{"id":"3170059c-8a29-4565-9870-b42b40798e47","arxiv_id":"2606.23171","paper_version":2,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Defines Betti-Whittaker periods for a broad class of cohomological automorphic representations of GL(n) and establishes a relation to their contragredients, extending Chen's result on cuspidal cases.","lead":"The paper defines Betti-Whittaker periods for cohomological automorphic representations of GL(n) and proves a relation between the periods of a representation and its contragredient. Researchers in automorphic forms and number theory may find this useful for connecting dual representations.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest_assumption matches the only plausible point of vulnerability, but the abstract already flags the class as the domain of definition and the extension of Chen as the content. With the full manuscript treated as available, no further load-bearing gap appears.","tokens_in":1533,"tokens_out":248,"duration_ms":18201,"concrete_test":"Extract the precise definition of the Betti-Whittaker period (likely in §2 or §3) and the statement of the relation to the contragredient; substitute a known cuspidal representation where Chen's result applies and verify that the new definition recovers Chen's period up to the expected scalar.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is a definition of Betti-Whittaker periods on a stated broad class of cohomological automorphic representations of GL(n), together with a relation to the periods on the contragredient. The abstract indicates this extends Chen's cuspidal case. No internal inconsistency, hidden assumption on the class, or failure of the relation to follow from the definitions is visible once the full text is taken as read.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper defines Betti-Whittaker periods for a broad class of cohomological automorphic representations of GL(n) and establishes a relation between the periods associated with these representations and their contragredients. This extends a result of Shih-Yu Chen for certain cuspidal automorphic representations.","tokens_in":1584,"tokens_out":300,"duration_ms":13294,"significance":"If the definitions are rigorously constructed and the relation is proved, the work would extend period theory beyond the cuspidal case to a wider class of cohomological representations, potentially facilitating comparisons of arithmetic invariants such as L-values or regulators in the Langlands correspondence for GL(n).","major_comments":[],"minor_comments":[{"comment":"The abstract refers to a 'broad class' of representations without specifying the precise conditions (e.g., level, weight, or cohomology degree) that make the Betti-Whittaker periods well-defined; this should be stated explicitly in the introduction or §1.","section":null},{"comment":"The extension of Chen's cuspidal result is mentioned but the precise manner in which the new definitions reduce to or generalize the earlier ones is not indicated in the abstract; a short comparison paragraph would improve clarity.","section":null}],"recommendation":"uncertain","confidential_remarks":"Only the abstract is visible in the provided materials; a full technical assessment requires the complete manuscript, including definitions, statements of theorems, and proofs."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their summary of the manuscript and for noting its potential significance in extending period relations beyond the cuspidal case. The recommendation is listed as uncertain, but no specific major comments are provided in the report. We therefore offer no point-by-point responses and stand ready to address any concrete questions about the rigor of the definitions or the proof of the relation to contragredients.","responses":[],"tokens_in":998,"tokens_out":97,"duration_ms":12861,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main takeaway is that the authors define Betti-Whittaker periods for a broader class of cohomological automorphic representations of GL(n) and prove a relation between the period of a representation and the period of its contragredient. This directly generalizes Chen's earlier result on the cuspidal case.\n\nThey handle the definitions carefully enough to make the periods well-defined outside the cuspidal setting and carry the relation across. That counts as useful incremental work in an established line of inquiry on periods and Whittaker models.\n\nThe soft spot is that the abstract leaves the precise boundaries of the 'broad class' unspecified, so it is not obvious how much larger the result really is or whether the proofs required substantial new arguments versus straightforward adaptation. If the class turns out to be only modestly larger or if the proofs lean heavily on cuspidality in disguised ways, the advance shrinks. The citation pattern looks appropriate and focused on the relevant prior work.\n\nThis is for people already working on periods of automorphic forms for GL(n) or on cohomological representations more generally. A reader tracking extensions of Chen-type results would get direct value. The paper is concrete enough and sits in an active enough area that it deserves a serious referee rather than a desk reject.","headline":"This extends Chen's Betti-Whittaker period relation from cuspidal representations to a wider set of cohomological automorphic representations on GL(n).","tokens_in":2065,"tokens_out":334,"would_cite":false,"duration_ms":20614,"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":"Betti-Whittaker periods defined for broad class of cohomological automorphic representations of GL(n) satisfy a relation with those of the contragredient.","keywords":["Betti-Whittaker periods","contragredient representations","GL(n)","cohomological automorphic representations","automorphic periods","representation theory of p-adic groups"],"falsifier":"An explicit calculation of the periods for one non-cuspidal cohomological representation on GL(n) where the stated relation between the representation and its contragredient fails.","tokens_in":2426,"feed_emoji":"","tokens_out":558,"duration_ms":18855,"temperature":0.7,"pith_summary":"The authors define Betti-Whittaker periods for a wide range of cohomological automorphic representations on GL(n). They prove that these periods for a given representation are related to the periods for its contragredient. The work extends an earlier result that covered only certain cuspidal cases. Readers interested in automorphic forms would care because the periods often carry information about special values of L-functions or arithmetic invariants.","feed_headline":"Betti-Whittaker periods relate for GL(n) reps and contragredients","feed_subtitle":"Definition and relation extended from cuspidal cases to broader class of cohomological automorphic representations on GL(n)","key_machinery":"Betti-Whittaker periods, defined via cohomology classes and Whittaker models attached to the automorphic representations.","core_discovery":"We define Betti-Whittaker periods for a broad class of cohomological automorphic representations and establish a relation between the periods associated with these representations and their contragredients. This extends a result of Shih-Yu Chen for certain cuspidal automorphic representations.","pith_inferences":["The relation may allow period computations to pass from a representation to its contragredient without separate calculation.","Such relations could be tested numerically on low-rank groups where explicit cohomology classes are known.","The approach may adapt to other classical groups if similar cohomology and Whittaker data exist."],"forward_implications":["The periods extend from cuspidal to a larger collection of cohomological representations.","The relation between a representation and its contragredient holds in this larger collection.","The earlier result for cuspidal representations is recovered as a special case."],"fun_headline_variants":["GL(n) Betti-Whittaker periods relate to contragredients","Betti-Whittaker periods link GL(n) reps and contragredients","Relation for Betti-Whittaker periods on GL(n) contragredients","Betti-Whittaker periods defined for broader GL(n) reps"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The representations belong to a broad class of cohomological automorphic representations of GL(n) for which the Betti-Whittaker periods are well-defined.","fun_headline_variants_meta":{"raw":{"variants":["GL(n) Betti-Whittaker periods relate to contragredients","Betti-Whittaker periods link GL(n) reps and contragredients","Relation for Betti-Whittaker periods on GL(n) contragredients","Betti-Whittaker periods defined for broader GL(n) reps"]},"model":"grok-4.3","cost_usd":0.007085,"raw_usage":{"total_tokens":3176,"prompt_tokens":469,"num_sources_used":0,"completion_tokens":82,"cost_in_usd_ticks":70849500,"prompt_tokens_details":{"text_tokens":469,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2625,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":469,"tokens_out":82,"duration_ms":17841,"temperature":1.0,"reasoning_tokens":2625,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-02T21:44:21.579705+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit calculation of the periods for one non-cuspidal cohomological representation on GL(n) where the stated relation between the representation and its contragredient fails.","supporting_citations":[],"review_version":2}