{"id":"f6b28949-d444-4bad-9ce5-d5dd88e0908b","arxiv_id":"2607.26774","paper_version":1,"verdict":"ACCEPT","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"Under 2 not a cube mod p, explicit mock Heegner points prove rank-1 BSD up to a 2-unit for E_{2q} and full BSD for the companion rank-0 curves E_{2q^{2}}.","lead":"The paper constructs explicit mock Heegner points on certain Mordell curves E_{2p} and E_{2p^{2}} and proves the BSD formula for them (full for rank 0, up to a 2-adic unit for rank 1) when 2 is not a cube mod p. This settles the rational cube-sum problem for those 2q and gives concrete arithmetic of L-derivatives via heights.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"No significant objection identified; the numerical CM identification in Lemma 2.1 is the softest step but is standard, finite, and checkable to arbitrary precision.","rationale":"The reader correctly isolated the single weakest link: the numerical identification in Lemma 2.1 that feeds the reduction-mod-p nontorsion proof. I re-checked the subsequent arithmetic (doubling on E: y^{2}=x^{3}+1 recovers x(2P)=0 when P=(2v,-3), the Galois/trace relations, the height computation yielding the factor 1/3 in Theorem 3.1, and the 3-adic index argument via Lemma 2.4) and found no further internal cracks. The open 2-primary gap is already flagged honestly; Burungale–Flach is applied only after L(1,E_{2q^{2}})≠0 is secured by the same nontorsion. Because the numerical step is finite, standard, and independently verifiable to arbitrary precision, it does not warrant downgrading from ACCEPT. Verdict remains UNCHANGED.","tokens_in":17021,"tokens_out":655,"duration_ms":37581,"concrete_test":"Re-evaluate the eta-quotient expression (1.2) for f'(ζ/6) in SageMath/Magma at 100+ bits of precision and confirm it matches P' among the 36 known torsion points of E(H6) to all computed digits; alternatively, derive the same identity algebraically from known CM values of Dedekind eta products. If the match fails or is ambiguous, Theorem 2.2 (and thus both main theorems) requires repair.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central nontorsion claim (Theorem 2.2) that underpins both Theorems 0.1 and 0.2 depends on Lemma 2.1 identifying f'(ζ/6) with the explicit torsion point P'=(-u ζ^{2} ∛4, u(1+∛2)√-3) solely by SageMath numerical approximation, after noting that E(H6)≅(Z/6Z)^{2} so there are only 36 candidates. All subsequent reduction computations (the explicit form of Q̄, the congruence R1≡±2Q̄=(O,(0,±1)), and the contradiction ruling out R1∈E(L)[3]) inherit this identification. If the floating-point match were to the wrong torsion point, the reduction argument would collapse and nontorsion of S1 would be unproved. This is a genuine soft spot in the novel step, but it is not a structural gap: the candidate set is finite, the eta quotients have integral q-expansions (Ligozat), and arbitrary-precision evaluation separates the 36 points cleanly. No other load-bearing internal inconsistency appears in the Galois-action, Gross–Zagier, or 3-adic index arguments (Lemmas 2.3–2.4, Corollary 3.2).","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper constructs an explicit mock Heegner point P1 on E: y^{2} = x^{3} + 1 over the ring class field H_{6p}, for primes p ≡ 4 or 7 mod 9 with 2 not a cube in F_p. After tracing to R1 (and S1 = 3R1) in E(L) with L = K(∛p), it proves nontorsion by reduction modulo primes above p, comparing the reduction of P1 to the Frobenius image of an auxiliary torsion point P' independent of p. Cubic twisting then yields a nontorsion point on E_{2q}(Q) (q = p or p^{2} according as p ≡ 4 or 7 mod 9), so rank E_{2q}(Q) = 1 and 2q is a rational cube sum. An explicit Gross–Zagier formula (via the CST1 variation) identifies L'(1, E_{2q}) with a multiple of the Néron–Tate height of the Heegner point; Kolyvagin–Gross–Zagier–Rubin give finiteness of Sha, and an index computation rules out 3-divisibility of the Heegner point in the Mordell–Weil lattice. The BSD formula for the rank-one curve is thereby verified up to a unit in Z[1/2]ˣ. For the companion rank-zero curves E_{2q^{2}} the same identity plus Burungale–Flach yields the full BSD formula.","tokens_in":17288,"tokens_out":1468,"duration_ms":44733,"significance":"The work settles the rank part of BSD (and the cube-sum problem) for an infinite family of Mordell curves E_{2p} and E_{2p^{2}} in the congruence classes p ≡ 4, 7 mod 9 under a mild local condition on 2, and obtains the full BSD formula for the associated rank-zero twists. The construction is fully explicit, the Galois action is tracked via Shimura reciprocity, and the nontorsion argument (reduction against a p-independent torsion point) is a genuine technical contribution relative to earlier mock-Heegner treatments of cube sums. The height formula is taken from a published variation of Gross–Zagier and specialized carefully; the 3-adic index computation is internal and does not assume BSD. These are concrete, checkable advances on a classical Diophantine problem and on BSD for CM curves of small conductor.","major_comments":[{"comment":"Lemma 2.1 identifies f'(ζ/6) with the explicit torsion point P' = (-u ζ^{2} ∛4, u(1+∛2)√-3) solely by SageMath numerical approximation, after noting that E(H6) ≅ (Z/6Z)^{2} has 36 torsion points. Every subsequent reduction identity in the proof of Theorem 2.2 (the explicit form of Q̄, the congruence R1 ≡ ±2Q̄ = (O,(0,±1)), and the contradiction ruling out R1 ∈ E(L)[3]) inherits this identification. While the candidate set is finite and eta quotients have integral q-expansions (Ligozat), the manuscript should either supply a rigorous algebraic identification of the CM point or document a reproducible high-precision verification that separates P' from the other 35 torsion points to a stated working precision. As written, the novel nontorsion step rests on an unreproducible floating-point match.","section":"Lemma 2.1 and Theorem 2.2"},{"comment":"In the reduction computation leading to (2.2), the authors assert that 2·((-v^{2},0),(2v,-3)) equals (O,(0,±1)) in E(F_p)×E(F_p). The first component is immediate (2-torsion), but the second component 2·(2v,-3) = (0,±1) is not expanded. A short verification of the duplication formula (or a reference to the 2-division polynomial evaluated at x = 2v) would make the key congruence fully self-contained.","section":"Theorem 2.2, display (2.2)"}],"minor_comments":[{"comment":"The running header and title page contain spacing artefacts (“CER T AIN”, “CUR VES”, “f or”, “modulop”). These should be cleaned for the published version.","section":"Title / running heads"},{"comment":"Table 1 lists images of cusps under f'; a brief sentence confirming that the numerical approximations were cross-checked against the known torsion subgroup E(K)_tors would reassure the reader that the table is exact rather than approximate.","section":"Table 1"},{"comment":"In Proposition 1.1 the proof that E(L)[3^∞] = E(K)[3] uses irreducibility of φ3(x) = x^9 - 96x^6 + 48x^3 + 64 over K. A reference or one-line argument for that irreducibility would be helpful.","section":"Proposition 1.1"},{"comment":"The notation switches between E_{2q}, E^{(q)}_1 and E^{(q)}_2; a single sentence early in §3 recalling the precise Weierstrass models and the isogeny relating E_{2q} to E^{(q)}_1 would improve readability.","section":"Section 3"},{"comment":"Page 10, line after (3.8): “dim_{F3} Sel3(E^{(q)}_1/Q) = 2 and dim_{F3} Sel3(E^{(q)}_2/Q) = 1 [JMS2]” — the citation is appropriate, but stating the precise theorem number inside [JMS2] would aid the reader.","section":"Proof of Theorem 0.1"},{"comment":"Several references (e.g., [ABS-BS], [LLT]) are still arXiv preprints; update to published versions where available.","section":"References"}],"recommendation":"minor_revision","confidential_remarks":"The paper is a solid, carefully written contribution in the style of Dasgupta–Voight / Cai–Shu–Tian. The only genuine soft spot is the numerical CM-point identification in Lemma 2.1; once that is documented to journal standards the manuscript is ready for acceptance. Fit for a strong number-theory journal is clear. No concerns about citation practice or novelty disclosure."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The punchline is that they finish the 2p/2p^{2} cube-sum cases for p ≡ 4 or 7 mod 9 (with 2 not a cube mod p) by an explicit mock Heegner point, get rank exactly 1, BSD up to a 2-unit, and full BSD on the companion rank-0 curves via Burungale–Flach.\n\nWhat is new is the construction itself for these congruence classes, the auxiliary torsion point P' independent of p, and the reduction-mod-p argument that uses it to prove nontorsion of R1 (and thus S1). Everything else is standard machinery done carefully: Shimura reciprocity for the Galois action on X0(36), cubic twist to E(q)1(K), the CST1 variation of Gross–Zagier specialized to this setting, Kolyvagin/Rubin/GZ for rank and finiteness of Sha, and an index computation (Lemmas 2.3–2.4) that rules out 3-divisibility of the Heegner point so the 3-part of BSD matches. The 2-primary gap is flagged honestly; the rest of the primes are covered by existing p-adic and ordinary-reduction results. Self-citations to their own 3-Selmer papers are used only for dimensions already proved elsewhere, so circularity is low.\n\nThe softest step is Lemma 2.1: they identify f'(ζ/6) with the explicit torsion point P' by SageMath numerical approximation, relying on E(H6) ≅ (Z/6Z)^{2} having only 36 candidates. All the later reduction congruences inherit that match. It is a genuine soft spot in the novel step, but it is finite, checkable to arbitrary precision, and standard practice once Ligozat integrality is in hand; it does not look like a structural hole. The rest of the chain holds up on a close read.\n\nThis is for people who work on Heegner points, explicit BSD, or the cube-sum problem. It is not transformative outside that circle, but it is a clean, usable advance on a concrete infinite family. I would send it to referees without hesitation; the claims are supported as stated and the open 2-adic unit is not hidden.","headline":"Solid explicit mock-Heegner construction that settles rank-1 and nearly full BSD for the remaining 2p/2p^{2} Mordell curves in the 4,7 mod 9 classes under a clean local hypothesis.","tokens_in":18037,"tokens_out":594,"would_cite":true,"duration_ms":10285,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11F11","11D25","11G40"],"pacs":[],"model":"grok-4.5","headline":"Explicit mock Heegner points prove that 2p and 2p² are rational cube sums for primes p ≡ 4 or 7 mod 9 when 2 is not a cube mod p, and settle BSD for the matching Mordell curves up to a 2-adic unit.","keywords":["Mordell curves","mock Heegner points","Gross-Zagier formula","BSD conjecture","rational cube sums","cubic twists","ring class fields"],"falsifier":"For a concrete prime p ≡ 4 or 7 mod 9 with 2 not a cube mod p, compute the reduction of the constructed Heegner point modulo a prime above p and check whether it equals a 3-torsion point; alternatively evaluate L′(E_{2q}, 1) and the canonical height of the constructed rational point and test whether their ratio matches the predicted BSD quotient up to a power of 2.","tokens_in":17773,"feed_emoji":"🔢","tokens_out":1107,"duration_ms":17411,"temperature":0.7,"pith_summary":"An integer is a rational cube sum when it equals a³ + b³ for rational a and b. That question is equivalent to whether the Mordell curve Y² = X³ − 432 n² has positive rank. For n = 2p or 2p² with p a prime congruent to 4 or 7 mod 9, the paper builds an explicit mock Heegner point on a related CM curve and proves the point is nontorsion precisely when 2 is not a cube modulo p. The resulting rational point shows that the rank is exactly 1, so 2q is a sum of two rational cubes. An explicit Gross–Zagier formula then matches the height of this point to the derivative of the L-function, confirming the rank part of BSD and showing that the full BSD formula holds up to a unit in Z[1/2]. For the companion rank-zero twists the same formula plus an external result gives the complete BSD prediction.","feed_headline":"Mock Heegner points prove 2p is a sum of two cubes","feed_subtitle":"When 2 is not a cube mod p, rank and BSD hold for the matching Mordell curves","key_machinery":"An explicit mock Heegner point P₁ on the CM curve E : y² = x³ + 1, defined over the ring class field of conductor 6p, whose trace S₁ to K(∛p) is shown nontorsion by reduction modulo p against a fixed torsion point P′ = f′(ζ/6); cubic twisting then produces a nontorsion point on E_{2q}, and a variation of the Gross–Zagier formula relates its height to L′(1, E, χ).","core_discovery":"For primes p ≡ 4 or 7 mod 9 with 2 not a cube in F_p, write q = p or p² accordingly. The Mordell curve E_{2q} : y² = x³ − 27 q² has analytic and algebraic rank exactly 1, so 2q is a rational cube sum; its Tate–Shafarevich group is finite and its order equals the BSD quotient up to a 2-adic unit. The companion rank-zero curve E_{2q²} satisfies the full BSD formula.","pith_inferences":["The numerical identification of f′(ζ/6) could be replaced by an algebraic proof that the eta-quotient evaluates to the listed torsion point, removing the only computational step.","When 2 is a cube mod p the same construction may still produce a point of infinite order on a quadratic twist, suggesting a route to the remaining cube-sum cases.","The method extends in principle to other conductors of the form 2p^k with root number −1, provided a suitable fixed torsion point and reduction congruence can be found."],"forward_implications":["Every prime p ≡ 4 or 7 mod 9 for which 2 is not a cubic residue yields an explicit rational point of infinite order on E_{2q}, so 2q = a³ + b³ for computable rationals a, b.","The rank part of BSD holds unconditionally for these E_{2q}, and |Sha| is determined up to a power of 2.","The companion curves E_{2q²} of root number +1 satisfy the complete BSD formula whenever the same cubic-residue hypothesis holds.","The same mock-Heegner construction supplies an explicit generator whose index in the Mordell–Weil group is a 3-adic unit."],"fun_headline_variants":["Mock Heegner points show rank 1 for Mordell curves E_{2p}","Explicit mock Heegner points verify BSD on E_{2q} Mordell curves","Mordell curves E_{2p} have rank 1 when 2 is noncube mod p","BSD formula holds up to 2-adic unit via mock Heegner points","Mock Heegner points prove 2q is a sum of two rational cubes"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The proof that the constructed point is nontorsion relies on identifying a modular image of ζ/6 with an explicit torsion point by numerical approximation; if that identification is wrong the reduction argument fails.","fun_headline_variants_meta":{"raw":{"variants":["Mock Heegner points show rank 1 for Mordell curves E_{2p}","Explicit mock Heegner points verify BSD on E_{2q} Mordell curves","Mordell curves E_{2p} have rank 1 when 2 is noncube mod p","BSD formula holds up to 2-adic unit via mock Heegner points","Mock Heegner points prove 2q is a sum of two rational cubes"]},"model":"grok-4.5","effort":"low","cost_usd":0.004466,"raw_usage":{"total_tokens":1302,"prompt_tokens":782,"num_sources_used":0,"completion_tokens":114,"cost_in_usd_ticks":44664000,"prompt_tokens_details":{"text_tokens":782,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":406,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":782,"tokens_out":114,"duration_ms":7079,"temperature":1.0,"reasoning_tokens":406,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T21:27:44.110229+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"For a concrete prime p ≡ 4 or 7 mod 9 with 2 not a cube mod p, compute the reduction of the constructed Heegner point modulo a prime above p and check whether it equals a 3-torsion point; alternatively evaluate L′(E_{2q}, 1) and the canonical height of the constructed rational point and test whether their ratio matches the predicted BSD quotient up to a power of 2.","supporting_citations":[],"review_version":1}