{"id":"0dc64c2b-9bcb-4dee-8771-f50cf34008d5","arxiv_id":"2412.17703","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Original Mazur-Tate refined BSD Conjectures 4 and 6 fail for many elliptic curves with split multiplicative reduction, but adding the torsion-order inverse to the coefficient ring appears to fix them.","lead":"A computational study of Mazur-Tate refined BSD conjectures finds that two original 1987 conjectures fail as stated for many elliptic curves, and proposes a corrected version that appears to hold in all tested cases. The paper tests over 425,000 curve-prime pairs using SageMath and provides explicit counterexamples.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The claim that Conjecture 4 also fails depends on the positive-rank status of the 367 Conjecture 2.12 failures, and the paper does not report those ranks.","rationale":"I read the paper as making two distinct negative claims: (i) Conjecture 6 of [MT87] fails, supported by 886 counterexamples to Conjecture 0.1; and (ii) Conjecture 4 also fails, supported by 367 counterexamples to Conjecture 2.12. The first claim is well supported: Conjecture 0.1 is equivalent to Conjecture 6 in the setting considered, and the cross-checks with three implementations give reasonable confidence in the numerical values. The second claim has a logical prerequisite that is not documented: Conjecture 2.12 is a consequence of Conjecture 2.4 only when rk_Z(E(Q)) > 0. The paper never states the ranks of the 367 failing pairs or of the samples it displays, and the example in Section 4 is not identified as positive-rank. If the database routine only checked Conjecture 2.12 for positive-rank curves, the conclusion is likely correct, but the paper should say so explicitly and report the rank data. Without that, the stronger statement that Conjecture 4 is false is underdocumented, although it may be true. The reader's verdict did not identify this particular gap, focusing instead on numerical reliability and the partly tautological nature of the modified conjecture. I therefore mark partial disagreement with the reader's weakest-assumption analysis and recommend a conditional acceptance pending the rank check.","tokens_in":13331,"tokens_out":15391,"duration_ms":146154,"concrete_test":"Recompute E.rank() (or an unconditional analytic rank verification) for all 367 pairs reported as Conjecture 2.12 failures, and in particular for the four displayed examples. If the displayed examples or a substantial subset of the 367 have rank 0, the claim that Conjecture 4 is false loses its stated support; if every reported failure has positive rank, or if at least one positive-rank pair can be exhibited as a counterexample, the Conjecture 4 conclusion stands and should be documented in the paper.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"Section 3 reports 367 pairs failing Conjecture 2.12 and the introduction concludes that Conjecture 4 of [MT87] is false. However, Conjecture 2.12 is stated only under the hypothesis rk_Z(E(Q)) > 0, and the implication from a failure of Conjecture 2.12 to failure of Conjecture 2.4 is valid only when the rank is positive. The paper does not give the ranks of the 367 pairs, nor of the displayed examples 377.a2, 832.f1, 4123.b1, and 7826.b1. If these pairs have rank 0, they are not admissible counterexamples to Conjecture 2.12 as a consequence of Conjecture 2.4, and the Conjecture 4 claim is unsupported. The Conjecture 6 counterexamples via Conjecture 0.1 do not require a rank hypothesis, so that part of the paper is not affected by this gap. What is needed is an explicit statement that the database tested Conjecture 2.12 only on positive-rank pairs, together with the rank of at least the displayed failing pairs.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper reports an extensive SageMath computation testing three Mazur–Tate refined BSD conjectures on 425,713 pairs (E, p), where E is a rational elliptic curve and p is a prime of split multiplicative reduction. The main negative finding is that the multiplicative reformulation of Conjecture 6 of [MT87] (Conjecture 0.1) fails for 886 pairs, and the positive-rank consequence Conjecture 2.12 fails for 367 pairs; no failures are reported for Conjecture 2.11. The author concludes that Conjectures 4 and 6 of [MT87] are false as stated when the coefficient ring R is taken to be minimal, and proposes modified conjectures (Conjectures 3.4–3.6) in which the inverse of the torsion order is adjoined to R, reporting that these modified statements hold for all tested pairs. The paper includes two worked examples and points to publicly available code.","tokens_in":13554,"tokens_out":9107,"duration_ms":95206,"significance":"If the negative claims are correct, the paper provides explicit numerical counterexamples to the original Mazur–Tate conjectures, which is an important and potentially surprising result. The computations appear to have been done carefully: for the failing pairs, modular symbols were recomputed with three independent SageMath implementations, and the code is available. The positive modified conjecture is weaker than it first appears, because adding (#E(Q)_Tor)^{-1} to R removes from S precisely the primes that were responsible for the failures; its confirmation is therefore in-sample and partly vacuous for the formerly failing cases. The paper is honest about several caveats, especially Remark 3.8 on the Sha values, but a few load-bearing points need to be addressed before the conclusions can be accepted as stated.","major_comments":[{"comment":"The claim that Conjecture 4 of [MT87] fails because 367 pairs do not satisfy Conjecture 2.12 is only valid for pairs with rk_Z(E(Q)) > 0, since Conjecture 2.12 is stated under that hypothesis. The manuscript does not state whether the database, or the Conjecture 2.12 computation, was restricted to positive-rank curves, and the ranks of the displayed failing pairs (377.a2, 832.f1, 4123.b1, 7826.b1) are not given. If any of these pairs have rank 0, they are not admissible as counterexamples to Conjecture 2.12 arising from Conjecture 2.4. Please report the rank for the displayed examples and state whether the 367 failures all have rk_Z(E(Q)) > 0; this is essential support for the Conjecture 4 conclusion.","section":"Section 3, Conjecture 2.12 and displayed examples"},{"comment":"The evidence for the modified Conjecture 3.4 is weaker than the phrase 'does appear to hold' suggests. Adjoining (#E(Q)_Tor)^{-1} to R makes every prime ℓ dividing #E(Q)_Tor invertible in R, so such ℓ are removed from the set S defined in Conjecture 3.4. Since the paper states that the failures of Conjecture 0.1 occur for primes ℓ ∈ S with ℓ | #E(Q)_Tor, the modification makes the conjecture vacuously satisfied for precisely the previously failing coordinates. The paper should explicitly quantify how many of the 886 failures are removed in this way, and should state whether any failures would remain at primes ℓ not dividing the torsion. As it stands, the numerical verification of Conjecture 3.4 provides only in-sample confirmation of a weakened statement, not independent support for the proposed modification.","section":"Section 3, Conjecture 3.4 and Remark 2.13"},{"comment":"The wording 'the failure of Conjecture 0.1 occurs when there exist primes ℓ ∈ S such that ℓ | #E(Q)_Tor' is ambiguous. It is not clear whether every failing pair (E,p) has at least one such ℓ, or whether the actual failing coordinates are always at such ℓ. A precise table of the failing prime ℓ for each of the 886 pairs, together with the torsion order and the set S, would settle whether the diagnosis is complete and would also clarify the contrast with the cases where ℓ divides the torsion and equation (1) still holds.","section":"Section 3, bullet list and Remark 3.1"}],"minor_comments":[{"comment":"Equation (9) appears to have lost exponent formatting and is not readable as printed: '5 ≡ 102− 131 ≡ 1 in (Z/7Z)^*/⟨−1⟩'. Please correct the typesetting so that the computation can be followed.","section":"Section 4, equation (9)"},{"comment":"The statement that 'No counter-examples were found for conjecture 2.11' is conditional on the SageMath/LMFDB values for the order of the Tate–Shafarevich group, which in some cases assume the strong BSD conjecture (Remark 3.8). This caveat should appear wherever the Conjecture 2.11 result is summarized, including the abstract or introduction if that result is advertised there.","section":"Abstract, Section 3, Conjecture 2.11"},{"comment":"The footnote contains a typo: 'mearly' should be 'merely'.","section":"Footnote 4"},{"comment":"It would help the reader if the paper explicitly noted that Conjecture 2.12 is a necessary condition for Conjecture 2.4 only when rk_Z(E(Q)) > 0, and that the displayed examples would need positive rank; this is the same point as the first major comment, and a sentence in Section 2 would prevent misunderstanding.","section":"Section 2, Conjecture 2.12 definition"}],"recommendation":"major_revision","confidential_remarks":"The negative result for Conjecture 6 appears solid and is the main contribution. The Conjecture 4 claim is plausible but requires the rank information described above. The modified Conjecture 3.4 is data-driven and partly tautological; I would not require out-of-sample testing for an experimental paper, but the authors should state the vacuity explicitly. The paper is a good candidate for publication after these points are addressed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline: this paper has real content. It reports 886 failures out of 425,713 pairs for the multiplicative form of Mazur-Tate Conjecture 6 (Conjecture 0.1), with explicit examples (e.g., 680.c1, p=5) and cross-checks across three SageMath implementations. That part is convincing and new. It also proposes a modified conjecture (Conjecture 3.4) that survives the full database, which is a reasonable new target even if in-sample.\n\nWhat's good: the authors translate Conjecture 6 into a multiplicative form (Lemma 2.9) that makes computation transparent, they find no counterexamples to Conjecture 2.11 (MT Conjecture 5), and they are honest about SageMath's Tate-Shafarevich dependence on BSD. Code and data are on GitHub, which is exactly what a numerical paper should ship.\n\nThe soft spot is the claim about Conjecture 4. The paper finds 367 failures of Conjecture 2.12 and concludes that Conjecture 4 also fails. But Conjecture 2.12 is only stated under rk(E(Q)) > 0, and the paper never reports the ranks of those 367 pairs or of the displayed examples (377.a2, 832.f1, 4123.b1, 7826.b1). If any have rank 0, they are not admissible: Conjecture 2.4 with rank 0 says the vanishing order is at least 1, not at least 2. So the Conjecture 4 claim is currently unsupported. This is fixable by reporting ranks and restricting the analysis to positive-rank curves. Without that, the introduction overstates what the data show.\n\nA smaller caveat: the modified Conjecture 3.4 is validated on the same data that motivated it. The authors do flag this (Remark 3.7), so it is a minor issue, but the conjecture should be labeled as post-hoc.\n\nWho is this for: arithmetic geometers and computational number theorists working on BSD-type refinements. It deserves a serious referee. My recommendation: engage with it, but ask for the rank data before accepting the Conjecture 4 conclusion.","headline":"Solid numerical counterexamples to Mazur-Tate Conjecture 6, but the Conjecture 4 claim depends on missing rank data.","tokens_in":14107,"tokens_out":4209,"would_cite":true,"duration_ms":36205,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G40","11G05","11F67","11Y99"],"pacs":[],"model":"deepseek-v4-flash","headline":"Large-scale computation finds that two refined Birch–Swinnerton-Dyer–type conjectures of Mazur and Tate fail as stated, while a torsion-modified version holds in every tested case.","keywords":["refined BSD conjectures","Mazur-Tate element","modular symbols","p-adic periods","elliptic curves over Q","group algebra augmentation quotient","numerical verification"],"falsifier":"For a reported failing pair such as 680.c1 at $p=5$ or 4123.b1 at $p=7$, compute the modular symbols and the p-adic period with an independent algorithm or implementation; if the resulting values make the conjectural congruence hold, that counterexample is void. Conversely, one pair $(E,p)$ with $(\\#E(\\mathbb{Q})_{\\mathrm{Tor}})^{-1}$ in the minimal ring $R$ for which Conjecture 3.4 fails would disprove the modified conjecture.","tokens_in":13123,"feed_emoji":"🧮","tokens_out":10982,"duration_ms":89750,"temperature":0.7,"pith_summary":"This paper reports a systematic numerical test of three refined Birch–Swinnerton-Dyer–type conjectures proposed by Mazur and Tate in 1987. On a database of 425,713 pairs $(E,p)$ consisting of a rational elliptic curve $E$ and a prime $p$ of split multiplicative reduction, the multiplicative form of Mazur–Tate Conjecture 6 (the paper's Conjecture 0.1) failed in 886 cases, and a related positive-rank statement (Conjecture 2.12) failed in 367 cases. The failures occur in the $\\ell$-Sylow subgroups of $G_p$ for primes $\\ell$ dividing the torsion order of $E(\\mathbb{Q})$. The paper then shows that adjoining $(\\#E(\\mathbb{Q})_{\\mathrm{Tor}})^{-1}$ to the coefficient ring $R$ yields a modified conjecture (Conjecture 3.4) that holds for every tested pair, and proposes analogous modifications of the original Conjectures 4 and 6. If correct, the original Mazur–Tate statements are false as stated, and the torsion-inverse hypothesis is an essential part of the refined BSD prediction.","feed_headline":"Mazur-Tate conjectures fail in 886 of 425,713 tested pairs","feed_subtitle":"Adding the inverse torsion order to the coefficient ring fixes the conjecture in all 425,713 tested cases","key_machinery":"The central object is the Mazur–Tate element $$\\theta_{E,M}=\\sum_{a\\in G_M}\\$\\lambda$(a,M)\\,[a]\\in R[G_M],$$ where $G_M=(\\mathbb{Z}/M\\mathbb{Z})^{*}/\\langle -1\\rangle$ and $\\lambda(a,M)$ is the plus modular symbol normalized by the real period. The conjectures assert that this element has prescribed vanishing order in the augmentation ideal and that its image in the augmentation quotient $Q_r(R,G_M)$ equals a 'corrected discriminant' built from the $p$-adic periods $\\tilde q_p$. The computational lever is the isomorphism $Q_1(R,G_p)\\cong\\bigoplus_{\\ell\\in S}\\mathrm{Syl}_\\ell(G_p)$ of Corollary 1.5, which rewrites the ideal-theoretic equality as one ordinary congruence per prime $\\ell$ dividing $\\#G_p$ with $\\ell^{-1}\\notin R$. Lemma 2.9 uses exactly this isomorphism to prove that, for split multiplicative $p$ and layer $M=p$, the multiplicative Conjecture 0.1 is equivalent to the original Conjecture 6 of [MT87].","core_discovery":"The paper's central claim is that Conjecture 0.1, shown in Lemma 2.9 to be equivalent to Conjecture 6 of [MT87] when the layer is a prime $p$ and $E$ has split multiplicative reduction at $p$, is not true in general. Across 425,713 computed pairs, the predicted congruence failed for 886 pairs; Conjecture 2.12, a consequence of Conjecture 4 in the positive-rank case, failed for 367 pairs. The same data showed no failure for Conjecture 2.11, which already includes the torsion inverse in its coefficient ring. When the minimal ring $R$ is enlarged to include $(\\#E(\\mathbb{Q})_{\\mathrm{Tor}})^{-1}$, the modified Conjecture 3.4 holds for all tested pairs, and the paper proposes Conjectures 3.5 and 3.6 as corrected versions of the original Conjectures 6 and 4.","pith_inferences":["Because the modified conjecture was validated on the same data that suggested the modification, its positive evidence is in-sample; testing Conjecture 3.4 on curves of conductor above 90,000 or on curves outside the standard tables would provide the out-of-sample check the paper does not include.","The failure pattern suggests that any future statement of refined BSD conjectures should build $(\\#E(\\mathbb{Q})_{\\mathrm{Tor}})^{-1}$ into the coefficient ring at the outset, rather than add it as a separate hypothesis.","A natural next calculation is to test whether the failure of Conjecture 4 shows up in higher augmentation quotients, such as $Q_2(R,G_p)$, where the rank contribution to the vanishing order should live; the current computations only probe the first quotient.","The verification of Conjecture 2.11 inherits the paper's caveat that the Tate–Shafarevich order used by the software may itself assume BSD, so a rigorous check of that conjecture would need an independently computed Sha."],"forward_implications":["The original Mazur–Tate Conjectures 6 and 4 are false as stated with the minimal coefficient ring; the failure is detected already in the first augmentation quotient.","The modified conjecture (Conjecture 3.4) passed all 425,713 tested pairs, so the torsion-inverse condition is a concrete correction to the refined BSD formulation.","The failure of the original conjectures is localized to the $\\ell$-primary part of $G_p$ for primes $\\ell$ dividing $\\#E(\\mathbb{Q})_{\\mathrm{Tor}}$, so future tests should focus on exactly those Sylow subgroups.","Conjecture 2.11, which already included the torsion inverse, survived the whole database, so the torsion-bearing version of Conjecture 5 remains numerically intact.","The numerical cross-checks found no discrepancies among the three implementations used for the failing pairs, so the paper does not attribute the reported failures to a software inconsistency."],"supporting_citations":[{"why":"states the Mazur–Tate element and Conjectures 4–6 that the paper tests and modifies.","marker":"[MT87]"},{"why":"establishes the p-adic period $q_p$ for an elliptic curve with split multiplicative reduction.","marker":"[Tat74]"},{"why":"one of the two sources for the Manin–Drinfeld theorem that the modular symbols $\\lambda(a,p)$ are rational.","marker":"[Man72]"},{"why":"the other source for rationality of the modular symbols used in the definition of the Mazur–Tate element.","marker":"[Dri73]"},{"why":"completes the Modularity Theorem that attaches the newform $f_E$ to every rational elliptic curve $E$, the starting point for the modular symbols.","marker":"[BCDT01]"},{"why":"supplies the list of elliptic curves up to conductor 90,134 from which the database of pairs $(E,p)$ was constructed.","marker":"[Cre22]"},{"why":"one of the three implementations of modular symbols used for the database and for cross-checking failing pairs.","marker":"[Cre97]"},{"why":"a second independent implementation used to recompute modular symbols for failing pairs.","marker":"[Ste07]"},{"why":"a third implementation used to cross-check the modular symbol values for the failing pairs.","marker":"[Wut18]"}],"fun_headline_variants":["Mazur-Tate conjectures fail in 886 pairs, but tweak fixes all","886 counterexamples to Mazur-Tate; adding torsion order fixes all","Numerical study shows BSD conjectures fail, then a tweak succeeds","Mazur-Tate conjecture has 886 failures; a torsion fix makes it hold","Torsion-order tweak repairs the Mazur-Tate conjecture after 886 fails"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The central negative claim rests on the correctness of the computed modular symbol and p-adic period values and on the database being representative enough to support a statement 'in general'; neither is independently proved.","fun_headline_variants_meta":{"raw":{"variants":["Mazur-Tate conjectures fail in 886 pairs, but tweak fixes all","886 counterexamples to Mazur-Tate; adding torsion order fixes all","Numerical study shows BSD conjectures fail, then a tweak succeeds","Mazur-Tate conjecture has 886 failures; a torsion fix makes it hold","Torsion-order tweak repairs the Mazur-Tate conjecture after 886 fails"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000464,"raw_usage":{"total_tokens":2253,"prompt_tokens":814,"completion_tokens":1439,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":430,"completion_tokens_details":{"reasoning_tokens":1333}},"tokens_in":430,"tokens_out":1439,"duration_ms":10186,"temperature":1.0,"reasoning_tokens":1333,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T05:15:27.707910+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a reported failing pair such as 680.c1 at $p=5$ or 4123.b1 at $p=7$, compute the modular symbols and the p-adic period with an independent algorithm or implementation; if the resulting values make the conjectural congruence hold, that counterexample is void. Conversely, one pair $(E,p)$ with $(\\#E(\\mathbb{Q})_{\\mathrm{Tor}})^{-1}$ in the minimal ring $R$ for which Conjecture 3.4 fails would disprove the modified conjecture.","supporting_citations":[],"review_version":1}