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arxiv: 2606.10149 · v1 · pith:ABAI55QSnew · submitted 2026-06-08 · 🧮 math.NT · math.AG· math.RT

A generic categorical local Langlands correspondence for quasi-split reductive groups

David Helm , Maarten Solleveld , Yujie Xu This is my paper

Pith reviewed 2026-06-27 14:33 UTC · model grok-4.3

classification 🧮 math.NT math.AGmath.RT
keywords categorical local LanglandsBernstein blocksL-parametersquasi-split reductive groupsp-adic groupsFargues-Scholzeind-coherent sheavesgeneric representations
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The pith

A fully faithful functor realizes the generic categorical arithmetic local Langlands correspondence for quasi-split reductive p-adic groups.

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

The paper constructs a natural fully faithful functor from the stable infinity-category of generic Bernstein blocks on the automorphic side to the stable infinity-category of ind-coherent sheaves on the moduli stack of arithmetic L-parameters. This holds for a large class of quasi-split reductive p-adic groups that includes all quasi-split classical groups. The construction generalizes the case of GL_n and provides a framework under which any classical local Langlands correspondence can be lifted to an infinity-categorical one. When combined with other recent results and an expected compatibility assumption, the same methods yield the full Fargues-Scholze categorical local Langlands equivalence without the genericity restriction for this class of groups.

Core claim

We construct a natural fully faithful functor from the stable ∞-category of generic Bernstein blocks on the automorphic side to the stable ∞-category of ind-coherent sheaves on the moduli stack of (arithmetic) L-parameters for quasi-split reductive p-adic groups, thereby proving a generic categorical local Langlands conjecture; we also formulate a classical local Langlands framework that allows lifting any classical correspondence to the ∞-categorical level.

What carries the argument

The natural fully faithful functor from the stable ∞-category of generic Bernstein blocks to the stable ∞-category of ind-coherent sheaves on the moduli stack of arithmetic L-parameters.

Load-bearing premise

The groups under consideration are quasi-split reductive p-adic groups, the Bernstein blocks are generic, and the expected compatibility of the Fargues-Scholze construction with spectral Eisenstein series holds when removing the genericity condition.

What would settle it

An explicit generic Bernstein block for one of the covered groups whose image under the functor fails to be fully faithful or fails to land in the expected ind-coherent sheaf on the L-parameter stack would disprove the claimed correspondence.

read the original abstract

We prove a generic categorical (arithmetic) local Langlands conjecture for a large class of quasi-split reductive $p$-adic groups $G$, including all quasi-split classical groups and some non-classical groups. More precisely, we construct a natural fully faithful functor from the stable $\infty$-category of generic Bernstein blocks on the automorphic side to the stable $\infty$-category of ind-coherent sheaves on the moduli stack of (arithmetic) $L$-parameters, generalizing earlier work of [BZCHN24] for $\mathrm{GL}_n$. Moreover, for an arbitrary quasi-split reductive $p$-adic group $G$, we formulate a classical local Langlands framework under which a classical correspondence can be lifted to an $\infty$-categorical correspondence. Furthermore, combined with the recent work of Hansen-Mann [HM26] and assuming the expected compatibility of Fargues-Scholze construction with spectral Eisenstein series, our results give the full Fargues--Scholze categorical local Langlands equivalence [FS24], without the genericity condition, for a large class of quasi-split reductive $p$-adic groups $G$.

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

0 major / 1 minor

Summary. The paper claims to prove a generic categorical (arithmetic) local Langlands conjecture for a large class of quasi-split reductive p-adic groups G (including all quasi-split classical groups and some non-classical groups). It constructs a natural fully faithful functor from the stable ∞-category of generic Bernstein blocks on the automorphic side to the stable ∞-category of ind-coherent sheaves on the moduli stack of (arithmetic) L-parameters, generalizing [BZCHN24] for GL_n. It also formulates a classical local Langlands framework under which a classical correspondence can be lifted to an ∞-categorical one, and (combined with Hansen-Mann and an assumed compatibility of Fargues-Scholze with spectral Eisenstein series) obtains the full Fargues-Scholze categorical equivalence without the genericity condition for the same class of groups.

Significance. If the construction of the fully faithful functor holds, the result would constitute a substantial advance in the categorical local Langlands program by extending the GL_n case to all quasi-split classical groups. The explicit formulation of a classical framework that lifts to the ∞-categorical setting and the conditional derivation of the full (non-generic) Fargues-Scholze equivalence are additional strengths that could serve as a template for further work.

minor comments (1)
  1. The abstract states that the groups include 'some non-classical groups' but does not list them; adding an explicit list or reference to the relevant section in the introduction would improve clarity.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript and for acknowledging its potential contribution to the categorical local Langlands program. The report lists the recommendation as uncertain but provides no specific major comments or points of criticism. We address this below and remain available to respond to any additional questions.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper constructs a natural fully faithful functor from generic Bernstein blocks to ind-coherent sheaves on the moduli stack of L-parameters, explicitly generalizing the GL_n case from the external citation [BZCHN24]. No load-bearing step reduces by definition or by fitted input to its own outputs; the genericity assumption and the conditional extension via [HM26] plus an external compatibility are stated as such rather than smuggled in. The derivation chain is a categorical construction relying on geometric data and prior independent results, remaining self-contained without self-citation chains or renaming of known patterns as new theorems.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The central claim rests on standard axioms of infinity-category theory, the existence of the moduli stack of L-parameters, and results from the cited literature on the local Langlands program; no free parameters or invented entities are introduced in the abstract.

axioms (3)
  • standard math Axioms of stable infinity-categories and fully faithful functors
    Invoked when defining the source and target categories and the functor between them.
  • domain assumption Existence and basic properties of the moduli stack of arithmetic L-parameters
    Required for the target category of ind-coherent sheaves.
  • domain assumption Results of [BZCHN24] for GL_n
    The generalization is stated to build directly on this prior construction.

pith-pipeline@v0.9.1-grok · 5739 in / 1231 out tokens · 44565 ms · 2026-06-27T14:33:23.457436+00:00 · methodology

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