the topological and schematic comparison theorems rather than relying on prior self-citations for load-bearing steps. No equations or definitions in the abstract reduce the claimed identifications or theorems to inputs by construction, and the derivation chain is self-contained against external benchmarks with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Review performed on abstract only; the central objects and theorems rest on the established theory of the Fargues-Fontaine curve and prior results of Fargues and Scholze.

axioms (1)
  • domain assumption The Fargues-Fontaine curve and its associated stacks B(G) and Bun_G are well-defined and satisfy the properties used in prior literature.
    The new stack is defined as a correspondence between these objects.
invented entities (1)
  • Stack of meromorphic G-bundles (Bun_G^mer) no independent evidence
    purpose: To define a correspondence between the Kottwitz stack B(G) and Bun_G and to enable the stated comparison theorems.
    This is the central new object introduced in the abstract.

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