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REVIEW 4 major objections 5 minor 26 references

Unveiling Arithmetic Statistics of Congruent Number Elliptic Curves via Data Science and Machine Learning

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A 3-million-integer census of congruent number curves tests Goldfeld's 50/50 split against data—and finds the asymptotic regime has not arrived.

desk verdict Useful 2-Selmer dataset and an honest body, but the abstract's Goldfeld 'verification' is contradicted by the paper's own data and needs to be fixed. read the letter →

arxiv 2509.03129 v1 pith:OAZ7J3GN submitted 2025-09-03 math.NT

Priyavrat Deshpande , Aditya Karnataki , Pratiksha Shingavekar This is my paper
classification math.NT MSC 11G0511Y3568T07
keywords congruentnumbersellipticcurves2-SelmergroupsGoldfeld'sconjectureHeath-BrownheuristicsarithmeticstatisticsmachinelearningFrobeniustraces
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper compiles the arithmetic statistics of the congruent-number elliptic curves E_D: y^2 = x^3 - D^2x for every square-free D up to 3 million, computing 2-Selmer ranks for all of them and Mordell-Weil, analytic, and 3-Selmer ranks, Frobenius traces, and BSD parameters for large subranges. Its aim is to confront the leading conjectures in the area—Heath-Brown's heuristics for 2-Selmer ranks, Poonen-Rains and Delaunay distributions, Smith's positive-density and BSD theorems, and Goldfeld's 50/50 analytic-rank split—with data large enough to see which parts of the theory hold and which parts show finite-sample drift. The headline claim is a verification of Goldfeld's conjecture, but the reported numbers are a split of roughly 44% rank 0 to 56% rank 1 (with some rank ≥2 curves) and p-values of 0, which the authors themselves read as "still in the finite-sample regime." Alongside this, the paper proves a small theorem: for a fixed prime, the average Frobenius trace over this family tends to 0 because it is an average of Legendre symbols, and it conjectures the same for any twist family. Finally, machine-learning models classify congruent versus non-congruent numbers almost perfectly from Selmer or BSD parameters but no better than chance from the first 1,000 Frobenius traces.

What carries the argument

The load-bearing object is the family E_D: y^2 = x^3 - D^2x, quadratic twists of E_1: y^2 = x^3 - x, and its 2-Selmer rank s(D), defined as the F_2-dimension of the 2-Selmer group minus 2 to strip off the constant rational 2-torsion. Because the Mordell-Weil rank r(D) is hard to compute directly, s(D) serves as a computable upper bound and, through the parity conjecture, as a parity oracle: s(D) is even exactly for D ≡ 1,2,3 (mod 8) and odd for D ≡ 5,6,7 (mod 8). The dataset construction—2-Selmer ranks for all D up to 3 million, descent bounds with parity and analytic-rank fallbacks for Mordell-Weil ranks beyond 1 million, analytic ranks cross-checked across algorithms, and GRH-conditional 3

What would settle it

Take a random sample of 1,000 square-free D < 3,000,000 that the external list declares non-congruent, compute their 2-Selmer ranks and, where possible, unconditional descent bounds on the Mordell-Weil rank without consulting the list; if any has positive rank, or if the rank proportions change materially, the dataset's labels—and with them the Heath-Brown, Goldfeld, and ML conclusions—are corrupted. Separately, recompute the analytic ranks of the 98,400 curves using code that does not share the same rank routines; if the 44/56 split moves toward 50/50, the claimed verification is an algorithm

Watch

Extended reading notes

Core claim

On its own terms, the central discovery is that the 2-Selmer ranks of the congruent-number family are now computed for all 1,823,773 square-free D ≤ 3,000,000, and these unconditional data reproduce the mod-8 parity dichotomy: s(D) is even for D ≡ 1,2,3 (mod 8) and odd for D ≡ 5,6,7 (mod 8). The empirical distribution is concentrated at s = 0 and s = 2 in the even classes and at s = 1 and s = 3 in the odd classes, with Heath-Brown's moment constants 3, 15, 35 matched for D ≡ 5,7 (mod 8) but not for D ≡ 1,3 (mod 8). For analytic ranks, the paper does not find Goldfeld's limit: among 98,400 curves, 43,529 have rank 0, 48,917 have rank 1, and 5,954 have rank ≥ 2, and the authors call this the f

Load-bearing premise

The entire rank and congruence label set rests on trusting an externally maintained list of which numbers up to 10 million are congruent, and on assuming the standard conjectures that equate 2-Selmer bounds with true ranks; if either is wrong for even a small fraction of numbers below 3 million, every statistic in the paper inherits that error.

Editorial extensions

If this is right

  • If Heath-Brown's moment predictions are correct, the residue-class-dependent deviations for D ≡ 1,3 (mod 8) must eventually decay; the paper's normalized-error plots make the rate of that decay a quantitative target for improved error terms.
  • If Goldfeld's conjecture is true, the 44/56 finite-sample split fixes a lower bound on how many twists are needed before asymptotic behavior appears; extending the analytic-rank computation beyond D ≈ 10^5 is a direct way to test convergence.
  • Because average Frobenius traces vanish for every fixed prime, the congruent-number family shows no aggregate murmuration; any signal would have to come from finer stratification, such as by rank or residue class, where the paper's smoothed plots hint at structure.
  • If the BSD-parameter machine-learning result generalizes, quantities like the regulator and special value encode congruent status so directly that near-perfect classification is expected; that makes feature-attribution analysis a natural next step.
  • The 3-Selmer parity and average-size findings suggest a Heath-Brown-style limit theorem for 3-Selmer ranks in this family is plausible; a larger 3-Selmer dataset would make the comparison quantitative.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper's own data support reading the Goldfeld section as a finite-sample diagnostic rather than a completed verification; the constructive next step is to model the observed rank proportions as a function of X and estimate the exponent at which they approach 1/2.
  • The pipeline's reliance on an externally maintained congruent-number list for D below 3 million is the single most fragile link; an independent sample-based check of that list would harden every downstream statistic, including the machine-learning labels.
  • The near-100% ML accuracy on BSD parameters likely measures how directly regulator and special value encode rank, not a new number-theoretic signal; a fairer test would withhold all quantities that are conjecturally determined by the L-function and train only on congruence-visible features such as D mod 8, prime factors, and Legendre symbols.
  • The absence of murmurations in the aggregate Frobenius averages is consistent with the theorem, but the residue-class smoothed plots show oscillations with p-dependent amplitude; averaging (D/p)a_p(E_1) separately by Mordell-Weil or Selmer rank may reveal structure that the overall mean cancels.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper compiles a large dataset for the congruent-number elliptic curves E_D: y^2 = x^3 - D^2x with square-free D: 2-Selmer ranks for D up to 3,000,000, Mordell-Weil ranks for about 1.73 million numbers, analytic ranks for 98,400 curves, 3-Selmer ranks for roughly 59,000-80,000 curves, BSD invariants, and Frobenius traces. It compares the observed distributions with the theorems of Heath-Brown, with Poonen-Rains and Delaunay heuristics, with Goldfeld's conjecture, and with Smith's results on congruent numbers and BSD. It also trains ML models to classify congruent numbers and proposes a new theorem and conjecture about averaged Frobenius traces in twist families.

Significance. The assembled dataset is a potentially useful resource: the 2-Selmer computations are unconditional, and the scripts and data are publicly available on GitHub/Zenodo. Several empirical comparisons, especially the residue-class dependence of 2-Selmer statistics and 3-Selmer moments, are suggestive and could inform future work. However, the headline claim of a rigorous verification of Goldfeld's conjecture is not supported by the paper's own Table 6, the MW-rank dataset depends on conditional assumptions and an unverified external list, and the proof of the new Theorem 5.3 has a gap. The paper is best viewed as an exploratory data-science contribution; its central assertions need substantial revision.

major comments (4)
  1. [Abstract and §4.1, Table 6] The abstract's claim of a 'rigorous verification' of Goldfeld's 50/50 split is contradicted by the paper's own data. Table 6 gives rank-0/rank-1 counts 4280/5720 at N=10000 through 21932/28068 at N=50000 with p-values 0; §4.1 reports totals 43,529 vs 48,917 among 98,400 curves, which is 47.1% vs 52.9% when restricted to ranks 0 and 1. The text itself says 'we are still in the finite-sample regime'. No convergence rate, error estimate, or statistical argument is supplied to justify the word 'rigorous'; the Bernoulli sampling in §7.2 concerns MW ranks and centers near 0.49, not 0.50. The abstract and Section 4 should be reworded to describe a finite-sample test that currently does not observe the predicted split.
  2. [§2.3] The MW-rank dataset, which underlies the Goldfeld/MW statistics and the ML labels, is built on several assumptions. For D absent from an externally maintained congruent-number list up to 10^7, the MW rank is set to 0; for 2-Selmer rank 1, the MW rank is set to 1 conditional on BSD; unresolved ranks are stored as -1. If that external list is incomplete for D < 3*10^6, or if the parity/BSD assumptions fail for individual curves, Table 8, Figure 28, and the ML labels inherit systematic errors. The paper should quantify how many rows have unresolved rank -1, state which rows are used in each statistic, and independently spot-check the external list.
  3. [§5, Theorem 5.3] The proof of Theorem 5.3 is incomplete. Lemma 5.2 controls averages over consecutive integers, while f_X(n) averages over square-free D; the proof passes to the sum over D with the factor mu(D)^2 and asserts that (1/X)*sum_{D<=X} chi(D) mu(D)^2 tends to 0 without invoking a square-free sieve or an equivalent argument. The normalization also changes from 1/#SF(X) to 1/X. The theorem may be true, but the proof as written does not establish it.
  4. [§4.2, Theorem 4.4 verification] The verification of Smith's Theorem 4.4 appears to use the wrong Selmer condition. Under the paper's convention #S2(D) = 2^{2+s(D)}, the assertion that Sel2 is generated by the rational 2-torsion subgroup is equivalent to s(D)=0 (dimension 2), not to '2-Selmer rank is 2'. The subsequent count of curves with even Selmer rank that are of rank 2 (48,305 of 97,893) therefore does not test the condition in Theorem 4.4 as stated. The authors should clarify the terminology and re-run the check with s(D)=0.
minor comments (5)
  1. [§3.1, after Theorem 3.4] The empirical constants '3, 15, 35' do not match the displayed product formula: if the product is (1+2^j), the third value is 135; if it is (1+2j), the third value is 105. Correct the formula or the numerical values.
  2. [§4.1] The paper does not explain how the 98,400 curves with analytic ranks were selected from the larger dataset, nor which subset had all four rank algorithms agree. Add a precise description of this subset and the selection criteria.
  3. [§6.2] The 100% accuracy obtained using regulator, special value, real period, and Tamagawa product is largely a restatement of BSD, since these invariants encode whether L(E,1)=0. This should be presented as a consistency check rather than as a new ML discovery.
  4. [Conjecture 5.4] The phrase 'the normalized trace a_p(E) is seen as a discrete random variable on {-1,0,+1}' is undefined; specify the normalization and the probability space.
  5. [Table 6] The p-values are reported as '0'; actual values should be given. The subsamples are nested prefixes, so the chi-square and Fisher tests are not independent across rows; this should be noted.

Circularity Check

1 steps flagged · score 2.0 of 10

One secondary self-definitional step in MW-rank dataset construction; core external benchmarks are independent.

  1. self definitional [Section 2.3 (Generation of the datasets) and Section 7.2 (The behavior of the MW rank)]
    "To speed up the calculation Magma recommends assuming GRH. Hence, we also consult 2-Selmer rank: if it is 0 then the MW rank is also zero. On the other hand, if it is 1 then the MW rank is also 1; this is conditional on BSD which implies that the dimension of Sha 2-torsion is even. We also consult the list of congruent numbers up to 10^7 ...; the numbers that do not appear in that list are not congruent and the MW rank of the corresponding curves is set to 0."

    The MW-rank labels for curves not resolved by RankBounds are assigned by the quoted rules: s(D)=0 -> MW 0; s(D)=1 -> MW 1 (conditional on BSD); absent from the external congruent-number list -> MW 0. Section 7.2 then reports as empirical findings that 'the parities of the Selmer rank and the MW rank match' and that residue classes 5,6,7 are essentially all rank 1; Section 4.2 reports 'our database shows that 100% of these numbers are congruent'. These observations are restatements of the dataset-construction rules, not independent checks. This is peripheral: the central Selmer-rank, analytic-rank, Frobenius-trace, and ML comparisons use externally computed data and held-out tests, so the overall derivation is not circular.

full rationale

The paper's principal comparisons are to external theorems and conjectures (Heath-Brown, Poonen-Rains, Delaunay, Goldfeld, Smith). The 2-Selmer ranks are computed unconditionally with Sage's selmer_rank(); analytic ranks are computed by multiple independent algorithms; ML experiments use disjoint train/test splits; the Frobenius-average theorem is proved from the definition of the Legendre symbol. No load-bearing self-citations appear (the reference list contains no author self-citations). The abstract's phrase 'rigorous verification' of Goldfeld is contradicted by the paper's own Table 6 (p-values 0 and ~44/50 split) and Section 4.1's admission of the finite-sample regime, but that is a correctness/consistency problem, not circularity. The only circular element is secondary: unresolved MW ranks were filled using Selmer-rank parity, BSD, and the external congruent-number list, and later sections report parity-matching and 100% congruent rates among 5,6,7 (mod 8) as empirical observations, which are restatements of those same filling rules.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new objects; its Conjecture 5.4 is an empirical conjecture, not an entity. The main assumptions are the standard conjectures (BSD, parity, GRH) and the completeness of an external dataset, all flagged in Section 2.3.

assumptions (5)
  • domain assumption Birch and Swinnerton-Dyer conjecture (rank equality and parity of Sha) is assumed when using MW rank to infer analytic rank and when setting MW rank from 2-Selmer rank 1.
    Invoked in Section 2.3 ('if it is 1 then the MW rank is also 1; this is conditional on BSD') and Section 4.1 (MW rank used as proxy for analytic rank).
  • domain assumption Parity conjecture (root number determines parity of rank) is assumed to resolve MW ranks when RankBounds do not match.
    Section 2.3: 'if the bounds do not match, then we compute the root number and use the parity conjecture to decide the rank.'
  • domain assumption Generalized Riemann Hypothesis is assumed in Magma's RankBounds and analytic_rank_upper_bound and in 3-Selmer computations via SetClassGroupBounds('GRH').
    Section 2.3: 'Magma recommends assuming GRH' and '3-Selmer ranks are conditional on GRH.'
  • domain assumption Externally maintained list of congruent numbers up to 10^7 is complete for D < 3×10^6, so absence from the list implies non-congruence.
    Section 2.3: 'the numbers that do not appear in that list are not congruent and the MW rank ... is set to 0.' This is an unverified completeness assumption.
  • standard math Standard properties of elliptic curves (torsion structure Z/2 x Z/2, Frobenius trace formulas, root number by Birch-Stephens) are used.
    Used throughout Section 2.2 without proof.

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Pith. "Pith review of Unveiling Arithmetic Statistics of Congruent Number Elliptic Curves via Data Science and Machine Learning." pith.science (2026). https://pith.science/paper/OAZ7J3GN

@misc{pith2026250903129,
  author       = {Pith},
  title        = {Pith review of: Unveiling Arithmetic Statistics of Congruent Number Elliptic Curves via Data Science and Machine Learning},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/OAZ7J3GN}},
  note         = {Machine review of arXiv:2509.03129}
}
abstract

This article presents a comprehensive data-scientific investigation into the arithmetic statistics of congruent number elliptic curves, leveraging a dataset of square-free integers up to $3$ million. We analyze the Mordell-Weil ranks, 2-Selmer ranks, and 3-Selmer ranks of the corresponding elliptic curves $E_D: y^2 = x^3 - D^2x$, where $D$ is a square-free number. Our study empirically examines the Heath-Brown heuristics, which predict the distribution of $2$-Selmer ranks as well as congruent numbers based on their residue modulo $8$. In particular, offering statistical insights into the proportion of numbers whose associated elliptic curves have positive rank. We provide a rigorous verification of Goldfeld's Conjecture in this context, analyzing the distribution of analytic ranks and demonstrating their alignment with the conjectured $50/50$ split for ranks $0$ and $1$. Furthermore, we explore the conjectural asymptotic distribution of $2-$ and $3$-torsion part of the Tate-Shafarevich group of these curves. Based on empirical evidence, we also suggest potential statistical distribution of $3$-Selmer and Mordell-Weil ranks. We also examine the averages of Frobenius traces and observe that they tend to zero without exhibiting any murmuration-like patterns. In addition to these number-theoretic analyses, we apply machine learning techniques to classify and predict congruent numbers, exploring the efficacy of computational methods in distinguishing congruent from non-congruent numbers based on the arithmetic properties of elliptic curves. This interdisciplinary approach blends advanced number theory with modern data science, providing empirical support for conjectures as well as discovery of new patterns.

Figures

Figures reproduced from arXiv: 2509.03129 by the authors.

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Figure 1. FIGURE 1 [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
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Figure 2. FIGURE 2 [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
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Figure 3. FIGURE 3 [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (29 more)
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Figure 4. Figure 4: FIGURE 4 [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
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Figure 5. Figure 5: FIGURE 5 [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]
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Figure 6. Figure 6: FIGURE 6 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
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Figure 7. Figure 7: FIGURE 7 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
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Figure 8. Figure 8: FIGURE 8 [PITH_FULL_IMAGE:figures/full_fig_p013_8.png]
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Figure 9. Figure 9: FIGURE 9 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]
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Figure 10. Figure 10: shows that when one studies the family of congruent number curves, the moments of their 2-Selmer sizes are fairly close to those of all elliptic curves over Q. FIGURE 10. Moments of 2-Selmer size for all congruent number elliptic curves 3.3. Delaunay Heuristics. We re…
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Figure 11. Figure 11: FIGURE 11 [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
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Figure 12. Figure 12: FIGURE 12 [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
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Figure 13. Figure 13: FIGURE 13 [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
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Figure 14. Figure 14: FIGURE 14 [PITH_FULL_IMAGE:figures/full_fig_p017_14.png]
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Figure 15. Figure 15: FIGURE 15 [PITH_FULL_IMAGE:figures/full_fig_p019_15.png]
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Figure 16. Figure 16: FIGURE 16 [PITH_FULL_IMAGE:figures/full_fig_p019_16.png]
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Figure 17. Figure 17: FIGURE 17 [PITH_FULL_IMAGE:figures/full_fig_p020_17.png]
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Figure 18. Figure 18: FIGURE 18 [PITH_FULL_IMAGE:figures/full_fig_p022_18.png]
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Figure 19. Figure 19: FIGURE 19 [PITH_FULL_IMAGE:figures/full_fig_p023_19.png]
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Figure 20. Figure 20: FIGURE 20 [PITH_FULL_IMAGE:figures/full_fig_p023_20.png]
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Figure 21. Figure 21: FIGURE 21 [PITH_FULL_IMAGE:figures/full_fig_p024_21.png]
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Figure 22. Figure 22: FIGURE 22 [PITH_FULL_IMAGE:figures/full_fig_p025_22.png]
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Figure 23. Figure 23: FIGURE 23 [PITH_FULL_IMAGE:figures/full_fig_p027_23.png]
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Figure 24. Figure 24: FIGURE 24 [PITH_FULL_IMAGE:figures/full_fig_p028_24.png]
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Figure 25. Figure 25: FIGURE 25 [PITH_FULL_IMAGE:figures/full_fig_p029_25.png]
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Figure 26. Figure 26: From the third moment onward empirical values are much less; probably indicating that the data is too less. Recall that the conjectured value the kth moment of size of 3-Selmer groups is Qk i=1(1 + 3i ). FIGURE 26. Comparison of moments of 3-Selmer sizes Empirical evi…
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Figure 27. Figure 27: FIGURE 27 [PITH_FULL_IMAGE:figures/full_fig_p030_27.png]
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Figure 28. Figure 28: FIGURE 28 [PITH_FULL_IMAGE:figures/full_fig_p031_28.png]
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Figure 29. Figure 29: FIGURE 29 [PITH_FULL_IMAGE:figures/full_fig_p031_29.png]
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Figure 30. Figure 30: FIGURE 30 [PITH_FULL_IMAGE:figures/full_fig_p032_30.png]
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Figure 31. Figure 31: FIGURE 31 [PITH_FULL_IMAGE:figures/full_fig_p033_31.png]
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Figure 32. Figure 32: FIGURE 32 [PITH_FULL_IMAGE:figures/full_fig_p033_32.png]

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