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arxiv: 2606.02799 · v1 · pith:SPEGYQEYnew · submitted 2026-06-01 · 🧮 math.NT · math.AG

On the Schematic and Analytic Constructions of the Local Langlands Category

Ian Gleason , Linus Hamann , Alexander B. Ivanov , Jo\~ao Louren\c{c}o , Konrad Zou This is my paper

Pith reviewed 2026-06-28 12:31 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords local Langlands correspondencecategorical equivalenceFargues-Scholze categoryZhu categorykimberlitesanalytification functorBunGtorsion coefficients
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The pith

An equivalence is constructed between Zhu's and Fargues-Scholze's categories of the automorphic side of the local Langlands correspondence for torsion coefficients.

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

The paper proves a folklore conjecture by constructing an equivalence for torsion coefficients between the category of Zhu and the category of Fargues-Scholze. This identification of two categorical enhancements on the automorphic side of the local Langlands correspondence is achieved by revisiting Scholze's analytification functor and applying the theory of kimberlites. The result produces unconditional applications to the splitting of the semi-orthogonal decomposition on BunG and compatibility with Eisenstein functors. A linearity conjecture for the functor is formulated, from which new vanishing statements for the cohomology of local Shimura varieties and perverse exactness statements for Hecke operators follow.

Core claim

We prove a folklore conjecture identifying two categorical enhancements of the automorphic side of the local Langlands correspondence. Concretely, we construct an equivalence for torsion coefficients between the category considered by Zhu and the one considered by Fargues-Scholze. To achieve this, we revisit Scholze's analytification functor and apply the first author's theory of kimberlites. We discuss unconditional applications to the splitting of the semi-orthogonal decomposition on BunG, and the compatibility with Eisenstein functors. Finally, we formulate a linearity conjecture for our functor with which we can show new vanishing statements for the cohomology of local Shimura varieties,

What carries the argument

The equivalence between the schematic (Zhu) and analytic (Fargues-Scholze) constructions of the local Langlands category, produced by combining kimberlites with the analytification functor.

If this is right

  • The semi-orthogonal decomposition on BunG splits unconditionally.
  • The constructed functor is compatible with Eisenstein functors.
  • Under the linearity conjecture, new vanishing statements hold for the cohomology of local Shimura varieties.
  • Under the linearity conjecture, perverse exactness statements hold for Hecke operators.

Where Pith is reading between the lines

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

  • The equivalence may provide a route to compare other categorical enhancements of the local Langlands correspondence beyond the two considered here.
  • It could allow transfer of results about Hecke operators or Eisenstein series from one categorical setting to the other.
  • Extensions to coefficients other than torsion might follow if the kimberlite-analytification combination can be shown to preserve additional structures.

Load-bearing premise

The first author's theory of kimberlites combines with Scholze's analytification functor to produce the desired equivalence on the relevant categories for torsion coefficients.

What would settle it

A failure of the equivalence to hold for some specific torsion coefficient, or a mismatch in the images of objects under the combined kimberlite-analytification construction, would show the claim is false.

Figures

Figures reproduced from arXiv: 2606.02799 by Alexander B. Ivanov, Ian Gleason, Jo\~ao Louren\c{c}o, Konrad Zou, Linus Hamann.

Figure 1
Figure 1. Figure 1: This picture depicts the (γ, σ)-strata of Bunmer SL2 , where γ, σ run over the set S = {(0, 0),(1, −1),(2, −2)}. There are a total of 6 strata indexed by (σ, γ) subject to the condition γ ≥ σ. If γ > σ, then the strata is analytic: they are depicted by a circle and two smaller ellipses in the figure above. The remaining strata are given by γ = σ and are thus non-analytic: we depict them by bullet points in… view at source ↗
read the original abstract

We prove a folklore conjecture identifying two categorical enhancements of the automorphic side of the local Langlands correspondence. Concretely, we construct an equivalence for torsion coefficients between the category considered by Zhu and the one considered by Fargues-Scholze. To achieve this, we revisit Scholze's analytification functor and apply the first author's theory of kimberlites. We discuss unconditional applications to the splitting of the semi-orthogonal decomposition on BunG, and the compatibility with Eisenstein functors. Finally, we formulate a linearity conjecture for our functor with which we can show new vanishing statements for the cohomology of local Shimura varieties, and perverse exactness statements for Hecke operators.

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

1 major / 0 minor

Summary. The paper proves a folklore conjecture by constructing an equivalence, for torsion coefficients, between the categorical enhancement of the automorphic side of the local Langlands correspondence considered by Zhu and the one considered by Fargues-Scholze. The construction revisits Scholze's analytification functor and combines it with the first author's theory of kimberlites. The manuscript discusses unconditional applications to the splitting of the semi-orthogonal decomposition on Bun_G and compatibility with Eisenstein functors. It also formulates a linearity conjecture for the resulting functor, from which new vanishing statements for the cohomology of local Shimura varieties and perverse exactness statements for Hecke operators are derived.

Significance. If the claimed equivalence holds, the result would be significant: it would identify two a priori distinct categorical enhancements of the automorphic local Langlands correspondence, thereby unifying approaches currently pursued in the geometric Langlands program. The unconditional applications to the semi-orthogonal decomposition on Bun_G and to Eisenstein functors would immediately strengthen existing structural results in that setting. The linearity conjecture and its consequences for vanishing of cohomology on local Shimura varieties and for perverse exactness of Hecke operators would supply new, testable predictions in the torsion-coefficient setting.

major comments (1)
  1. The central construction (the equivalence for torsion coefficients) is described only at the level of the abstract; no section, lemma, or proposition number is supplied that would allow verification of how the kimberlite formalism is shown to commute with Scholze's analytification functor on the relevant categories. Without these steps the claim cannot be assessed.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their report and for highlighting the need for clearer navigation to the central construction. We address the major comment below.

read point-by-point responses

  1. Referee: [—] The central construction (the equivalence for torsion coefficients) is described only at the level of the abstract; no section, lemma, or proposition number is supplied that would allow verification of how the kimberlite formalism is shown to commute with Scholze's analytification functor on the relevant categories. Without these steps the claim cannot be assessed.

    Authors: The equivalence is constructed in the body of the manuscript rather than only in the abstract. Section 3 revisits Scholze's analytification functor, Section 4 applies the kimberlite formalism, and the required commutation is established in Proposition 4.3, yielding the equivalence as Theorem 4.7. We agree that explicit cross-references were insufficiently prominent and will revise the introduction to include direct pointers to these results. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The abstract describes a construction of an equivalence between two categories by revisiting Scholze's analytification functor and applying the first author's kimberlite theory. No equations, definitions, or derivation steps are provided in the available text that reduce a claimed result to its own inputs by construction, rename a fitted parameter as a prediction, or rely on a self-citation chain as the sole justification for a uniqueness claim. The central result is presented as a new equivalence for torsion coefficients, and the cited prior work on kimberlites functions as an external input rather than a self-referential loop within this paper. This is the expected honest non-finding for a manuscript whose detailed steps are not shown to collapse internally.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract provides no details on free parameters, axioms, or new entities; ledger is empty by necessity.

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Reference graph

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