{"id":"b8de7e5a-891e-4a6b-85a3-063afb577e7a","arxiv_id":"2606.31698","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves local-global compatibility at p ≠ ℓ for Scholze's automorphic group determinants, generalizing Varma's result to torsion classes in Betti cohomology via p-adic GL representations with Z_ℓ coefficients.","lead":"The paper proves local-global compatibility results at primes p not equal to ℓ for automorphic group determinants from Scholze's constructions. It extends prior work by Varma to include torsion classes in Betti cohomology by combining existing tools from p-adic representation theory.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader correctly flagged the combination step as the key assumption and correctly set low confidence due to abstract-only access. With the full text now available per instructions, that assumption is presented as the content of the paper rather than an unexamined gap; no separate load-bearing flaw appears.","tokens_in":1563,"tokens_out":287,"duration_ms":32414,"concrete_test":"Locate the statement of the main theorem (likely Theorem A or 1.1) and the paragraph that invokes the Z_ℓ-representation theory; confirm that the proof reduces the torsion case to the known smooth case via a cited reference or explicit functor without introducing new hypotheses on the cohomology class.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a direct generalization of Varma's result via Scholze's group determinants combined with the existing theory of p-adic GL_n representations over Z_ℓ. The abstract states that the argument proceeds by this combination for torsion classes in Betti cohomology; no internal inconsistency, hidden assumption on torsion vanishing, or failure of the local correspondence at p ≠ ℓ is visible from the claim structure. The reader's weakest assumption (successful combination without extra obstructions) is exactly the point the paper asserts it resolves, and nothing in the given description indicates that assertion is circular or unsupported within the paper's own logic.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","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.","tokens_in":1633,"tokens_out":311,"duration_ms":29785,"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.","major_comments":[],"minor_comments":[{"comment":"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.","section":null},{"comment":"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.","section":null}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"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.","responses":[],"tokens_in":1042,"tokens_out":48,"duration_ms":12751,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","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.\n\nWhat 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.\n\nThe 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.\n\nThis 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.\n\nIt deserves a serious referee because it addresses a central compatibility question with a concrete, falsifiable claim. I would send it to peer review.","headline":"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.","tokens_in":2108,"tokens_out":396,"would_cite":false,"duration_ms":34336,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"grok-4.3","headline":"Local-global compatibility at p ≠ ℓ holds for torsion classes in Betti cohomology using Scholze's automorphic group determinants.","keywords":["local-global compatibility","torsion automorphic forms","Betti cohomology","p-adic general linear groups","group determinants","Scholze construction","number theory"],"falsifier":"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.","tokens_in":2437,"feed_emoji":"","tokens_out":630,"duration_ms":28446,"temperature":0.7,"pith_summary":"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.","feed_headline":"Local-global compatibility holds for torsion classes at p ≠ ℓ","feed_subtitle":"Extends Varma's result to Betti cohomology torsion via Scholze determinants and Z_ℓ representations of p-adic groups.","key_machinery":"Scholze's automorphic group determinants combined with the representation theory of p-adic general linear groups with Z_ℓ coefficients.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["Local-global compatibility at p ≠ ℓ for torsion automorphic forms","Scholze determinants compatible at p ≠ ℓ for torsion Betti classes","Local-global compatibility for Betti torsion classes at p ≠ ℓ","Local-global compatibility at p ≠ ℓ for torsion in Betti cohomology"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Local-global compatibility at p ≠ ℓ for torsion automorphic forms","Scholze determinants compatible at p ≠ ℓ for torsion Betti classes","Local-global compatibility for Betti torsion classes at p ≠ ℓ","Local-global compatibility at p ≠ ℓ for torsion in Betti cohomology"]},"model":"grok-4.3","cost_usd":0.010318,"raw_usage":{"total_tokens":4386,"prompt_tokens":464,"num_sources_used":0,"completion_tokens":72,"cost_in_usd_ticks":103178000,"prompt_tokens_details":{"text_tokens":464,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":3850,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":464,"tokens_out":72,"duration_ms":51294,"temperature":1.0,"reasoning_tokens":3850,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T03:43:17.692563+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"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.","supporting_citations":[],"review_version":1}