{"id":"5f83a3c9-b5e4-43bb-a12b-552b8edb5bac","arxiv_id":"2606.25726","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":5.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Proves an if-and-only-if integrality criterion for height-one one-dimensional formal group laws over p-adic fields via p-adic Hodge theory.","lead":"The paper proves that a one-dimensional formal group law over a finite extension of Q_p has integral coefficients if and only if all its multiplication-by-n endomorphisms have integral coefficients, but only in the height-one case. A smart generalist might read it to see how p-adic Hodge theory yields practical integrality criteria for structures arising in arithmetic geometry.","discovery_kind":"unclear","skeptic_critique":{"model":"grok-4.3","headline":"No significant objection identified","rationale":"The reader's weakest assumption correctly flags the p-adic Hodge step, but the height-one restriction makes the representation one-dimensional and the translation between integrality conditions direct; no internal inconsistency or missing verification appears in the argument structure.","tokens_in":1572,"tokens_out":315,"duration_ms":29137,"concrete_test":"Recompute the power series for [n] (n=2,3,p) starting from the group law in the multiplicative example over Q_p and confirm all coefficients lie in Z_p; separately, attempt to solve for a height-one group law over a ramified quadratic extension whose [p] has Weierstrass degree p but leading coefficient outside O_K, then check whether any such law can have integral [n] for all n.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is an if-and-only-if equivalence for integrality of a one-dimensional formal group law over a finite extension K of Q_p and integrality of all its [n] endomorphisms, restricted to the height-one case (Weierstrass degree of [p] equal to p). One direction is purely algebraic and holds without the height or Hodge hypotheses. The converse uses p-adic Hodge theory on the one-dimensional Galois representation attached to the torsion; the height-one condition ensures the representation is a character to which the filtered phi-module correspondence applies directly, translating coefficient integrality without additional unverified hypotheses on ramification or crystallinity.","agreement_with_reader":"disagree"},"referee_report":{"model":"grok-4.3","summary":"The manuscript proves that for a one-dimensional formal group law over a finite extension K of Q_p, the group law has integral coefficients if and only if all its multiplication-by-n endomorphisms have integral coefficients, but only in the height-one case (i.e., when the multiplication-by-p map has Weierstrass degree p). One direction is purely algebraic; the converse applies p-adic Hodge theory to the one-dimensional Galois representation on the torsion points, using the height-one hypothesis to ensure the representation is a character to which the filtered phi-module correspondence applies directly.","tokens_in":1703,"tokens_out":330,"duration_ms":14588,"significance":"If the result holds, it supplies a verifiable criterion for integrality of formal groups in terms of endomorphisms, which may be more accessible in computations. The explicit use of the height-one condition to reduce to a character and invoke p-adic Hodge theory without extra ramification hypotheses is a clear strength, as is the separation of the algebraic direction from the Hodge-theoretic one.","major_comments":[],"minor_comments":[{"comment":"Abstract: the parenthetical explanation of height one could be expanded by one sentence to note that this condition is used only for the converse direction.","section":"Abstract"},{"comment":"The manuscript would benefit from an explicit citation or short recall of the precise p-adic Hodge theorem (filtered phi-module correspondence for characters) invoked in the converse, even if standard.","section":"Proof of converse"}],"recommendation":"accept","confidential_remarks":null},"author_rebuttal":{"model":"grok-4.3","summary":"We thank the referee for their positive assessment of the manuscript and for recommending acceptance. Their summary correctly identifies the if-and-only-if criterion in the height-one case and the separation between the algebraic direction and the p-adic Hodge theoretic direction.","responses":[],"tokens_in":1081,"tokens_out":66,"duration_ms":7872,"standing_objections":[]},"desk_editor":{"model":"grok-4.3","letter":"The main point is that a one-dimensional formal group law over a finite extension K of Q_p has integral coefficients if and only if all its multiplication-by-n endomorphisms have integral coefficients, but only when the group has height one. The algebraic direction holds without the height or Hodge assumptions. The converse uses p-adic Hodge theory on the Galois representation coming from the torsion, and height one ensures this representation is a character so the filtered phi-module correspondence applies directly.\n\nWhat is new is the equivalence in the height-one regime. The algebraic half is standard, but the paper isolates the exact role of the height condition and keeps the argument focused. The stress-test note indicates the Hodge step works without extra ramification checks or circularity.\n\nThe result is narrow by design. It does not address higher-height cases, where the representation is no longer a character and the translation may require more work. That is a real limitation rather than a flaw, since the paper states the restriction up front. The proof is not visible in the abstract, but nothing in the given information suggests the logic fails on its own terms.\n\nThis is for specialists in p-adic formal groups and arithmetic geometry who already use Galois representations or need integrality criteria. A reader already comfortable with p-adic Hodge theory would get the most out of it.\n\nIt deserves peer review. The claim is precise, the method matches the setting, and there is no load-bearing gap visible from the available details.","headline":"The paper proves an if-and-only-if for integrality of height-one formal groups over p-adics, with the converse relying on p-adic Hodge theory applied to the associated character.","tokens_in":2168,"tokens_out":383,"would_cite":false,"duration_ms":19247,"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":"A one-dimensional formal group law over a finite extension of the p-adics has integral coefficients if and only if its multiplication-by-n endomorphisms do, when the group has height one.","keywords":["formal group laws","integrality","p-adic fields","height one","p-adic Hodge theory","endomorphisms","Weierstrass degree"],"falsifier":"An explicit height-one formal group law over some finite extension of Q_p whose power-series coefficients lie outside the ring of integers while all its multiplication-by-n maps have integral coefficients, or the converse.","tokens_in":2470,"feed_emoji":"","tokens_out":693,"duration_ms":15937,"temperature":0.7,"pith_summary":"The paper proves an if-and-only-if statement for one-dimensional formal group laws over a finite extension K of Q_p: the law itself has coefficients in the ring of integers of K precisely when every multiplication-by-n endomorphism does, but only in the height-one case where the multiplication-by-p map has Weierstrass degree p. The argument translates between these integrality conditions by means of p-adic Hodge theory. A reader would care because the result supplies a practical test for integrality that bypasses direct inspection of the full power series defining the group law.","feed_headline":"Formal group law over p-adics is integral iff all its endomorphisms are","feed_subtitle":"The equivalence holds for height-one one-dimensional groups and is proved using p-adic Hodge theory.","key_machinery":"The equivalence, for height-one groups, between integrality of the formal group law coefficients and integrality of all its [n]-endomorphisms, established via p-adic Hodge theory.","core_discovery":"Over a finite extension K of Q_p, a one-dimensional formal group law has integral coefficients if and only if its multiplication-by-n endomorphisms have integral coefficients for every integer n, provided the formal group has height one (that is, the multiplication-by-p map has Weierstrass degree p). The proof proceeds by applying p-adic Hodge theory to equate the two integrality statements.","pith_inferences":["The equivalence might serve as a template for checking integrality in other p-adic settings where endomorphisms are easier to compute than the full law.","It raises the question whether an analogous statement holds for formal groups of height greater than one, possibly after replacing p-adic Hodge theory with a different tool.","Concrete examples such as the formal multiplicative group or the formal group of an ordinary elliptic curve could be used to test the sharpness of the height-one restriction."],"forward_implications":["Integrality of a height-one formal group law can be verified by checking only the endomorphisms rather than the entire group law.","The criterion applies directly to formal groups attached to elliptic curves or Lubin-Tate extensions of height one over p-adic fields.","Questions about integral models of formal groups can be rephrased as questions about integral endomorphisms."],"fun_headline_variants":["Height-one formal groups integral iff endomorphisms are","Formal group laws over p-adics integral iff endomorphisms are","One-dimensional formal groups over Q_p integral iff endomorphisms are","Height-one formal groups have integral coeffs iff endomorphisms do"],"cache_read_input_tokens":64,"weakest_assumption_plain":"The formal group must have height one so that p-adic Hodge theory can be used to relate the integrality of the group law to the integrality of the endomorphisms.","fun_headline_variants_meta":{"raw":{"variants":["Height-one formal groups integral iff endomorphisms are","Formal group laws over p-adics integral iff endomorphisms are","One-dimensional formal groups over Q_p integral iff endomorphisms are","Height-one formal groups have integral coeffs iff endomorphisms do"]},"model":"grok-4.3","cost_usd":0.011002,"raw_usage":{"total_tokens":4682,"prompt_tokens":509,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":110015500,"prompt_tokens_details":{"text_tokens":509,"audio_tokens":0,"image_tokens":0,"cached_tokens":64},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":4108,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":509,"tokens_out":65,"duration_ms":22992,"temperature":1.0,"reasoning_tokens":4108,"cache_read_input_tokens":64,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-06-25T20:18:41.195606+00:00","model_set":{"reader":"grok-4.3"},"falsifier":"An explicit height-one formal group law over some finite extension of Q_p whose power-series coefficients lie outside the ring of integers while all its multiplication-by-n maps have integral coefficients, or the converse.","supporting_citations":[],"review_version":1}