{"id":"736dbe2a-1f14-4c27-b13e-e426e5c2e4eb","arxiv_id":"2508.19947","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"For rank 0 once-punctured elliptic curves, the quadratic Chabauty locus equals the p-adic points of a fixed finite subscheme of torsion points described explicitly by residues and local heights.","lead":"This paper proves a non-abelian generalization of Stoll's conjecture in the rank 0 quadratic case: the quadratic Chabauty locus of a curve is cut out by a single finite subscheme independent of the prime p. It introduces a geometric method for Chabauty-Kim and gives an explicit characterization of the locus for once-punctured rank 0 elliptic curves, sharpening earlier results of Bianchi.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem A for Y=E\\{∞} depends on the unverified exact applicability of [BMPZ16, Thm 2(ii)] to the semiabelian scheme B in §5.4; if that theorem carries an extra hypothesis not checked for B, the trichotomy may be incomplete.","rationale":"The reader's verdict identifies the same weakest assumption: the application of the Bertrand-Masser-Pillay-Zannier trichotomy in the exceptional case of Theorem D. I agree that this is the single most load-bearing step for the headline result, since every other part of the proof is either a detailed internal construction or a more standard Manin-Mumford input. The rest of the paper appears internally coherent: the isogeny geometric quadratic Chabauty locus is set up carefully, the comparison with quadratic Chabauty is proved through Selmer schemes, and the reduction to finiteness of quadratic torsion packets is explicit. No internal inconsistency or obvious computational error was found in the sections I examined. However, because the exceptional case is decided by quoting a substantial external theorem without reproducing its hypotheses, the correctness of the central claim is conditional on [BMPZ16, Theorem 2(ii)] applying verbatim to the particular semiabelian scheme B = \\hat f^*P' and section s constructed there. This is a genuine verification step, not a manufactured objection: if the quoted theorem has a non-degeneracy or properness hypothesis that B does not satisfy, the trichotomy could omit cases, and the argument that every infinite quadratic torsion packet forces containment in a strict AT subvariety would break. I therefore recommend a conditional acceptance rather than outright acceptance or rejection.","tokens_in":58776,"tokens_out":27484,"duration_ms":343152,"concrete_test":"Obtain the exact statement of [BMPZ16, Theorem 2(ii)] including every hypothesis, and verify each one for the pair (B, s) in Section 5.4. In particular: (1) if the theorem requires the base curve to be complete/projective, prove that s either extends to the smooth compactification of Y = E\\{∞} or that the theorem has an affine-curve version; (2) if it assumes p and q are nonconstant or q is non-torsion, check that the omitted constant/torsion cases are handled by the earlier Manin-Mumford arguments; (3) if it assumes the section is not contained in a proper algebraic subgroup scheme, verify that the 'image not contained in any strict AT subvariety' hypothesis in Theorem D implies this. If a hypothesis fails, construct a concrete section of a pullback of the universal extension over E with infinitely many torsion points outside the trichotomy; if all hypotheses hold, the proof of Theorem","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central theorem for once-punctured elliptic curves funnels through Theorem D's exceptional case (Section 5.4, dim(A)=1, r=1), and that case rests entirely on the relative Manin-Mumford theorem [BMPZ16, Theorem 2(ii)], quoted as Theorem 5.14. The paper constructs B = \\hat f^*P' over Y with section s and assumes the quoted trichotomy (p torsion, q torsion, or q = α(p) modulo torsion with α antisymmetric) is exhaustive for this B. This is load-bearing: if [BMPZ16] has an additional hypothesis not verified for B—for example that the base curve is proper/complete, that the section is not contained in a proper semiabelian subscheme, that the extension is non-isotrivial, or that p and q are nonconstant—then the paper's conclusion after the trichotomy, that λ_P ∘ \\hat f is torsion in every case, could fail. In that event the finiteness of quadratic torsion packets for the once-punctured elliptic curve case, and hence Theorem 6.13 / Theorem A, would not follow from the cited theorem. The paper states the conclusion of [BMPZ16, Theorem 2(ii)] but does not quote its full hypotheses or check them line by line for the specific pair (B, s). This is not an observed contradiction, but it is the least secure external link in the chain establishing the main claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper formulates a non-abelian analogue of Stoll's conjecture (Conjecture 1.2) and proves the rank-zero quadratic case. The main theorems are: construction of the (split) quadratic Albanese variety of a smooth curve (Theorems B and B'), an exact comparison between the isogeny geometric quadratic Chabauty locus and the usual quadratic Chabauty locus under rank-zero hypotheses (Theorem C / Theorem 4.1), a Manin-Mumford-type finiteness theorem for quadratic torsion packets in AT varieties (Theorem D), and an explicit description of the finite subscheme cutting out the quadratic Chabauty locus for once-punctured elliptic curves of Mordell-Weil rank zero (Theorem 6.13). The strategy is to replace the p-adic Chabauty condition by a purely geometric condition — membership in the inverse image of a finite set W under the map f∞ — and then to prove finiteness of the relevant fibres via unlikely-intersections results, with the elliptic-curve case ultimately depending on a relative Manin-Mumford theorem.","tokens_in":59169,"tokens_out":20873,"duration_ms":223813,"significance":"If correct, the paper is a substantial advance: for once-punctured elliptic curves of rank 0 it gives a p-independent algebraic description of the quadratic Chabauty locus, uniformly bounded in p, and a simultaneous necessary and sufficient criterion for a torsion point to lie in the locus. The paper introduces a useful category of AT varieties, contains detailed conceptual proofs, and involves no fitted parameters; the computations in §6 are explicit and checkable. The exact equality in Theorem 4.1 addresses the known strictness gap between geometric quadratic Chabauty and quadratic Chabauty. I regard the central claims as very likely correct, but one load-bearing external-input verification is missing (see major comment 1).","major_comments":[{"comment":"The exceptional case dim(A)=1, r=1 of Theorem D rests entirely on the quoted theorem [BMPZ16, Theorem 2(ii)], but the paper neither gives the full hypotheses of that theorem nor verifies them for the specific semiabelian scheme B = \\hat f^*P' over Y and section s. In particular, the paper should state and check any standing assumptions on the base field, on Y (e.g. properness or non-isotriviality), on the extension class q, and on the section s (e.g. non-degeneracy or not being contained in a proper semiabelian subscheme). Without this verification, the trichotomy in Theorem 5.14 is not formally justified, and the finiteness of quadratic torsion packets for the once-punctured elliptic curve case, and hence Theorem 6.13 / Theorem A, does not follow from the cited theorem as written.","section":"§5.4, Theorem 5.14"},{"comment":"The notation Z(Q) and the domain of the function H^st are inconsistent. The theorem states Z(Q) ⊆ Y(Q)_tors, but the proof and the worked examples require points in Y(Qbar)_tors; Definition 6.11 only defines H^st on Q-rational torsion points. Consequently condition (i), 'H^st(Q) ∈ Q ⊗ Q^×', is not well-formed for arbitrary algebraic torsion points, and condition (ii) is not meaningful unless H^st(Q) is known to be a Galois-invariant element of K ⊗ Q^×. The intended meaning is clear — H^st should be defined on all torsion points and condition (i) should be the assertion that its class lies in the rational subspace Q ⊗ Q^× — but the statement of the main explicit theorem should be corrected.","section":"§6, Definition 6.11 and Theorem 6.13"}],"minor_comments":[{"comment":"The term 'of motivic origin' is deliberately left undefined. This is acceptable as a convention for the conjecture, but the paper should make explicit that Theorem A' and Theorem A are theorems about quotients realized by morphisms, not about all quotients of motivic origin.","section":"§1.1, Remark 1.3"},{"comment":"In the row for I_m^*, the entry for c=4 is '-(m+4)/8'; due to the line break in the table this could be misread as '-m+4/8'. Please reformat for clarity.","section":"§6, Table 1"},{"comment":"The sentence '8+3√7 is a principal unit' would be clearer as '8+3√7 is a unit of norm 1 in Q(√7)' — the latter formulation directly explains why its valuations depend only on the rational prime below each place.","section":"§6.3, Example 6.16"},{"comment":"The phrase 'not realised by any morphism' relies on the paper's convention that a quotient is realised only when it is isomorphic to the target variety's full pro-unipotent fundamental group. A brief reminder of this convention at the start of the appendix would prevent misunderstanding.","section":"Appendix A, Corollary A.2"}],"recommendation":"major_revision","confidential_remarks":"The paper is strong and the main theorems appear sound. My recommendation of major revision is driven by the unverified applicability of [BMPZ16, Theorem 2(ii)] in §5.4, which is load-bearing for the once-punctured elliptic curve case. I believe this is fixable by carefully stating and checking the hypotheses of the cited theorem; after that, I would be happy to accept. The ambiguity in Theorem 6.13 about the domain of H^st should also be corrected."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First thing to know: this paper delivers new theorems, not just a framework. It proves the rank-0 quadratic case of a non-abelian Stoll conjecture and, for a once-punctured elliptic curve of Mordell-Weil rank 0, gives an explicit finite subscheme Z such that the quadratic Chabauty locus Y(Z_p)^{U^p_T} equals Z(Q_p) for all good primes p. That is a genuinely p-independent description, and it sharpens Bianchi's results. The construction of AT varieties and the quadratic Albanese variety, and the isogeny geometric quadratic Chabauty method, are new and do what the earlier Edixhoven-Lido variant could not: Theorem 4.1 shows exact equality with quadratic Chabauty, not just containment. The explicit criterion in Theorem 6.13 involving H^st, local heights, and Kodaira-Tamagawa data is computable, and the two worked examples are convincing.\n\nThe reader's accept verdict is about right, with one caveat. The least secure step is §5.4, where Theorem D's exceptional case (dim A = 1, r = 1) rests on [BMPZ16, Thm 2(ii)]. The paper states the trichotomy but does not quote the full hypotheses of that theorem, and I could not tell from the text alone whether all of them hold for the semiabelian scheme B = f-hat^* P' over Y. If the original theorem requires properness of the base curve, a non-degeneracy condition on the section, or something about isotriviality, the trichotomy may not apply as stated. This is not an observed error, but it is load-bearing for the once-punctured elliptic curve case. A referee should check [BMPZ16] line by line. The rest of the chain—Theorems B/B', 4.1, D except that case, and 6.13—is detailed and coherent. The deliberately undefined 'motivic origin' in Conjecture 1.2 is a bit unsatisfying but harmless, since the unconditional theorems do not rely on it.\n\nWho this is for: arithmetic geometers working on Chabauty-Kim, unlikely intersections, or explicit Diophantine geometry. It deserves a serious referee. I would send it out, with a request to verify the BMPZ application specifically; if that checks out, it is a strong paper.","headline":"Proves a real new case of non-abelian Stoll and makes the rank-zero punctured-elliptic-curve quadratic Chabauty locus explicit; one external application in §5.4 needs a careful hypothesis check.","tokens_in":59592,"tokens_out":3121,"would_cite":true,"duration_ms":33512,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","14G05","14K12","11G30"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that for a rank-0 elliptic curve E over Q, the quadratic Chabauty locus of E minus the origin is, at every good prime p, exactly the Q_p-points of a single finite subscheme defined over Q.","keywords":["non-abelian Chabauty","quadratic Chabauty","elliptic curves","Manin-Mumford","unlikely intersections","AT varieties","Mordell-Weil rank zero","p-adic heights"],"falsifier":"Build the semiabelian scheme B→Y from the once-punctured elliptic curve construction and check whether its section meets the torsion locus in infinitely many points outside the three cases of the relative Manin-Mumford trichotomy; any such intersection would break Theorem D and Theorem A. Alternatively, on a rank-0 curve with a torsion point Q satisfying both criteria of Theorem 6.13, evaluate the actual Coleman functions defining quadratic Chabauty at a good prime p; a contradiction would disprove the claimed equality.","tokens_in":58663,"feed_emoji":"🔢","tokens_out":10124,"duration_ms":105737,"temperature":0.7,"pith_summary":"The paper's target is a non-abelian analogue of the conjecture that the abelian Chabauty locus is interpolated by a p-independent finite subscheme. It proves the rank-zero quadratic case, including the case where Y is an elliptic curve minus its origin and E/Q has Mordell-Weil rank 0. In that case the quadratic Chabauty locus at every good prime is shown to equal Z(Q_p) for one explicit finite Galois-stable subscheme Z consisting of torsion points; the apparent p-adic dependence collapses into the valuations of a single function H^st. This matters because it turns quadratic Chabauty for these curves from a p-adic analytic construction into an algebraic subset of bounded size, uniformly in p.","feed_headline":"Quadratic Chabauty locus: p-independent for rank-0 elliptic curves","feed_subtitle":"For E/Q of rank 0 minus the origin, every good prime sees the same finite subscheme as the quadratic Chabauty locus.","key_machinery":"The central object is the split quadratic Albanese variety, the universal G_m^r-torsor over the relevant abelian variety (an AT variety: abelian-by-torus) which realises the maximal quotient of the pro-unipotent fundamental group that is an extension of V_pJ by Q_p(1)^r. From it the paper builds an isogeny system β_n : P → P_n using the identity P_n = P^{⊗(1+n)} ⊗ [-1]^*P^{⊗(1-n)} and the cube formula [n]^*P_n ≅ P^{⊗2n^2}; these are isogenies of AT varieties. Taking the filtered colimit gives a map f_∞ : Y(Z_p) → lim P_n(Q_p), and the isogeny geometric quadratic Chabauty locus is f_∞^{-1}(W) for a finite set W of integral points. The proof's load-bearing finiteness result is a Manin-Mumford","core_discovery":"On the paper's own terms, the central discovery is that the 'skimmed' quadratic Chabauty locus for a once-punctured rank-0 elliptic curve is a p-independent algebraic object. The paper constructs an isogeny-geometric quadratic Chabauty locus Y(Z_p)^f from the split quadratic Albanese variety P of Y and proves it coincides with the usual non-abelian quadratic Chabauty locus (Theorem 4.1). Writing the finite set W in the rational colimit lim P_n(Q), the isogeny locus is the inverse image of W, and the main finiteness theorem says quadratic torsion packets are finite unless the image lies in a strict AT subvariety (Theorem D). For Y = E minus the origin, the resulting subscheme Z is explicit: Z","pith_inferences":["A testable extension the paper does not spell out: because Theorem 6.13 is a finite arithmetic criterion, one can enumerate torsion points up to growing height and search for the subscheme Z purely from H^st and the W^st_ℓ table, without p-adic integration.","The structural reading this suggests: Stoll-type p-independence is governed by Manin-Mumford for quadratic torsion packets in AT varieties, so any future counterexample would most plausibly arise from a curve whose image lies in a strict AT subvariety, not from the generic locus.","For computational practice, the isogeny formula indicates that H^st can be evaluated through isogenies rather than division polynomials; using cyclic isogenies of small degree might make the criterion feasible on large torsion subgroups, but the paper leaves that implementation open."],"forward_implications":["For every rank-0 elliptic curve E/Q, the quadratic Chabauty locus of E minus the origin has size bounded independently of p, and is explicitly computable from torsion points and local heights.","A torsion point lies in the locus exactly when H^st(Q) is rational and its valuations match the Kodaira-type table; this is a simultaneous necessary and sufficient test, refining previous sufficient tests.","The isogeny-geometric variant gives the equality Y(Z_p)^f = Z(Q_p) without requiring a rational base point or good reduction at p, so in rank 0 the p-adic method can be replaced by a geometric description.","Under the same rank-zero hypothesis, the Selmer Section Conjecture for hyperbolic curves follows once such a finite p-independent subscheme exists.","The appendix shows the full depth-2 unipotent quotient is not realisable by a morphism to a smooth variety for projective curves of genus at least 4, so the AT quotient is near the limit of geometric realisability."],"supporting_citations":[{"why":"The conjecture being generalised: for Jacobian rank at most g-2, the abelian Chabauty locus is contained in a fixed finite subscheme.","marker":"[Sto]"},{"why":"Supplies the Selmer-scheme formulation of non-abelian Chabauty and the conjecture that large quotients cut out rational points.","marker":"[BDCKW18]"},{"why":"Prior treatment of once-punctured elliptic curves; proves torsion-point algebraicity and gives local-height criteria that the paper sharpens to necessary-and-sufficient form.","marker":"[Bia20]"},{"why":"Introduces geometric quadratic Chabauty; the paper's isogeny-geometric variant is built on its P_n/β_n construction.","marker":"[EL23]"},{"why":"Proves containment of geometric quadratic Chabauty in the quadratic Chabauty locus and gives examples of strict containment, motivating the isogeny refinement.","marker":"[DRHS23]"},{"why":"Supplies the relative Manin-Mumford trichotomy used in the exceptional dimension-one case.","marker":"[BMPZ16]"},{"why":"Manin-Mumford for abelian varieties, used for the dimension-at-least-two cases of Theorem D.","marker":"[Ray83]"},{"why":"Computes the possible values of the local height on integral points, yielding the explicit W^st_ℓ table used in Theorem 6.13.","marker":"[CPS06]"}],"fun_headline_variants":["Quadratic Chabauty locus is p-independent for rank-0 once-punctured elliptic curves","Rank-0 once-punctured elliptic curves: quadratic Chabauty locus p-independent","P-independent quadratic Chabauty locus for rank-0 once-punctured elliptic curves","Quadratic Chabauty: rank-0 once-punctured elliptic curves have p-independent locus"],"cache_read_input_tokens":2688,"weakest_assumption_plain":"The rank-0 punctured-elliptic-curve case depends on a classification of when a family of curves can contain infinitely many torsion points, and the paper assumes that classification is complete for the particular semiabelian family it constructs; if a non-listed example could occur, the main theorem for punctured elliptic curves would not follow.","fun_headline_variants_meta":{"raw":{"variants":["Quadratic Chabauty locus is p-independent for rank-0 once-punctured elliptic curves","Rank-0 once-punctured elliptic curves: quadratic Chabauty locus p-independent","P-independent quadratic Chabauty locus for rank-0 once-punctured elliptic curves","Quadratic Chabauty: rank-0 once-punctured elliptic curves have p-independent locus"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001334,"raw_usage":{"total_tokens":5219,"prompt_tokens":658,"completion_tokens":4561,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":402,"completion_tokens_details":{"reasoning_tokens":4466}},"tokens_in":402,"tokens_out":4561,"duration_ms":33868,"temperature":1.0,"reasoning_tokens":4466,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T15:21:05.626003+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Build the semiabelian scheme B→Y from the once-punctured elliptic curve construction and check whether its section meets the torsion locus in infinitely many points outside the three cases of the relative Manin-Mumford trichotomy; any such intersection would break Theorem D and Theorem A. Alternatively, on a rank-0 curve with a torsion point Q satisfying both criteria of Theorem 6.13, evaluate the actual Coleman functions defining quadratic Chabauty at a good prime p; a contradiction would disprove the claimed equality.","supporting_citations":[],"review_version":1}