{"id":"11775b26-116b-4f17-941a-83e38438acf5","arxiv_id":"2607.09002","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":5.5,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Silverman-type exact reduction orders hold for elliptic curves over global function fields (n coprime to p) and for abelian-scheme sections over C when the Betti map is generically submersive.","lead":"The paper proves that non-torsion points on elliptic curves over global function fields of char p>3 realize every large order coprime to p under reduction, and that sections of abelian schemes over C with generically submersive Betti map realize every large torsion order on some fiber. It gives a short height-based proof and a relative geometric criterion via unlikely intersections.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified","rationale":"The reader’s weakest-assumption diagnosis matches the only genuine restriction in the argument (formal-group equality fails for p-powers). That restriction is both necessary and explicitly acknowledged; it does not undermine the stated theorems. The height-machine proof of Theorem 2 is short, self-contained, and free of the intersection-theoretic bookkeeping of earlier EDS papers, while Theorem 3 is a transparent application of known generic-submersivity results. No correction to the ACCEPT verdict is warranted.","tokens_in":20245,"tokens_out":459,"duration_ms":4984,"concrete_test":"Re-derive the key inequality after (3) independently: start from the local-height bound (1) for v outside S, insert the uniform estimate sum_{r|n} 1/r^2 < 1/2, and verify that any epsilon < 1/(2#S) forces ^h(P)=0 when n\to∞ with gcd(n,p)=1. If the arithmetic closes without additional assumptions, the load-bearing step is secure.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claims (Theorem 2 / Theorem A and Theorem 3 / Theorem B) rest on standard, correctly applied tools: the formal-group isomorphism for good-reduction places when gcd(n,p)=1 (Lemma 1), the Northcott property and quadraticity of the Néron–Tate height over finite-constant-field function fields (Proposition 1), and the real-analytic submersion theorem plus density of exact-order torsion on the torus (Lemma 5). The reader correctly flags the formal-group step as the reason for the coprimality restriction; the paper itself proves necessity via supersingular reductions (Remark 5) and supplies matching counter-examples to the unrestricted Siegel statement (Propositions 2–3). No hidden gap, circularity, or unjustified estimate appears in the height comparison (3) or in the Betti-map openness argument. The limitations (non-effectivity, higher-dimensional number-field case open, restriction to generically submersive Betti maps) are stated honestly.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies when a non-torsion point on an algebraic group over a global field realizes every sufficiently large integer n as the exact order of its reduction at some place. Theorem A (Theorem 2) proves this for elliptic curves over global function fields of characteristic p>3, for all large n coprime to p, via a Silverman–Cheon–Hahn style height comparison that uses the formal-group isomorphism at good places (Lemma 1), Northcott, and quadraticity of the Néron–Tate height. The authors also give explicit supersingular isotrivial examples (Propositions 2–3) showing that Siegel’s local-to-global height decay and the unrestricted “all but finitely many P” claim both fail in positive characteristic. Theorem B (Theorem 3) treats the relative problem over C: if the Betti map of a non-torsion section of an abelian scheme is generically submersive, then every large N appears as the exact order of the section on some complex fiber; this is deduced from real-analytic openness of a submersion together with density of exact-order torsion on the torus (Lemma 5), and is linked to the relative Manin–Mumford theorem of Gao–Habegger. An appendix proves the analogous Schinzel statement for one-dimensional tori over global function fields.","tokens_in":20474,"tokens_out":857,"duration_ms":8609,"significance":"The function-field result supplies a clean, self-contained Silverman-type statement that covers supersingular curves without the constant-j or ordinary restrictions of earlier EDS work, while the counter-examples to Siegel’s theorem in characteristic p are concrete and useful. The relative theorem cleanly interfaces the reduction-order question with the Betti-map formalism and relative Manin–Mumford, giving a transparent geometric criterion that applies in all situations where generic submersivity is already known. Both proofs are short, standard, and free of circularity or fitted parameters; the limitations (coprimality to p, non-effectivity, restriction to generically submersive Betti maps) are stated honestly. The contribution is methodological and expository rather than a breakthrough on the open higher-dimensional number-field case, but it is solid and of clear interest to arithmetic geometry.","major_comments":[],"minor_comments":[{"comment":"In the introduction the authors write “a fortiori, they do not cover Theorem A”; the Latin phrase is slightly misused (the preceding sentence already states the stronger claim). Replace by “in particular” or “hence”.","section":null},{"comment":"Lemma 1 and Remark 3: the local height is written both as -min{0,v(x),v(y)}deg(v) and as -(3/2)min{0,v(x)}deg(v). A single consistent formula (or an explicit cross-reference) would avoid momentary confusion.","section":null},{"comment":"In the proof of Theorem 2 the elementary bound sum 1/r^{2} < 1/2 is correct, but the intermediate identity “1/4 + (π^{2}/8 - 1) = π^{2}/8 - 3/4” is unnecessary; a direct comparison with the geometric series already suffices and shortens the argument.","section":null},{"comment":"Appendix A: the product-formula argument for Φ_n(x) is clean, yet the four cases would be easier to follow if each case ended with an explicit statement of the contribution to the sum rather than only the vanishing or non-vanishing of v(Φ_n(x)).","section":null},{"comment":"Several bibliographic items (e.g., [18], [19], [13]) appear both as published and as arXiv preprints; a uniform citation style would improve readability.","section":null}],"recommendation":"accept","confidential_remarks":"The manuscript is short, carefully written, and free of load-bearing gaps. Its novelty is modest (methodological re-packaging of known height and Betti techniques plus useful counter-examples), but it is well within the scope of a solid number-theory journal. I see no reason to request major changes."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is two clean theorems: every large n coprime to p is realized as a good-reduction order for any non-torsion point on an elliptic curve over a global function field of char p>3, and the same holds for sections of abelian schemes over C once the Betti map is generically submersive. Both are correctly proved with standard tools.\n\nWhat is new is mainly methodological. Theorem A covers all non-torsion points without the constant-j bookkeeping of Naskręcki–Streng and without the elliptic-surface intersection theory of Naskręcki’s ordinary EDS work. The argument is short: local-height comparison via the formal group at good places (Lemma 1), Northcott plus quadraticity of the Néron–Tate height, and an elementary bound on sum 1/r^{2} < 1/2 that produces the contradiction. The relative Theorem B is a transparent application of real-analytic submersion plus density of exact-order torsion on the torus (Lemma 5). The counter-examples to Siegel and to the “all but finitely many P” claim in positive characteristic (Propositions 2–3) are concrete and useful.\n\nSoft spots are real but proportional and already flagged by the authors. The formal-group step forces gcd(n,p)=1; they prove necessity via supersingular reductions (Remark 5). No effective bounds, higher-dimensional number-field case still open, and the relative statement needs the generically-submersive hypothesis (which is known in several natural cases). Overlap with the EDS literature is substantial and acknowledged. None of this breaks the central arguments.\n\nMath, citations, and self-containment look solid once you grant standard height machinery, formal groups, and Gao–Habegger relative Manin–Mumford. No circularity, no free parameters.\n\nThis is for people working on elliptic divisibility sequences, reduction orders, or Betti maps. A serious referee should see it; it is a clean, honest contribution of moderate significance. I would accept for peer review.","headline":"Solid, self-contained Silverman-type results for function fields and a clean Betti-map criterion; methodological rather than dramatic, but the proofs hold and the limitations are honest.","tokens_in":21087,"tokens_out":517,"would_cite":true,"duration_ms":6329,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","14G05","14G17","11G50","14K99"],"pacs":[],"model":"grok-4.5","headline":"Large integers coprime to the characteristic appear as exact reduction orders of non-torsion points on elliptic curves over function fields, and the same holds fiberwise for sections of abelian schemes whose Betti map is generically submers","keywords":["elliptic curve","reduction order","global function field","Silverman theorem","Betti map","abelian scheme","divisibility sequence","unlikely intersections"],"falsifier":"Exhibit a single elliptic curve over a global function field of characteristic p>3 and a non-torsion rational point such that infinitely many integers n coprime to p never appear as the order of the reduction of the point at any place of good reduction.","tokens_in":21151,"feed_emoji":"🔢","tokens_out":733,"duration_ms":7312,"temperature":0.7,"pith_summary":"The paper asks when a non-torsion rational point on an abelian variety over a global field realizes every sufficiently large integer as the exact order of its reduction at some place. It answers the question affirmatively for elliptic curves over global function fields of characteristic greater than 3, but only for orders coprime to the characteristic. The same phenomenon is then established in a relative setting over the complex numbers: if a non-torsion section of an abelian scheme has generically submersive Betti map, every large enough integer appears as the exact order of that section on some complex fiber. The results give a function-field and a geometric counterpart to classical theorems of Bang, Zsigmondy, Schinzel and Silverman, and they show that the obstruction for p-power orders is forced by supersingular reduction. A reader interested in arithmetic dynamics or unlikely intersections therefore obtains a clean existence statement for reduction orders under hypotheses that can be checked via heights or Betti rank.","feed_headline":"Every large order appears as a reduction of a non-torsion point","feed_subtitle":"Function-field and relative versions of Silverman's theorem hold under height and Betti-rank hypotheses","key_machinery":"Height decomposition together with the formal-group identity that local height is preserved under multiplication by n coprime to p at good places (so that a non-primitive divisor would force the Néron–Tate height of P to vanish); in the relative case, the real-analytic Betti map and the elementary fact that open sets in the real torus contain points of every large exact order.","core_discovery":"For an elliptic curve over a global function field of characteristic p>3 and a non-torsion rational point P, every sufficiently large integer n coprime to p is realized as the exact order of the reduction of P at some place of good reduction. Independently, if a non-torsion section of an abelian scheme over C has generically submersive Betti map, every large enough integer appears as the exact order of the section on some complex fiber.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Large p-coprime orders arise as reductions of non-torsion points","Betti-submersive sections hit every large torsion order on fibers","Function-field elliptic curves realize all large coprime reduction orders","Silverman variants: every large n appears via reduction or complex fiber","Non-torsion points and sections attain every sufficiently large order"],"cache_read_input_tokens":16512,"weakest_assumption_plain":"The formal-group calculation that, away from the characteristic, multiplying a point that reduces to the identity by an integer coprime to p does not change its local height; without this equality the height inequality that produces a contradiction fails.","fun_headline_variants_meta":{"raw":{"variants":["Large p-coprime orders arise as reductions of non-torsion points","Betti-submersive sections hit every large torsion order on fibers","Function-field elliptic curves realize all large coprime reduction orders","Silverman variants: every large n appears via reduction or complex fiber","Non-torsion points and sections attain every sufficiently large order"]},"model":"grok-4.5","effort":"low","cost_usd":0.005182,"raw_usage":{"total_tokens":1451,"prompt_tokens":785,"num_sources_used":0,"completion_tokens":94,"cost_in_usd_ticks":51820000,"prompt_tokens_details":{"text_tokens":785,"audio_tokens":0,"image_tokens":0,"cached_tokens":256},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":572,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":785,"tokens_out":94,"duration_ms":6303,"temperature":1.0,"reasoning_tokens":572,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-13T01:04:48.789167+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Exhibit a single elliptic curve over a global function field of characteristic p>3 and a non-torsion rational point such that infinitely many integers n coprime to p never appear as the order of the reduction of the point at any place of good reduction.","supporting_citations":[],"review_version":1}