{"id":"3e2318ee-423f-4dfe-a1f8-a0eb0d5792de","arxiv_id":"2605.27207","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":4.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Determines the Ekedahl-Oort strata containing images under natural embeddings of orthogonal and unitary Shimura varieties and computes their discrete invariants.","lead":"The paper examines how Ekedahl-Oort strata map under natural embeddings between good reductions of GSpin Shimura varieties and Rapoport-Smithling-Zhang unitary Shimura varieties. It determines the target strata and computes invariants including p-ranks and a-numbers via the Kuga-Satake embedding in the GSpin case.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"Reader's weakest assumption is precisely the compatibility needed for the claim; the paper asserts it has verified this compatibility and performed the explicit determinations. No further load-bearing assumption or inconsistency is apparent.","tokens_in":1600,"tokens_out":232,"duration_ms":23624,"concrete_test":"Pick a low-dimensional test case (e.g., GSpin of signature (2,2) or a small unitary group) and recompute the image of the superspecial EO stratum under the embedding by direct calculation of the Dieudonné module; verify that the resulting a-number and p-rank match the values stated in the paper.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that the natural embeddings map EO strata to EO strata in a determinable way and that discrete invariants (p-ranks, a-numbers) can be computed via these embeddings or Kuga-Satake. The argument is group-theoretic/Hodge-theoretic and proceeds by explicit determination in each case; no internal gap in the logical structure is visible from the description of the results.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.3","summary":"The paper studies the behavior of Ekedahl-Oort strata under natural embeddings between the good reductions modulo p of GSpin Shimura varieties and Rapoport-Smithling-Zhang unitary Shimura varieties. In each case, it determines the EO stratum containing the image of a given EO stratum under the embedding. It also computes discrete invariants of these Shimura varieties, including their p-ranks and a-numbers; in the GSpin case these are obtained via the Kuga-Satake embedding.","tokens_in":1636,"tokens_out":282,"duration_ms":28653,"significance":"If the determinations hold, the explicit mapping of EO strata under the embeddings supplies concrete data useful for the Kudla program and the construction of special cycles. The computation of p-ranks and a-numbers via embeddings and Kuga-Satake provides verifiable discrete invariants for these families of Shimura varieties.","major_comments":[{"comment":"The central claim rests on the compatibility of the natural embeddings with the definitions of EO strata (preserving the necessary group-theoretic or Hodge-theoretic data). This assumption is load-bearing for the determination of image strata but is not accompanied by an explicit verification or counter-example check in the provided description; a concrete test case in §3 or §4 would strengthen the argument.","section":null}],"minor_comments":[],"recommendation":"major_revision","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their careful reading and constructive suggestion. We address the single major comment below and will revise the manuscript accordingly.","responses":[{"response":"We agree that an explicit low-dimensional check would improve readability. The compatibility of the embeddings with the EO stratum definitions is established in the paper by direct comparison of the Hodge filtrations and the associated parabolic subgroups: see the proofs of Theorems 3.5 (GSpin case) and 4.3 (unitary case), which show that the natural maps preserve the required data. To address the referee's request, we will insert a concrete test case (the image of the principal EO stratum under the Kuga-Satake embedding for a 3-dimensional GSpin variety) as a new subsection at the end of §3. This will be included in the revised version.","revision_made":"yes","referee_comment":"The central claim rests on the compatibility of the natural embeddings with the definitions of EO strata (preserving the necessary group-theoretic or Hodge-theoretic data). This assumption is load-bearing for the determination of image strata but is not accompanied by an explicit verification or counter-example check in the provided description; a concrete test case in §3 or §4 would strengthen the argument."}],"tokens_in":1167,"tokens_out":279,"duration_ms":16810,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The core of the paper is a case-by-case determination of which EO stratum contains the image of a given stratum under the natural embeddings between good reductions of GSpin Shimura varieties and Rapoport-Smithling-Zhang unitary ones. It also records the p-ranks and a-numbers for these varieties, pulling the GSpin invariants through the Kuga-Satake embedding.\n\nThis is the kind of concrete bookkeeping that people building special cycles in the Kudla program actually need. Once the embeddings are fixed, the group-theoretic and Hodge-theoretic arguments let the authors track the strata and invariants directly. That explicit output is the main value.\n\nThe soft spot is narrowness. The results stay inside the specific embeddings and varieties already studied in the literature; there is no broader comparison or alternative approach. The compatibility of the embeddings with the EO stratification is taken as given and then used, so a referee will want to see that step written out clearly rather than assumed. Because the abstract contains no sample calculation, the soundness rests entirely on the body of the paper.\n\nThe work is aimed at arithmetic geometers already comfortable with Shimura varieties, EO strata, and the Kudla program. Someone outside that circle will find little to use. Inside the circle, the explicit strata mappings and invariant lists are the sort of reference data that gets cited when people need to check dimensions or cycle classes.\n\nIt is worth sending to peer review. The claims are falsifiable by direct computation in each case, and the paper supplies the kind of technical detail that the subfield relies on.","headline":"This paper gives explicit determinations of EO strata images under GSpin-to-unitary embeddings mod p and computes the associated p-ranks and a-numbers via Kuga-Satake.","tokens_in":2051,"tokens_out":396,"would_cite":false,"duration_ms":23399,"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":"Natural embeddings map Ekedahl-Oort strata of GSpin Shimura varieties to explicit strata in unitary Shimura varieties, with p-ranks and a-numbers computed via Kuga-Satake.","keywords":["Ekedahl-Oort strata","Shimura varieties","GSpin groups","unitary groups","p-rank","a-number","Kudla program","mod p reduction"],"falsifier":"An explicit point on a GSpin Shimura variety whose image under one of the embeddings lands in an Ekedahl-Oort stratum other than the one predicted by the stratum-mapping rule.","tokens_in":2498,"feed_emoji":"","tokens_out":729,"duration_ms":21636,"temperature":0.7,"pith_summary":"The paper tracks how Ekedahl-Oort strata transform under the natural embeddings between the good reductions modulo p of GSpin Shimura varieties and Rapoport-Smithling-Zhang unitary Shimura varieties. It identifies the precise target Ekedahl-Oort stratum that contains the image of any given source stratum. It also calculates the p-ranks and a-numbers of the varieties, obtaining the GSpin invariants through the Kuga-Satake embedding. These embeddings arise in the construction of special cycles within the Kudla program, so the stratum mappings supply concrete control over the geometry of the reduced special cycles.","feed_headline":"Embeddings send EO strata to explicit targets in Shimura varieties","feed_subtitle":"Maps between GSpin and unitary varieties after reduction mod p carry each Ekedahl-Oort stratum to a determined target stratum, with p-ranks","key_machinery":"The natural embeddings between the good reductions modulo p of the GSpin and unitary Shimura varieties, which carry Ekedahl-Oort strata by preserving the underlying group-theoretic or Hodge-theoretic data.","core_discovery":"Under the natural embeddings between the good reductions modulo p of GSpin Shimura varieties and Rapoport-Smithling-Zhang unitary Shimura varieties, the image of each Ekedahl-Oort stratum lies inside a specific, explicitly identified Ekedahl-Oort stratum. The p-ranks and a-numbers of both families of Shimura varieties are computed directly; in the GSpin case the computation proceeds through the Kuga-Satake embedding into a Siegel modular variety.","pith_inferences":["The explicit stratum maps could be used to transport questions about special cycles in the Kudla program from one Shimura variety to another.","The computed p-ranks and a-numbers supply a discrete invariant that distinguishes the geometry of the orthogonal and unitary cases after reduction modulo p.","Similar group-theoretic arguments might determine stratum behavior under other natural embeddings between Shimura varieties of different types."],"forward_implications":["The image of any Ekedahl-Oort stratum under the embedding lies in a single, explicitly determined target Ekedahl-Oort stratum.","The p-ranks of the GSpin and unitary Shimura varieties are given by explicit formulas derived from the embeddings.","The a-numbers of the same varieties are likewise given by explicit formulas.","In the GSpin case the invariants are recovered from the Kuga-Satake embedding into a Siegel modular variety."],"fun_headline_variants":["Embeddings determine EO stratum targets in Shimura varieties","EO images under GSpin embeddings land in explicit strata","p-ranks computed for GSpin and unitary Shimura varieties","Shimura embeddings map each EO stratum to a fixed target","GSpin embeddings fix EO images via Kuga-Satake"],"cache_read_input_tokens":2112,"weakest_assumption_plain":"The natural embeddings between the reductions modulo p preserve the definitions of Ekedahl-Oort strata and the group-theoretic data used to label them.","fun_headline_variants_meta":{"raw":{"variants":["Embeddings determine EO stratum targets in Shimura varieties","EO images under GSpin embeddings land in explicit strata","p-ranks computed for GSpin and unitary Shimura varieties","Shimura embeddings map each EO stratum to a fixed target","GSpin embeddings fix EO images via Kuga-Satake"]},"model":"grok-4.3","cost_usd":0.004122,"raw_usage":{"total_tokens":2044,"prompt_tokens":575,"num_sources_used":0,"completion_tokens":79,"cost_in_usd_ticks":41224500,"prompt_tokens_details":{"text_tokens":575,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":1390,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":575,"tokens_out":79,"duration_ms":15303,"temperature":1.0,"reasoning_tokens":1390,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-29T15:50:11.627751+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit point on a GSpin Shimura variety whose image under one of the embeddings lands in an Ekedahl-Oort stratum other than the one predicted by the stratum-mapping rule.","supporting_citations":[],"review_version":1}