REVIEW 2 minor 146 references
Local-global compatibility at p ≠ ℓ holds for torsion classes in Betti cohomology using Scholze's automorphic group determinants.
Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →
T0 review · grok-4.3
2026-07-01 03:43 UTC pith:KMWRTQYM
load-bearing objection 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.
Local-global compatibility at pneqell for torsion automorphic forms
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
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.
What carries the argument
Scholze's automorphic group determinants combined with the representation theory of p-adic general linear groups with Z_ℓ coefficients.
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.
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.
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.
Where Pith is reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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
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
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.
Axiom & Free-Parameter Ledger
read the original 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.
Reference graph
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discussion (0)
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