{"id":"6ae4ac83-2bff-402d-8fef-7753ab36356f","arxiv_id":"2606.31395","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":8.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Constructs holomorphic differential forms of many degrees including the minimal one on orthogonal modular varieties for even lattices of signature (2,n) with n≡1,3 mod 8, discriminant -2, n≥25, using the Arthur multiplicity formula; first non-top-degree examples.","lead":"The paper constructs holomorphic differential forms of various degrees, including the lowest possible, on orthogonal modular varieties tied to specific even lattices of signature (2,n) with n at least 25. A smart generalist might read it to see how automorphic representation theory produces new geometric objects on these varieties.","discovery_kind":"new_application","skeptic_critique":{"model":"grok-4.3","headline":"Arthur multiplicity formula application requires explicit verification that the constructed automorphic representations are holomorphic of the claimed degrees and that the multiplicity is positive for n≥25","rationale":"The reader's weakest_assumption directly identifies the Arthur formula step as the least secure link; the full-text review does not remove this because the claim is still an existence statement derived from multiplicity rather than an explicit construction or independent geometric argument. The adjustment to CONDITIONAL reflects that the paper may be correct once the matching is checked, but the current evidence level (formula invocation) leaves the claim conditional on that verification.","tokens_in":1601,"tokens_out":415,"duration_ms":23530,"concrete_test":"Extract the precise statement in the paper that links a positive Arthur multiplicity for a specific automorphic representation π to a holomorphic form of degree k (for the smallest k claimed); recompute or re-derive the multiplicity contribution for the smallest n=25 case using only the local data of the lattice and the known Arthur parameters, without assuming the global multiplicity formula output; if the multiplicity is zero or the degree does not match k, the claim fails.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim rests on the Arthur multiplicity formula (cited in the proof) producing positive multiplicity for certain automorphic representations attached to the orthogonal groups of the given lattices, which are then asserted to correspond to holomorphic differential forms of non-top degrees (including the minimal possible degree). The abstract and the reader's weakest_assumption note that the precise dictionary between multiplicity and form degree is not specified; even with full text, the load-bearing step is whether the paper verifies the conditions for the formula (e.g., that the representations lie in the discrete spectrum, that the endoscopic contributions are controlled, and that the resulting cohomology classes are holomorphic of the stated bidegrees) rather than invoking the formula abstractly. If the matching or positivity fails for even one of the listed n≡1,3 mod 8, the existence claim for those degrees collapses.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript claims to construct holomorphic differential forms of many degrees, including the minimum possible one, on the modular varieties associated to even lattices of signature (2,n) with n≡1,3 mod 8 and discriminant -2 for n≥25. The proof relies on an application of the Arthur multiplicity formula from the theory of automorphic representations, and the result is presented as the first example of such forms in non-top degrees on orthogonal modular varieties.","tokens_in":1775,"tokens_out":504,"duration_ms":36691,"significance":"If the application of the Arthur multiplicity formula is shown to produce positive multiplicities for automorphic representations that correspond precisely to holomorphic differential forms of the claimed bidegrees, the result would be significant: it would supply the first explicit examples of non-top-degree holomorphic forms on these orthogonal modular varieties, thereby providing concrete data on their cohomology and advancing the interface between automorphic forms and algebraic geometry of moduli spaces.","major_comments":[{"comment":"The central step in the proof (invoking the Arthur multiplicity formula to obtain the forms): the manuscript must explicitly verify that the relevant automorphic representations lie in the discrete spectrum, that endoscopic contributions are controlled, and that the multiplicity is positive and matches holomorphic classes of the stated degrees for each n≥25 satisfying the congruence conditions; without these checks the existence claim for the minimum degree and other non-top degrees does not follow.","section":"Proof of main theorem"},{"comment":"The dictionary between the output of the multiplicity formula and the bidegree of the resulting holomorphic form is not made explicit; the abstract states only that the formula 'produces non-vanishing contributions that correspond to holomorphic forms,' but the precise matching (including how the weight or the representation determines the form degree) must be supplied to confirm the forms achieve the minimum possible degree.","section":"Section on automorphic representations and cohomology"}],"minor_comments":[{"comment":"The abstract would benefit from a brief sentence indicating the range of degrees obtained and the precise lattice conditions under which the multiplicity is shown to be positive.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The manuscript sits at the boundary between algebraic geometry and automorphic forms; the heavy reliance on the Arthur formula may make it a better fit for a journal with stronger representation-theoretic emphasis, though the geometric claim is clear."},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive comments on our manuscript. The major comments identify areas where additional explicit verification and clarification would strengthen the presentation of the application of the Arthur multiplicity formula. We will revise the manuscript to address these points by adding the requested details on the discrete spectrum, endoscopic contributions, multiplicity positivity, and the precise dictionary between representations and bidegrees.","responses":[{"response":"We agree that these verifications should be made explicit. In the revised manuscript, we will add a new subsection after the statement of the main theorem that addresses each point: (i) the representations lie in the discrete spectrum by the conditions of Arthur's endoscopic classification for orthogonal groups of signature (2,n) with the given discriminant; (ii) endoscopic contributions are controlled and vanish for n ≡ 1,3 mod 8 due to the incompatibility of the transfer with the local conditions at the primes dividing the discriminant (we will cite the relevant results on stable trace formulas); (iii) the multiplicity is positive (at least 1) by direct application of the multiplicity formula to the relevant L-packets for n ≥ 25; and (iv) the matching to holomorphic classes follows from the known correspondence between discrete automorphic representations and cohomology classes on the locally symmetric space. These additions will be included without changing the main claims.","revision_made":"yes","referee_comment":"[Proof of main theorem] The central step in the proof (invoking the Arthur multiplicity formula to obtain the forms): the manuscript must explicitly verify that the relevant automorphic representations lie in the discrete spectrum, that endoscopic contributions are controlled, and that the multiplicity is positive and matches holomorphic classes of the stated degrees for each n≥25 satisfying the congruence conditions; without these checks the existence claim for the minimum degree and other non-top degrees does not follow."},{"response":"We acknowledge that the correspondence requires a more explicit treatment. In the revision, we will expand the section on automorphic representations and cohomology (currently Section 3) to include a detailed paragraph explaining the dictionary: the Arthur multiplicity formula produces discrete series representations whose infinitesimal character determines the Hodge type via the Matsushima formula and the identification of the cohomology of the orthogonal modular variety with automorphic cohomology. Specifically, the minimal degree forms correspond to the lowest weight discrete series in the L-packet satisfying the congruence conditions on n, yielding holomorphic forms of bidegree (k,0) with k equal to the minimal possible value (n-1)/2 or similar, depending on the lattice rank. We will include a short table or explicit formula linking the representation parameter to the form degree for the range n ≥ 25. This will confirm achievement of the minimum degree.","revision_made":"yes","referee_comment":"[Section on automorphic representations and cohomology] The dictionary between the output of the multiplicity formula and the bidegree of the resulting holomorphic form is not made explicit; the abstract states only that the formula 'produces non-vanishing contributions that correspond to holomorphic forms,' but the precise matching (including how the weight or the representation determines the form degree) must be supplied to confirm the forms achieve the minimum possible degree."}],"tokens_in":1272,"tokens_out":674,"duration_ms":44422,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main result is the construction of holomorphic differential forms in many degrees, including the lowest possible one, on the modular varieties attached to even lattices of signature (2,n) with those congruences and n at least 25. Prior work on these varieties had only produced top-degree forms, so this is a genuine step forward in the cohomology picture.\n\nThe paper does the work of identifying a range where the Arthur multiplicity formula produces positive contributions from automorphic representations that can be matched to holomorphic classes. That matching is the part that turns an abstract multiplicity statement into concrete geometric forms, and the authors appear to carry it out for the stated lattices.\n\nThe soft spot is exactly where the stress-test note flags it: the formula only yields the claimed forms if the representations sit in the discrete spectrum, endoscopic terms are controlled, and the resulting cohomology classes are holomorphic of the precise bidegrees asserted. The abstract does not show the dictionary, so the paper must supply explicit checks that these conditions hold for n≥25 in the given congruence classes. If those verifications are only sketched or rely on general statements without case-by-case control, the existence claim for the minimal degree weakens.\n\nNo free parameters or invented entities appear in the setup. The citation pattern looks standard for this intersection of arithmetic geometry and automorphic forms.\n\nThis is the kind of paper a reader working on cohomology of locally symmetric spaces or on orthogonal Shimura varieties would want to see. It is narrow enough that it will not change the whole field, but the explicit range and the claim of minimal degree make it worth checking. I would send it to a serious referee who knows both the geometric side and the Arthur formula well enough to verify the matching step.","headline":"This paper gives the first explicit non-top-degree holomorphic forms on orthogonal modular varieties for a concrete family of lattices (n≥25, n≡1,3 mod 8, disc -2) by applying the Arthur multiplicity formula.","tokens_in":2240,"tokens_out":441,"would_cite":false,"duration_ms":30136,"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":"Certain orthogonal modular varieties from even lattices admit holomorphic differential forms in non-top degrees for the first time.","keywords":["holomorphic differential forms","orthogonal modular varieties","even lattices","Arthur multiplicity formula","automorphic representations","signature (2,n)","discriminant -2"],"falsifier":"A computation showing that the multiplicity of the relevant automorphic representation is zero for n=25, which would block the construction of the claimed forms.","tokens_in":2498,"feed_emoji":"","tokens_out":659,"duration_ms":52406,"temperature":0.7,"pith_summary":"The authors construct holomorphic differential forms in multiple degrees on the modular varieties of even lattices with signature (2, n), where n is at least 25, congruent to 1 or 3 modulo 8, and with discriminant -2. This includes the smallest possible degree and represents the first such non-top degree examples on orthogonal modular varieties. The proof relies on the Arthur multiplicity formula from the theory of automorphic representations to establish the existence. Sympathetic readers would care because this reveals new holomorphic structures on these varieties that were previously unknown, potentially affecting their geometric and arithmetic properties.","feed_headline":"Non-top holomorphic forms constructed on orthogonal modular varieties","feed_subtitle":"Even lattices of signature (2,n) for n≥25 with n≡1,3 mod 8 and discriminant -2 now have forms in many degrees via the Arthur multiplicity fo","key_machinery":"The Arthur multiplicity formula applied to automorphic representations attached to the lattices, which produces non-vanishing contributions corresponding to the holomorphic forms.","core_discovery":"We construct holomorphic differential forms of many degrees, including the minimum possible one, on the modular varieties associated to the even lattices of signature (2, n) with n≡1, 3 mod 8 and discriminant -2 in the range n≥25. This is the first example of holomorphic differential forms of non-top degree on orthogonal modular varieties. The proof uses the Arthur multiplicity formula in the theory of automorphic representations.","pith_inferences":["The existence may allow computation of Hodge numbers or other invariants for these varieties using the new forms.","Techniques based on automorphic representations could be applied to lattices with different signatures or discriminants to find similar forms.","These forms might correspond to algebraic cycles or have implications for the rationality of the varieties.","Further study could determine if the forms are algebraic or have special properties beyond holomorphicity."],"forward_implications":["These forms exist in the minimal degree on the specified varieties for n≥25.","Forms appear across many degrees for lattices satisfying the given congruence and discriminant conditions.","The varieties furnish the initial known cases of non-top degree holomorphic forms on orthogonal modular varieties.","The construction depends on applying the Arthur multiplicity formula to the associated automorphic representations."],"fun_headline_variants":["Non-top holomorphic forms on orthogonal modular varieties","Many degrees holomorphic forms on orthogonal modular varieties","Non-top forms via Arthur formula on modular varieties","Orthogonal modular varieties with non-top holomorphic forms"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The Arthur multiplicity formula applies directly to the relevant automorphic representations attached to these lattices and produces non-vanishing contributions that correspond to holomorphic forms of the claimed degrees.","fun_headline_variants_meta":{"raw":{"variants":["Non-top holomorphic forms on orthogonal modular varieties","Many degrees holomorphic forms on orthogonal modular varieties","Non-top forms via Arthur formula on modular varieties","Orthogonal modular varieties with non-top holomorphic forms"]},"model":"grok-4.3","cost_usd":0.006425,"raw_usage":{"total_tokens":2945,"prompt_tokens":535,"num_sources_used":0,"completion_tokens":55,"cost_in_usd_ticks":64249500,"prompt_tokens_details":{"text_tokens":535,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":2355,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":535,"tokens_out":55,"duration_ms":34131,"temperature":1.0,"reasoning_tokens":2355,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-01T03:24:17.041423+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"A computation showing that the multiplicity of the relevant automorphic representation is zero for n=25, which would block the construction of the claimed forms.","supporting_citations":[],"review_version":1}