{"id":"d9634796-4ecd-4a87-996e-cc708ac9e595","arxiv_id":"1908.09050","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Torsion points of a section of an abelian scheme over a p-adic field have rigid-analytic neighborhoods isolating each torsion order; a nodal-fiber example shows that bad reduction destroys this separation.","lead":"For an abelian scheme over a p-adic field, this paper proves that torsion values of a section of different orders stay p-adically separated, giving each order a rigid-analytic neighborhood that contains no torsion of other orders. An explicit example shows that allowing one bad fiber makes torsion points of many orders accumulate, so the abelian scheme hypothesis is needed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified; the proof of Theorem 1.2 is sound after scrutiny of Lemma 2.1 and the local separation argument.","rationale":"The reader identified Lemma 2.1 as the weakest assumption, and I agree that it is the most load-bearing step. However, on detailed scrutiny the lemma is correct: the construction via invariant differentials and the formal logarithm yields a rigid-analytic open subgroup E whose group law is additive in suitable coordinates, and property (4) is justified by compactness of A_{x0}(K). The proof of Theorem 1.2 uses exactly the needed consequences of Lemma 2.1, and the neighborhood choices respect the rigid-analytic topology. I also considered and dismissed the apparent counterexample where torsion points of increasing p-power order accumulate at the identity of a constant abelian scheme: in the p-adic topology these torsion points approach the boundary of the formal-group convergence disk, not the identity, so the theorem's local separation remains consistent. The counterexample in Section 3 illustrates failure with bad reduction and is not a counterexample to the theorem. Overall, the paper's central claim is correct; no adjustment to the reader's ACCEPT verdict is needed.","tokens_in":7992,"tokens_out":36853,"duration_ms":378922,"concrete_test":"Verify Lemma 2.1 explicitly for a non-isotrivial elliptic scheme over a p-adic disk: choose a global invariant differential, compute the formal logarithm on the identity component, and confirm that the resulting E is an open subgroup isomorphic to a unit ball and of finite index in each fiber over the relevant base points.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I examined the pivotal Lemma 2.1 and the proof of Theorem 1.2. The finite-index property (4) of Lemma 2.1 follows from p-adic compactness: A_{x0}(K) is compact, the subgroup E∩A_{x0}(K) is open because E is a rigid-analytic open containing the identity, so its cosets give a finite open cover. The torsion-freeness of E(L)≅O_L^g is immediate. The proof of Theorem 1.2 correctly uses ns(x)∈E and torsion-freeness to force any torsion point in U0 to have order dividing n, then excludes proper divisors by Zariski closedness. A potential concern about p^n-torsion points accumulating at the identity of a constant abelian scheme is resolved: in the formal group, p^n-torsion points have residue valuations tending to 0, hence their absolute values tend to 1; they move to the boundary of the formal power-series disk, not to the center, so a sufficiently small ball E excludes them. No circularity, no unsupported parameter fitting, and the counterexample does not threaten the abelian-scheme theorem. I find no load-bearing flaw.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"Let K be a p-adic field, S a quasi-projective K-variety, A→S an abelian scheme, and s:S→A a section. For each n, let S_n be the locus where s is torsion of exact order n. The paper proves (Theorem 1.2) that every L-point of S, for a finite extension L/K, has a rigid-analytic neighborhood on which the exact torsion order is constant if the point is torsion, or on which there is no torsion at all if the point is not torsion. The proof relies on Lemma 2.1, which states that around any point of S there is a rigid-analytic open subgroup E of A, isomorphic over a neighborhood U to U×B^g with additive group law, and of finite index in the fiber over that point. Choosing n so that ns(x0)∈E, the torsion-freeness of E(L) forces any torsion point in U to be killed by n, and Zariski closedness of the proper-order torsion loci excludes smaller orders. The paper ends with an explicit elliptic curve family over a p-adic disk, with a singular fiber at t=0 and a section whose torsion orders n occur at points t_n tending to 0, showing that the abelian scheme hypothesis is essential.","tokens_in":8185,"tokens_out":28085,"duration_ms":246960,"significance":"The paper is a short, well-written note that establishes a clean p-adic separation property for torsion values of sections of abelian schemes. The main theorem is a natural counterpart to complex-density results and is proved by an elegant combination of p-adic compactness, logarithmic coordinates, and Zariski closedness of torsion loci; the argument is self-contained once standard rigid-analytic facts are admitted. The explicit counterexample via Tate uniformization is valuable and, despite some small typographical issues, convincingly shows optimality of the hypotheses. I also checked the potential concern that high p-power torsion points might accumulate at the identity and violate the torsion-freeness of the subgroup E: in the logarithmic coordinates E is a small ball where the group law is additive, and p^n-torsion points have coordinates tending to the boundary of the formal group rather than to zero, so they are excluded by taking E sufficiently small. The result should be of interest to researchers working on p-adic unlikely intersections and p-adic dynamics.","major_comments":[],"minor_comments":[{"comment":"The phrase 'the order of the image of s(x0) in the finite group A_{x0}(K)/((E ∩ A_{x0})(K))' is imprecise because the quotient is a finite set of cosets and need not be a group; the intended meaning is the least positive n with ns(x0) ∈ E, which exists by finiteness.","section":"§2, proof of Theorem 1.2"},{"comment":"In Lemma 3.3 the conclusion should include the constant term a2: the inverse is a2 + rZ_p[[f/r^2, (x1-a1)/r^2]] in general, not rZ_p[[...]]. In the application to Proposition 3.4, a2 = p lies in rZ_p because r has valuation 1, so the printed conclusion is correct there; please state the lemma with the constant term or with the hypothesis a2 ∈ rZ_p.","section":"§3, Lemma 3.3"},{"comment":"The power series ring is stated as pZ_p[[(X-X(0,p))/p^3, q/p^3]], but Lemma 3.3, applied with r = ∂X/∂z(0,p)·p (a unit times p), gives the first variable normalized by p^2 rather than p^3; the convergence domain X ∈ X0 + p^4Z_p, q ∈ p^4Z_p is stricter than necessary. Please correct the exponent or the stated radii.","section":"§3, Proposition 3.4"},{"comment":"The congruence for the solution of φ(t) = ŝ(t)^n, stated as t ∈ p^n + p^{n+1}Z_p, appears off by a factor of p: the displayed leading terms φ(t) = -(p/(1-p)^2)t + O(t^2) and ŝ(t)^n = p^n(1+O(t)) give t ∈ p^{n-1} + p^nZ_p. Since the argument only needs t_n → 0, the conclusion is unaffected.","section":"§3, proof of Proposition 3.1"},{"comment":"There are sign inconsistencies in the displayed family: with the given equation y^2 = (x-1/12)^2(x+1/6) + t(x - p/(1-p)^2 - 1/12), the coefficient B2 should be 1/864 - t(p/(1-p)^2 + 1/12), not 1/864 - t(p/(1-p)^2 - 1/12); consequently the stated leading term of 1/j(E_t) should be +p/(1-p)^2 t + O(t^2), not -p/(1-p)^2 t + O(t^2). These signs should be corrected consistently.","section":"§3, equation of E_t"},{"comment":"In part (2), when taking U = U0 - S_n, it would be helpful to state explicitly that S_n ∩ U0 is closed in U0 (it is the zero locus of ns after the proper-order loci are excluded), so that the complement is a rigid-analytic open neighborhood.","section":"§2, proof of Theorem 1.2(2)"},{"comment":"Reference [1] appears to be a preprint; please add the arXiv identifier or publication data.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The main theorem is sound and the paper is well within the journal's scope. The counterexample in Section 3 contains several small typos and sign inconsistencies, but they are localized and clearly fixable, and they do not affect the central result."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: Theorem 1.2 is correct and the paper is a clean contribution. The proof is short: Lemma 2.1 gives, around any point, a rigid-analytic open subgroup E of A that is fiberwise isomorphic to the unit ball and has finite index in the fiber at the base point. That finite-index property is the load-bearing step, and it follows from p-adic compactness; once you have ns(x) in E for nearby x, torsion-freeness of E forces exact order n, and Zariski closedness excludes proper divisors. No gap I can find.\n\nWhat's actually new: Scanlon's theorem is for a fixed abelian variety; this handles a family and gives a uniform local statement about exact torsion orders. The proof via the finite-index open subgroup is a neat adaptation. The paper also gives a sharpness example: when the fiber degenerates to a nodal cubic, torsion points of exact order n accumulate at the singular fiber. The construction via Tate uniformization is explicit and convincing; it shows the abelian hypothesis is needed.\n\nSoft spots: these are minor. The power series computations in Section 3 are partly asserted rather than derived. I trust them, but a referee should ask for one or two intermediate steps. The notation n is overloaded (order of torsion vs. index in the proof of Theorem 1.2); that will trip up a careful reader, but it is not a mathematical issue. The paper admits computations are left to the reader; that is fine for a short note.\n\nI do not see a hidden load-bearing flaw. The circularity concern is nil: Lemma 2.1 is proved from smoothness and compactness, and the counterexample is independent. One might ask whether the main theorem could be derived from Scanlon directly by looking at subvarieties of the universal abelian scheme, but the finite-index subgroup argument is genuinely different and handles exact-order separation in a way that would not follow automatically.\n\nI would send this to a reading group on p-adic unlikely intersections or rigid-analytic geometry. It deserves a serious referee, and I would accept.","headline":"A clean, correct note: the torsion locus of a section of an abelian scheme separates p-adically by exact torsion order, with a sharpness example that shows the abelian hypothesis is needed.","tokens_in":8726,"tokens_out":1493,"would_cite":true,"duration_ms":15325,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G10","14G22","14K15"],"pacs":[],"model":"deepseek-v4-flash","headline":"For a section of an abelian scheme over a p-adic field, torsion values of different exact orders are separated by rigid-analytic neighborhoods, and non-torsion points are p-adically isolated from the torsion locus.","keywords":["p-adic fields","abelian schemes","torsion locus","rigid analytic spaces","formal groups","Tate uniformization","bad reduction","unlikely intersections"],"falsifier":"Look for a smooth fiber at which torsion values of infinitely many exact orders accumulate: for a section of an abelian scheme over a p-adic field, if there were a sequence x_m converging to x_0 with x_0 in the smooth locus and s(x_m) of pairwise distinct exact torsion orders, Theorem 1.2 would fail. The paper's Section 3 example shows the analogous accumulation does occur at a singular fiber, so smoothness is the decisive distinction.","tokens_in":7781,"feed_emoji":"🧮","tokens_out":10414,"duration_ms":97123,"temperature":0.7,"pith_summary":"The paper establishes a p-adic separation theorem for torsion values of a section of an abelian scheme. The main theorem says that around any point of the base variety, torsion values have a single exact order: a point that is torsion of order n has a rigid-analytic neighborhood containing no torsion of any other order, and a non-torsion point has a neighborhood with no torsion at all. This gives a precise sense in which torsion points of different orders stay away from each other over p-adic fields, in contrast to the complex-analytic setting where torsion values can be dense. The proof works by constructing, near each point, a rigid-analytic open subgroup of the abelian scheme that is isomorphic to the unit ball and fiberwise of finite index; torsion detection then reduces to a single linear equation. The paper closes with an example showing the conclusion fails if the abelian scheme is allowed to have bad reduction.","feed_headline":"Torsion values separate p-adically by exact order","feed_subtitle":"For p-adic abelian schemes, each torsion order has its own rigid-analytic neighborhood.","key_machinery":"The machinery is a local structure theorem for abelian schemes over p-adic fields. It says that around each K-point of the base there is a rigid-analytic open set E in the total space, containing the identity section, that is isomorphic to U times the g-dimensional unit ball in such a way that the group law is coordinatewise addition and the intersection of E with each fiber is a subgroup; moreover this subgroup has finite index in the fiber over the chosen point. Relative translation-invariant differentials are used to write the group law as addition in power-series coordinates, and p-adic compactness gives the finite-index statement. Once E is available, a single multiple n brings the section into the torsion-free group E throughout a neighborhood, so the torsion condition becomes ns(x)=0 and all nearby torsion has the same exact order.","core_discovery":"Write S_n for the subvariety of the base where the section is torsion of exact order n. The central claim is that for any finite extension L of the p-adic field K, every point x0 in S_n(L) has a rigid-analytic neighborhood U with U disjoint from S_n' for n' different from n, and every point x0 not in any S_n has a rigid-analytic neighborhood disjoint from all S_n. In particular, the torsion locus is a disjoint union of p-adically open pieces indexed by exact torsion order, and no torsion point of one order can be approached by torsion points of other orders. The same conclusion is shown to be false for families with bad reduction: the paper constructs an explicit elliptic scheme over the p-adic disk with a section whose torsion points of every sufficiently large exact order accumulate at the singular fiber.","pith_inferences":["One might expect a quantitative version: over a fixed quasicompact rigid subspace with good reduction, the p-adic distance between torsion values of different orders could be bounded below uniformly, with the bound depending on the heights of the orders; the theorem gives the qualitative separation but no such bound.","The same local-group argument would likely work for multisections or finite collections of sections, provided each section satisfies the finite-index condition near a point; this would give a p-adic analogue of simultaneous separation.","The counterexample at bad reduction suggests that torsion orders can concentrate only where the identity component of the special fiber degenerates, so the natural global formulation of this phenomenon may live on the smooth locus of a Néron model.","If the theorem is right, complex-density results for torsion values have no direct p-adic analogue: in the rigid topology the torsion locus is locally confined to at most one exact order per open set."],"forward_implications":["No sequence of torsion points with pairwise distinct exact orders can converge in S(K): any limit would be torsion of one order or non-torsion, and both cases are excluded by the theorem.","The exact torsion order is locally constant on the torsion locus: each S_n is open in the rigid topology of the union of the S_n's.","A non-torsion point has a rigid-analytic neighborhood free of torsion values, so the torsion locus has empty interior and is not topologically dense in S(K).","The hypothesis that the fibers are abelian varieties is necessary in general: the explicit family in Section 3 has torsion of every large order accumulating at a singular fiber."],"supporting_citations":[{"why":"Supplies the rigid-analytic foundations, including Tate algebras, affinoids, and open subspaces, used to state and prove the local structure theorem.","marker":"[3]"},{"why":"Provides the jumping-locus theorem that the authors explicitly take as the model for their separation statement.","marker":"[7]"},{"why":"Supplies the Tate curve uniformization and its explicit formulas, used in Section 3 to construct the accumulation counterexample at bad reduction.","marker":"[11]"}],"fun_headline_variants":["Torsion orders p-adically separated on abelian schemes","Each exact torsion order has its own p-adic open set","Torsion locus is disjoint union over p-adic orders","Bad reduction breaks p-adic separation of torsion"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The premise that carries the argument is that at every point the abelian scheme contains a rigid-analytic open subgroup isomorphic to the unit ball whose intersection with the given fiber has finite index; if this finite-index condition fails, nearby torsion orders could vary and the theorem's conclusion would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Torsion orders p-adically separated on abelian schemes","Each exact torsion order has its own p-adic open set","Torsion locus is disjoint union over p-adic orders","Bad reduction breaks p-adic separation of torsion"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000468,"raw_usage":{"total_tokens":2245,"prompt_tokens":770,"completion_tokens":1475,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":386,"completion_tokens_details":{"reasoning_tokens":1407}},"tokens_in":386,"tokens_out":1475,"duration_ms":11326,"temperature":1.0,"reasoning_tokens":1407,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:23:39.063368+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Look for a smooth fiber at which torsion values of infinitely many exact orders accumulate: for a section of an abelian scheme over a p-adic field, if there were a sequence x_m converging to x_0 with x_0 in the smooth locus and s(x_m) of pairwise distinct exact torsion orders, Theorem 1.2 would fail. The paper's Section 3 example shows the analogous accumulation does occur at a singular fiber, so smoothness is the decisive distinction.","supporting_citations":[{"cited_title":"Bosch, U","cited_arxiv_id":null,"evidence_quote":"Supplies the rigid-analytic foundations, including Tate algebras, affinoids, and open subspaces, used to state and prove the local structure theorem."},{"cited_title":"Maulik and B","cited_arxiv_id":null,"evidence_quote":"Provides the jumping-locus theorem that the authors explicitly take as the model for their separation statement."},{"cited_title":"Silverman","cited_arxiv_id":null,"evidence_quote":"Supplies the Tate curve uniformization and its explicit formulas, used in Section 3 to construct the accumulation counterexample at bad reduction."}],"review_version":1}