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arxiv: 2606.29478 · v1 · pith:MFG6TYH2new · submitted 2026-06-28 · 🧮 math.NT · math.AG

The Categorical Local Langlands Correspondence and Anabelomorphy

Kirti Joshi This is my paper

Pith reviewed 2026-06-30 02:08 UTC · model grok-4.3

classification 🧮 math.NT math.AG
keywords anabelomorphyp-adic fieldsLanglands parametersabsolute Galois groupscategorical local Langlandsreductive groupssplit tori
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The pith

Anabelomorphic p-adic fields have isomorphic stacks of Langlands parameters.

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

The paper shows that whenever two p-adic fields have topologically isomorphic absolute Galois groups, the stacks of Langlands parameters attached to them in the Fargues-Scholze construction are isomorphic. This identification links the categorical local Langlands correspondence directly to anabelomorphy of fields. A reader would care because the result indicates that these stacks are determined by the Galois group topology alone, so arithmetic data encoded in the stacks can transfer between such fields. The author formulates a conjecture that relates this isomorphism to the main conjecture of Fargues and Scholze and proves the conjecture when the group is a split torus.

Core claim

If K and L are anabelomorphic p-adic fields, then the stacks of Langlands parameters for K and L considered in Fargues and Scholze are isomorphic. This leads to a conjecture providing a precise relationship between the main conjecture of Fargues and Scholze and anabelomorphy of p-adic fields, and the conjecture is established when the reductive group is a split torus.

What carries the argument

The stack of Langlands parameters, constructed so that it depends only on the topological isomorphism type of the absolute Galois group.

If this is right

  • The isomorphism transfers the structure of Langlands parameters between anabelomorphic fields.
  • The main conjecture of Fargues and Scholze for one field corresponds to the same conjecture for any anabelomorphic field.
  • The relationship between the two conjectures holds when the group is a split torus.

Where Pith is reading between the lines

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

  • The result suggests that any statement about Langlands parameters that can be phrased in terms of the stack would be invariant under anabelomorphy.
  • It raises the question of whether other objects in the local Langlands correspondence, such as the category of representations, also descend to the Galois group topology.

Load-bearing premise

The stacks of Langlands parameters are constructed in a manner that depends only on the topological isomorphism type of the absolute Galois group.

What would settle it

A pair of anabelomorphic p-adic fields whose stacks of Langlands parameters are not isomorphic would falsify the central claim.

read the original abstract

Let $G/\mathbb{Q}_p$ be a connected, split, reductive group over $\mathbb{Q}_p$. In this paper I show that if $K$ and $L$ are anabelomorphic $p$-adic fields i.e. $K$ and $L$ have topologically isomorphic absolute Galois groups, then the stacks of Langlands parameters (for the fields $K$ and $L$) considered in [Fargues and Scholze, 2024], are also isomorphic (Theorem 2.2.1). This leads to Conjecture 3.3.1 which provides a precise relationship between the main conjecture of [Fargues and Scholze, 2024] and anabelomorphy of $p$-adic fields considered in [Joshi, 2020a]. I establish my conjecture for a split torus in Theorem 4.1.

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 manuscript claims that if K and L are anabelomorphic p-adic fields (topologically isomorphic absolute Galois groups), then the stacks of G-Langlands parameters constructed in Fargues-Scholze (2024) are isomorphic (Theorem 2.2.1). It formulates Conjecture 3.3.1 relating the Fargues-Scholze main conjecture to anabelomorphy results from the author's prior work, and establishes the conjecture for a split torus in Theorem 4.1.

Significance. If the central claim holds, the result would establish invariance of the categorical local Langlands correspondence under anabelomorphy of local fields, providing a mechanism to transfer statements across fields with isomorphic Galois groups. The explicit verification for split tori supplies a concrete special case where the correspondence is shown to be compatible with anabelomorphy.

major comments (2)
  1. [§2.2, Theorem 2.2.1] §2.2, Theorem 2.2.1: The claim that a topological isomorphism Gal(K) ≅ Gal(L) induces an isomorphism of the Fargues-Scholze stacks is asserted without an explicit construction showing how the isomorphism of Galois groups produces an isomorphism of the underlying Fargues-Fontaine curves (which depend on the specific ring of integers, residue field, and untilts of each field); the manuscript invokes the Fargues-Scholze framework but does not verify that the stack isomorphism follows from the group isomorphism alone.
  2. [Theorem 4.1] Theorem 4.1: The verification for split tori is presented as a partial check of Conjecture 3.3.1, but the argument relies on the same unelaborated invariance of the stacks under Galois-group isomorphisms; it is unclear whether the torus case avoids the field-specific data in the curve construction or merely specializes it.
minor comments (2)
  1. The notation for the stacks of Langlands parameters could be introduced with a brief reminder of the precise functoriality with respect to the base field in the Fargues-Scholze reference.
  2. References to the author's prior anabelomorphy results [Joshi, 2020a] are frequent; a short self-contained summary of the relevant anabelomorphy statement would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address each major comment below and will revise the manuscript to improve the explicitness of the arguments.

read point-by-point responses

  1. Referee: [§2.2, Theorem 2.2.1] §2.2, Theorem 2.2.1: The claim that a topological isomorphism Gal(K) ≅ Gal(L) induces an isomorphism of the Fargues-Scholze stacks is asserted without an explicit construction showing how the isomorphism of Galois groups produces an isomorphism of the underlying Fargues-Fontaine curves (which depend on the specific ring of integers, residue field, and untilts of each field); the manuscript invokes the Fargues-Scholze framework but does not verify that the stack isomorphism follows from the group isomorphism alone.

    Authors: We agree that the proof of Theorem 2.2.1 would be strengthened by an explicit construction. The manuscript derives the stack isomorphism from the functoriality of the Fargues-Scholze construction with respect to isomorphisms of absolute Galois groups, which induce equivalences on the categories of untilts and hence on the Fargues-Fontaine curves. However, we acknowledge that this step is not elaborated in detail. In the revised version we will add a dedicated paragraph (or subsection) providing the explicit correspondence between the Galois-group isomorphism and the induced isomorphism of curves and stacks. revision: yes

  2. Referee: [Theorem 4.1] Theorem 4.1: The verification for split tori is presented as a partial check of Conjecture 3.3.1, but the argument relies on the same unelaborated invariance of the stacks under Galois-group isomorphisms; it is unclear whether the torus case avoids the field-specific data in the curve construction or merely specializes it.

    Authors: Theorem 4.1 applies the stack isomorphism of Theorem 2.2.1 to the split-torus case, where Langlands parameters reduce to continuous homomorphisms from the Galois group into the dual torus. This description depends only on the Galois group, but the underlying geometric data of the Fargues-Fontaine curve remains. We will revise the proof of Theorem 4.1 to explicitly trace how the Galois-group isomorphism produces the required equivalence in the torus setting, thereby clarifying the relationship to the curve construction. revision: yes

Circularity Check

0 steps flagged

Minor self-citation in conjecture framing; central theorem independent of self-cited inputs

full rationale

The paper states Theorem 2.2.1 as a result shown within the manuscript by invoking the external Fargues-Scholze construction and the definition of anabelomorphy (topological Galois isomorphism). Conjecture 3.3.1 then relates this to the author's prior work on anabelomorphy, but this is framing rather than a load-bearing step that reduces the theorem itself to a self-citation or tautology. No equation or derivation in the provided text equates the claimed isomorphism to an input by construction; the FS framework is cited as external and the proof of the theorem is presented as new content. This is the most common honest outcome for papers that cite their own prior definitions without making the new claim equivalent to that citation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available; no explicit free parameters, invented entities, or non-standard axioms are visible. The work relies on standard background from algebraic geometry and number theory as referenced.

axioms (1)
  • domain assumption The stacks of Langlands parameters are functorial with respect to topological isomorphisms of absolute Galois groups.
    Invoked to obtain the isomorphism in Theorem 2.2.1 from the definition of anabelomorphy.

pith-pipeline@v0.9.1-grok · 5675 in / 1348 out tokens · 39691 ms · 2026-06-30T02:08:14.928531+00:00 · methodology

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Reference graph

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