Local-global compatibility at $p\neq\ell$ for torsion automorphic forms

Bence Hevesi

Local-global compatibility at $p\neq\ell$ for torsion automorphic forms

T0 review · 0 major / 2 minor · reviewed 2026-07-01 · grok-4.3

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Pith's one-line read Local-global compatibility at p ≠ ℓ holds for torsion classes in Betti cohomology using Scholze's automorphic group determinants.

desk verdict This extends Varma's local-global compatibility to torsion Betti classes by combining Scholze's determinants with Z_ℓ-coefficient p-adic GL_n representations, and the stated logic shows no internal breaks. read the letter →

arxiv 2606.31698 v1 pith:KMWRTQYM submitted 2026-06-30 math.NT math.RT

Bence Hevesi This is my paper · ORCID

classification math.NTmath.RT

keywords local-globalcompatibilitytorsionautomorphicformsBetticohomologyp-adicgenerallineargroupsgroupdeterminantsScholzeconstructionnumbertheory

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves that local-global compatibility results at primes p different from ℓ extend to torsion automorphic forms appearing in Betti cohomology. It generalizes an earlier result of Varma by applying the same compatibility to these torsion classes. A sympathetic reader would care because this strengthens the connection between local representations at p and global automorphic data even when torsion is present. The argument works by combining Scholze's construction of group determinants with the known theory of representations of p-adic general linear groups over Z_ℓ coefficients.

What carries the argument

Scholze's automorphic group determinants combined with the representation theory of p-adic general linear groups with Z_ℓ coefficients.

What would settle it

A concrete counter-example consisting of a torsion class in Betti cohomology for which the local component at some p ≠ ℓ fails to match the global determinant in the expected way.

Watch

Extended reading notes

Core claim

Local-global compatibility at p ≠ ℓ holds for the automorphic group determinants of Scholze when these are applied to torsion classes in Betti cohomology, generalizing the corresponding statement for non-torsion classes.

Load-bearing premise

The theory of representations of p-adic general linear groups with Z_ℓ coefficients combines with Scholze's construction without extra obstructions when torsion classes are involved.

Editorial extensions

If this is right

Torsion classes in Betti cohomology obey the same local-global compatibility as ordinary classes at places away from ℓ.
The compatibility statement applies uniformly to the group determinants constructed by Scholze.
The result covers all torsion classes that appear in the relevant cohomology, not merely those arising from ordinary or crystalline sources.
The argument relies only on the existing theory of Z_ℓ-coefficient representations of p-adic GL_n and does not require new local input.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If the compatibility holds for Betti cohomology, analogous statements may be testable in étale or de Rham cohomology settings with the same coefficient ring.
The result suggests that local-global matching for torsion can be used to constrain possible Galois representations attached to torsion automorphic forms.
Extending the same technique to other groups or to places p = ℓ would require checking whether the Z_ℓ-representation theory still combines without obstruction.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The main point is that the paper proves local-global compatibility at p ≠ ℓ for Scholze's automorphic group determinants, extending Varma's result to torsion classes in Betti cohomology. The argument is built by taking Scholze's construction and pairing it directly with the existing theory of representations of p-adic general linear groups over Z_ℓ.

What is new is the torsion extension itself. The paper does a clean job of framing the work as a targeted generalization rather than a wholesale new setup, and it keeps the focus on how the two pieces fit together for the torsion case.

The soft spots are limited. The abstract is short, so the details of how torsion is carried through the local correspondence matter, but nothing in the claim structure points to circularity or an unstated vanishing assumption that would collapse the argument. The key assumption—that the combination works without extra obstructions for torsion—is exactly the assertion the paper makes, and the stress-test note finds no visible inconsistency in that logic. If the full text shows the local side matching without new conditions on the torsion, the result holds; if not, that would be the place to press.

This is for people already working on p-adic aspects of the Langlands correspondence or on Galois representations attached to torsion cohomology. A reader who knows Varma and Scholze would get direct value from seeing the extension written out.

It deserves a serious referee because it addresses a central compatibility question with a concrete, falsifiable claim. I would send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript proves local-global compatibility results at p ≠ ℓ for the automorphic group determinants constructed by Scholze, generalising Varma's result to torsion classes appearing in Betti cohomology. The argument proceeds by combining Scholze's construction with the existing theory of representations of p-adic general linear groups with Z_ℓ-coefficients.

Significance. If the result holds, it would extend a key compatibility statement in the Langlands program to the torsion setting, which is relevant for integral structures, modularity lifting, and the study of automorphic forms over Z_ℓ. The approach leverages two established frameworks (Scholze's determinants and Z_ℓ-representations of GL_n) without introducing new ad-hoc parameters or invented entities.

minor comments (2)

The abstract and introduction should explicitly state the precise range of p and ℓ (e.g., whether p is unramified or any prime distinct from ℓ) and the level of the torsion classes considered, to make the scope of the generalisation from Varma immediately clear.
Notation for the group determinant and the local Langlands correspondence at p should be introduced once and used consistently; cross-references to the relevant statements in Scholze's work would aid readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, assessment of significance, and recommendation of minor revision. No specific major comments were provided in the report.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity; derivation combines independent prior constructions

full rationale

The paper asserts local-global compatibility by combining Scholze's group determinants (external construction) with the established theory of p-adic GL_n representations over Z_ℓ coefficients (also external). No equations, parameters, or uniqueness claims are shown to reduce to self-defined quantities, fitted inputs renamed as predictions, or load-bearing self-citations. The generalization of Varma's result is presented as an application of these two independent inputs to torsion classes, with no internal reduction or ansatz smuggling visible in the stated argument structure. The derivation is therefore self-contained against external benchmarks.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Abstract provides no information on free parameters, axioms, or invented entities.

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Cite this review

Pith. "Pith review of Local-global compatibility at $p\neq\ell$ for torsion automorphic forms." pith.science (2026). https://pith.science/paper/KMWRTQYM

@misc{pith2026260631698,
  author       = {Pith},
  title        = {Pith review of: Local-global compatibility at $p\neq\ell$ for torsion automorphic forms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KMWRTQYM}},
  note         = {Machine review of arXiv:2606.31698}
}

abstract

We prove local-global compatibility results at $p \neq \ell$ for the automorphic group determinants constructed by Scholze, generalising the result of Varma to torsion classes appearing in Betti cohomology. Our argument combines the construction of Scholze with the theory of representations of $p$-adic general linear groups with $\mathbf{Z}_{\ell}$-coefficients.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed July 1, 2026 · model on record in the stance chip above.

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work page 2014

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1999 , publisher=

Companion forms and weight one forms , author=. 1999 , publisher=

work page 1999