REVIEW 3 major objections 4 minor 22 references
Numerical study of refined conjectures of the BSD type
T0 review · 3 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash
Pith's one-line read 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.
desk verdict Solid numerical counterexamples to Mazur-Tate Conjecture 6, but the Conjecture 4 claim depends on missing rank data. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
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].
What would settle it
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.
Extended reading notes
Core claim
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.
Load-bearing premise
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.
Editorial extensions
If this is right
- 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.
Reading between the lines
- 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.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
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.
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 (3)
- [Section 3, Conjecture 2.12 and displayed examples] 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 3, Conjecture 3.4 and Remark 2.13] 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 3, bullet list and Remark 3.1] 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.
minor comments (4)
- [Section 4, equation (9)] 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.
- [Abstract, Section 3, Conjecture 2.11] 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.
- [Footnote 4] The footnote contains a typo: 'mearly' should be 'merely'.
- [Section 2, Conjecture 2.12 definition] 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.
Circularity Check
Central counterexample claims are independent numerical tests; the modified Conjecture 3.4 is partially self-confirming because adding (#E(Q)_Tor)^{-1} to R removes the failing primes from the tested set S by definition.
-
self definitional
[Section 3, paragraph beginning 'Now, we reverify Conjecture 0.1...' and Conjecture 3.4 statement]
"Now, we reverify Conjecture 0.1 and Conjecture 2.12 after adding the hypothesis that (# E(Q)Tor)−1 ∈ R (where E(Q)Tor denotes the torsion subgroup of E(Q)) ... However, with this additional hypothesis, we observed that all the conjectures did hold for all the pairs (E, p) in the database, even for the pairs (E, p) that previously did not satisfy Conjecture 0.1 or Conjecture 2.12. This can be thought as the failure of Conjecture 0.1 and Conjecture 2.12 lies in the ℓ-Sylow subgroups of Gp for the primes ℓ | #E(Q)Tor ."
The apparent confirmation of Conjecture 3.4 for the previously failing pairs is built into the definition of S and R. By Conjecture 3.4, S is defined by 'ℓ ∈ S if and only if ℓ | #Gp and ℓ−1 /∈ R', and R now contains (#E(Q)Tor)^{-1}. Hence any prime ℓ dividing #E(Q)Tor satisfies ℓ^{-1} ∈ R, so ℓ is removed from S. The observed failures of Conjecture 0.1 and Conjecture 2.12 occurred precisely at primes ℓ | #E(Q)Tor. Therefore the previously failing pairs satisfy Conjecture 3.4 vacuously on the relevant Sylow component; the statement that they 'did hold' after adding the torsion-inverse hypothesis is a consequence of the definition of S and R rather than an independent numerical verification.
full rationale
The paper's main results are numerical counterexamples to Mazur-Tate Conjectures 4 and 6. These are obtained by computing modular symbols and p-adic periods independently and checking the stated equalities; no parameter is fitted to make the equalities hold, and the failure claims are not derived from the conjectures themselves. The equivalence in Lemma 2.9 is a mathematical derivation, not circularity. The reliance on SageMath/LMFDB data, including BSD-assumed Tate-Shafarevich orders, is an acknowledged reliability caveat (Remark 3.8), not a circular step. The rank issue noted by a skeptic is a correctness/evidence gap in the Conjecture 2.12-to-Conjecture 4 implication, but it is not circular reasoning. The only structurally circular element is the validation of the modified Conjecture 3.4: the added hypothesis (#E(Q)_Tor)^{-1} ∈ R removes from S exactly the primes that caused the original failures, so those pairs satisfy the modified conjecture by definition rather than by new evidence. This partial self-confirmation warrants a moderate score, but the central counterexample claims remain independent and self-contained.
Assumptions & free parameters
assumptions (7)
- standard math E is modular, so the newform f_E and modular symbols exist (Modularity Theorem).
- standard math Manin-Drinfeld theorem: modular symbols are rational.
- standard math Tate's p-adic uniformization gives a unique q_p for split multiplicative reduction.
- standard math The number of components C_p equals ord_p(q_p) (Silverman).
- standard math Q1(R,G) is isomorphic to the direct sum of the ℓ-Sylow subgroups of G for ℓ that are not invertible in R.
- standard math Bergunde-Gehrmann theorem on vanishing of the sum of modular symbols.
- domain assumption SageMath 10.1/10.4 and LMFDB data are correct for modular symbols, p-adic periods, and Tate-Shafarevich orders.
Cite this review
Pith. "Pith review of Numerical study of refined conjectures of the BSD type." pith.science (2026). https://pith.science/paper/HAPMUDPU
@misc{pith2026241217703,
author = {Pith},
title = {Pith review of: Numerical study of refined conjectures of the BSD type},
year = {2026},
howpublished = {\url{https://pith.science/paper/HAPMUDPU}},
note = {Machine review of arXiv:2412.17703}
}
abstract
In 1987, Mazur and Tate stated conjectures which, in some cases, resemble the classical Birch-Swinnerton-Dyer conjecture and its $p$-adic analog. We study experimentally three conjectures stated by Mazur and Tate using SageMath. Our findings indicate discrepancies in some of the original statements of some of the conjectures presented by Mazur and Tate. However, a slight modification on the statement of these conjectures does appear to hold.
Reference graph
Works this paper leans on
-
[1]
On the modularity of elliptic curves over Q : wild 3-adic exercises
Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor. On the modularity of elliptic curves over Q : wild 3-adic exercises. J. Amer. Math. Soc. , 14(4):843--939, 2001
2001
-
[2]
On the order of vanishing of S tickelberger elements of H ilbert modular forms
Felix Bergunde and Lennart Gehrmann. On the order of vanishing of S tickelberger elements of H ilbert modular forms. Proc. Lond. Math. Soc. (3) , 114(1):103--132, 2017
work page 2017
-
[3]
B. J. Birch and H. P. F. Swinnerton-Dyer. Notes on elliptic curves. II . J. Reine Angew. Math. , 218:79--108, 1965
work page 1965
-
[4]
J. E. Cremona. Algorithms for modular elliptic curves . Cambridge University Press, Cambridge, second edition, 1997
work page 1997
-
[5]
Johncremona/ecdata: 2022-10-13, October 2022
John Cremona. Johncremona/ecdata: 2022-10-13, October 2022
work page 2022
-
[6]
A basis for augmentation quotients of finite abelian groups
Shan Chang and Guoping Tang. A basis for augmentation quotients of finite abelian groups. Journal of Algebra , 327(1):466--488, 2011
work page 2011
-
[7]
V. G. Drinfel'd. Two theorems on modular curves. Funkcional. Anal. i Prilov z en. , 7(2):83--84, 1973
work page 1973
-
[8]
p -adic periods and modular symbols of elliptic curves of prime conductor
Ehud de Shalit. p -adic periods and modular symbols of elliptic curves of prime conductor. Invent. Math. , 121(2):225--255, 1995
work page 1995
Show all 22 references
-
[9]
D. L. Johnson. Presentations of groups , volume 15 of London Mathematical Society Student Texts . Cambridge University Press, Cambridge, second edition, 1997
1997
-
[10]
Ju. I. Manin. Parabolic points and zeta functions of modular curves. Izv. Akad. Nauk SSSR Ser. Mat. , 36:19--66, 1972
1972
-
[11]
Arithmetic conjectures suggested by the statistical behavior of modular symbols
Barry Mazur and Karl Rubin. Arithmetic conjectures suggested by the statistical behavior of modular symbols. Exp. Math. , 32(4):657--672, 2023
2023
-
[12]
Mazur and J
B. Mazur and J. Tate. Refined conjectures of the `` B irch and S winnerton- D yer type''. Duke Math. J. , 54(2):711--750, 1987
1987
-
[13]
Mazur, J
B. Mazur, J. Tate, and J. Teitelbaum. On p -adic analogues of the conjectures of B irch and S winnerton- D yer. Invent. Math. , 84(1):1--48, 1986
1986
-
[14]
Computations on an equation of the B irch and S winnerton- D yer type
Francisco Xavier Portillo-Bobadilla. Computations on an equation of the B irch and S winnerton- D yer type . ProQuest LLC, Ann Arbor, MI, 2004. Thesis (Ph.D.)--The University of Texas at Austin
2004
-
[15]
Portillo-Bobadilla
Francisco X. Portillo-Bobadilla. Experimental evidence on a refined conjecture of the BSD type. Bol. Soc. Mat. Mex. (3) , 25(3):529--541, 2019
2019
-
[16]
Silverman
Joseph H. Silverman. Advanced Topics in the Arithmetic of Elliptic Curves . Graduate Studies in Mathematics, 1994
1994
-
[17]
Silverman
Joseph H. Silverman. The arithmetic of elliptic curves , volume 106 of Graduate Texts in Mathematics . Springer, Dordrecht, second edition, 2009
2009
-
[18]
W.A. Stein. Modular Forms, a Computational Approach . Graduate studies in mathematics. American Mathematical Society, 2007
2007
-
[19]
John T. Tate. The arithmetic of elliptic curves. Invent. Math. , 23:179--206, 1974
1974
-
[20]
Ring-theoretic properties of certain H ecke algebras
Richard Taylor and Andrew Wiles. Ring-theoretic properties of certain H ecke algebras. Ann. of Math. (2) , 141(3):553--572, 1995
1995
-
[21]
Modular elliptic curves and F ermat's last theorem
Andrew Wiles. Modular elliptic curves and F ermat's last theorem. Ann. of Math. (2) , 141(3):443--551, 1995
1995
-
[22]
Numerical modular symbols for elliptic curves
Christian Wuthrich. Numerical modular symbols for elliptic curves. Math. Comp. , 87(313):2393--2423, 2018
2018
Reviewed August 11, 2026 · model on record in the stance chip above.
Discussion (0). Continue with ORCID to comment.