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REVIEW 2 major objections 6 minor 17 references

Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums

T0 review · 2 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read This paper claims that feeding a neural network multiple Mestre–Nagao sums, S0 and S5 at several bounds, improves elliptic-curve rank classification, and that learned prime weights add a smaller gain.

desk verdict The multi-value feature set genuinely improves elliptic curve rank classification; the learned-sum claim is a single-run near-tie that needs replication. read the letter →

arxiv 2506.07967 v1 pith:QLETOOCV submitted 2025-06-09 math.NT

Zvonimir Bujanović , Matija Kazalicki , Domagoj Vlah This is my paper
classification math.NT MSC 11G0568T07
keywords ellipticcurvesMordell-WeilrankMestre-NagaosumsclassificationneuralnetworksmachinelearningFrobeniustracesmurmurations
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

The paper tries to establish that the classical Mestre–Nagao heuristic for guessing the rank of an elliptic curve over $\mathbb{Q}$ can be made substantially more accurate by giving a neural network several related sums at once: $S_0(B)$ and $S_5(B)$, each evaluated at eight bounds $B$, together with the conductor. On the paper's source databases of 17,388,429 curves, the best such model reaches a Matthews correlation coefficient (MCC) of 0.856 on a uniform conductor range and 0.795 when trained on smaller conductors and tested on larger ones, compared with 0.712 and 0.686 using only $S_0(10^5)$. A second method learns prime weights $w_p$ in sums of the form $\sum_{p<10^5} w_p a_p/\sqrt{p}$, giving a slight additional edge (MCC 0.7322 versus 0.712). If correct, these results show that rank heuristics are not a single number but a feature vector, and that data-driven weighting can modestly improve on fixed analytical weighting.

What carries the argument

The load-bearing object is the multi-value Mestre–Nagao feature vector. For a curve $E$, $S_0(B)=\frac{1}{\log B}\sum_{p\le B, p\nmid N} a_p\frac{\log p}{p}$ and $S_5(B)=\sum_{p\le B, p\nmid N}\log\frac{p+1-a_p(E)}{p}+\sum_{p\le B, p\mid N}\log\frac{1.5p-1}{p}$; the inputs are the values of both sums at several bounds, together with $\log_{10}N$, fed to a four-hidden-layer ReLU network with a weighted cross-entropy loss. The learned-sum variant replaces the fixed $\log p/p$ weighting with weights $w_p$ produced by 1D convolutional layers acting on a positional encoding of the primes, so the network forms $S_{\mathrm{opt}}(E)=\sum_{p<10^5} w_p a_p/\sqrt{p}$ and classifies from $(\log_{10}N, S_{\mathrm{opt}})$. The paper's argument is carried by comparing MCC across these feature configurations; the multi-bound combination is what produces the largest gain, and the learned weighting produces a smaller one.

What would settle it

Retrain the best multi-sum model and the $S_0(10^5)$ baseline on the subset of curves whose ranks have been independently verified by two different unconditional methods, and test on a held-out verified subset; the central claim is contradicted if the multi-bound model no longer beats the single-sum baseline on that verified subset.

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Extended reading notes

Core claim

The central claim is that rank classification accuracy is governed by the multiplicity of bound values, not just their size: a fully connected network trained on the vector of $(S_0(B), S_5(B))$ for $B \in \{1000,5000,\dots,100000\}$ plus $\log_{10} N$ outperforms every single-sum baseline, reaching MCC 0.856 on the uniform split and 0.795 on the transfer split. The same pattern holds for each sum family separately, and feeding two values of $S_5$ at different bounds already beats one value, which the paper explains by the heuristic $S_5(B) \approx \log C_E + r\log\log B$: two bounds let the network estimate the curve-dependent constant $C_E$. The paper further claims that a network which learns weights $w_p$ in $S_{\mathrm{opt}}=\sum_{p<10^5} w_p a_p/\sqrt{p}$, optionally conditioned on the conductor, reaches MCC 0.7322 and slightly beats the $S_0(10^5)$ baseline. These are heuristic classifications; the paper does not claim to compute or prove ranks.

Load-bearing premise

The entire comparison assumes the rank labels in the paper's source databases are correct for all 17,388,429 curves; if even a small fraction of labels are wrong, the reported MCC values and the size of the claimed improvements are biased.

Editorial extensions

If this is right

  • The best multi-sum model, using $S_0$, $S_5$, and all eight bounds, reaches MCC 0.856 on the uniform conductor range and 0.795 on the top-range transfer test, beating $S_0(10^5)$ alone (0.712 and 0.686).
  • Using two values of $S_5$ (at $10^3$ and $10^5$) already improves over a single value, consistent with the heuristic $S_5(B)\approx \log C_E + r\log\log B$, where the second bound helps estimate the curve-dependent constant $C_E$.
  • Simple rectangular decision regions in the $(S_0(1000), S_0(100000))$ plane outperform the single-sum baseline in fixed conductor ranges, and the neural network improves on those rectangles, showing the two bounds carry complementary information.
  • The learned-weight network that also sees the conductor reaches MCC 0.7322, a modest but consistent gain over the $S_0(10^5)$ baseline of 0.712; both learned variants beat the baseline.
  • Both approaches remain heuristic classifiers: they sharpen the ranking of candidate curves but do not certify the true rank.

Reading between the lines

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

  • Because the improvement from multiple bounds is tied to the oscillatory, or murmuration, behavior of Mestre–Nagao sums, one could test whether choosing bounds near local extrema of the oscillation gives a further accuracy gain; the paper does not explore this.
  • The learned weights $w_p$, if stable across conductor ranges, could be compared against the analytic $\log p/p$ weighting to identify which primes carry the most rank signal; the paper stops short of deriving such formulas.
  • A natural downstream use is as a prefilter: a cheap multi-sum classifier could prioritise curves for expensive unconditional rank computation, though the paper does not discuss this application.
  • Because the rank labels come from large databases, a decisive follow-up would retrain on a smaller set of curves whose ranks are proven by two independent methods and check whether the reported MCC gaps persist.
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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

2 major / 6 minor

Summary. The manuscript proposes two methods to improve elliptic curve rank classification using Mestre-Nagao sums. The first, termed the multi-value approach, feeds a neural network with the sums S0(B) and S5(B) evaluated at several bounds B, together with the logarithmic conductor. The second learns weights wp in a generalized sum Sopt(E)=∑_{p<10^5} w_p a_p(E)/√p, with or without the conductor as an additional input. Experiments on a dataset of 17,388,429 curves from LMFDB and the Balakrishnan et al. database report that the multi-value model with all bounds achieves MCC 0.856 in the uniform test range and 0.795 in the top-range transfer test, versus 0.712 and 0.686 for the S0(10^5) baseline. The learned-sum networks achieve MCC 0.7283 (conductor-independent) and 0.7322 (conductor-dependent), compared with the same baseline.

Significance. The main multi-value result is credible and potentially useful: the gains over single-sum baselines are large and are computed on held-out test sets, including a transfer experiment to a higher conductor range. The paper's first contribution therefore appears sound and provides a simple, practical way to improve Mestre-Nagao heuristics. The second contribution, however, is supported by a single training run with a small accuracy gap and no estimate of variance; the paper's own figures for the learned weights are explicitly labeled conceptual. If the learned-sum claim is intended as a substantive result, it requires repeated-trial validation. The paper does not mention code or data availability, which limits reproducibility.

major comments (2)
  1. [Section 3 and Table 7] The claimed advantage of learned Mestre-Nagao sums rests on a single training configuration (5 epochs, One Cycle Policy, no random seeds) yielding MCC 0.7322 versus 0.712 for the S0(10^5) baseline. With a gap of only 0.0202 MCC, seed-to-seed variability in fully connected and convolutional networks on imbalanced data is typically of this size or larger, so the improvement cannot be distinguished from training noise. Figures 3 and 4 do not provide actual learned coefficients, as they are labeled 'conceptual'. I request repeated training runs with several seeds reporting mean and standard deviation, or an ablation/permutation test, before the learned-sum claim is accepted.
  2. [Section 2.3, Figures 1 and 2] The rectangular-region classification is presented as outperforming the single-sum baseline (MCC 0.9826 to 0.9921 and 0.7530 to 0.7649), but the rectangles are optimized on the same curves used to compute the MCC. This in-sample optimization makes the comparison with the neural network, which is evaluated out-of-sample on a separate training split, unfair. I recommend either fitting the rectangles on a training subset and evaluating on a held-out subset, or explicitly labeling these numbers as in-sample and excluding them from the main quantitative comparison.
minor comments (6)
  1. [Section 1] The statement that lim_{B→∞} S0(B) = -r_an + 1/2 appears to conflict with the standard Nagao conjecture, which is typically stated as lim_{B→∞} S0(B) = -r_an (see, e.g., [KM23]); please verify the sign and constant or clarify the normalization.
  2. [Section 2.1, Table 1] The notation for the bounds is inconsistent: the table writes '103,10^5' where the text means '10^3,10^5'. Please format all bounds uniformly.
  3. [Section 2, Eq. (2)] The factor 1.5 in the conductor term of S5 is introduced without derivation or citation; a reference to the Elkies-Klagsbrun variant or a brief explanation of this constant would help the reader.
  4. [Section 3, Eq. (3)] Calling the trained weights 'optimal' overstates what the network computes; the training procedure finds an empirical-loss minimum, not necessarily the global optimum. I suggest using 'learned' or 'trained' weights.
  5. [Tables 1 and 4] The main tables report MCC values without error bars or multiple seeds; although the multi-value gains are large, reporting the mean and standard deviation over several training runs would strengthen the empirical claims.
  6. [Figures 1 and 2] The figures lack axis labels and a color legend; since they are central to the Section 2.3 discussion, adding these elements would improve clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the multi-sum and learned-sum claims are evaluated on held-out test data, so the improvements are out-of-sample empirical findings rather than constructions of the inputs.

full rationale

The derivation chain is self-contained supervised learning. The rank labels come from the external LMFDB and Balakrishnan et al. databases, while the features S0(B), S5(B), and the ap/sqrt(p) sequence are computed from elliptic curve data independently of those labels. In both the top-range and uniform tests, models are trained and validated on one portion of the data and evaluated on a disjoint held-out portion, so the reported MCC improvements (e.g., 0.856 vs 0.712 or 0.795 vs 0.686) are not forced by fitting to the test labels. The learned weights wp in Section 3 are optimized on the training split only, and the reported test MCC of 0.7322 is an out-of-sample number. Cited prior work involving current authors, such as [BKN24] and [KV23], is motivational background about murmurations and earlier neural-network rank classification, not a load-bearing premise for the quantitative claims. Section 2.3 fits rectangular regions in-sample to maximize MCC, but the text explicitly presents these as illustrative classification regions rather than as held-out predictions, so this is not a fitted input renamed as a prediction. The missing seed-level error bars on the small learned-sum improvement are a robustness concern, not circularity.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central claims are empirical; the main free parameters are the learned weights, the optimized rectangles in section 2.3, the fixed 1.5 constant in S5, and the network hyperparameters. The paper relies on the correctness of database ranks and on the Nagao/BSD heuristic as motivation, but the classification benchmarks do not depend on BSD being true.

free parameters (4)
  • Learned prime weights wp = not reported numerically; figures labeled conceptual
    Section 3, Eq. (3): weights wp are optimized by the neural network on the training set to maximize rank classification accuracy.
  • Rectangular region boundaries (Section 2.3) = not reported; varies by conductor range
    In Figures 1-2, rectangles are chosen to maximize MCC on the same curves used for evaluation.
  • Conductor term factor 1.5 in S5 = 1.5
    Equation (2): the factor 1.5 in the p|N term is taken from Elkies-Klagsbrun [EK20] and is not derived or re-fit here.
  • Neural network hyperparameters = 4 hidden layers; 64-256 neurons; learning rate 1e-4; 5 epochs for section 3
    Selected by hyperparameter search in Sections 2 and 3; these are standard model-selection choices.
assumptions (4)
  • domain assumption The Mestre-Nagao sum S0(B) converges to -r + 1/2 under BSD / Nagao's conjecture.
    Section 1 and Equation (1), citing [KM23]; used as motivation for using these sums, not required for the empirical classification benchmark.
  • domain assumption Rank labels from LMFDB and Balakrishnan et al. are correct for all curves in the datasets.
    Sections 2.1 and 2.2: these labels define the supervised targets and the test accuracy.
  • domain assumption The datasets are split randomly and have no duplicate curves or leakage between train and test.
    Sections 2.2 and 3: uniform random split 60/20/20 is assumed valid.
  • domain assumption Frobenius traces ap(E) are computed correctly for all primes up to the chosen bounds.
    Needed for all S0, S5, and wp sums; the paper does not describe the computation details.

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Cite this review

Pith. "Pith review of Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums." pith.science (2026). https://pith.science/paper/QLETOOCV

@misc{pith2026250607967,
  author       = {Pith},
  title        = {Pith review of: Improving elliptic curve rank classification using multi-value and learned Mestre-Nagao sums},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QLETOOCV}},
  note         = {Machine review of arXiv:2506.07967}
}
read the original abstract

Determining the rank of an elliptic curve E/Q remains a central challenge in number theory. Heuristics such as Mestre--Nagao sums are widely used to estimate ranks, but there is considerable room for improving their predictive power. This paper introduces two novel methods for enhancing rank classification using Mestre--Nagao sums. First, we propose a ``multi-value'' approach that simultaneously uses two distinct sums, S_0 and S_5, evaluated over multiple ranges. This multi-sum perspective significantly improves classification accuracy over traditional single-sum heuristics. Second, we employ machine learning -- specifically deep neural networks -- to learn optimal, potentially conductor-dependent weightings for Mestre--Nagao sums directly from data. Our results indicate that adaptively weighted sums offer a slight edge in rank classification over traditional methods.

Figures

Figures reproduced from arXiv: 2506.07967 by the authors.

Figure 1
Figure 1. Rank cutoff areas for elliptic curves of conductor [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Rank cutoff areas for elliptic curves of conductor [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Conceptual visualization of learned optimal coefficients [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Conceptual visualization of learned optimal coefficients [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]

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Reference graph

Works this paper leans on

17 extracted references · 13 canonical work pages

  1. [1]

    Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curves

    Babei, A., Banwait, B. S., Fong, A. J., Huang, X., Singh, D. (2024). Machine learning approaches to the Shafarevich–Tate group of elliptic curves. arXiv preprint arXiv:2412.18576

    work page Pith review arXiv 2024

  2. [2]

    Babei, A., Charton, F., Costa, E., Huang, X., Lee, K.-H., Lowry-Duda, D., Narayanan, A., Pozdnyakov, A. (2025). Learning Euler factors of elliptic curves. arXiv preprint arXiv:2502.10357

    arXiv 2025

  3. [3]

    Bieri, J., Butbaia, G., Costa, E., Deines, A., Lee, K.-H., Lowry-Duda, D., Oliver, T., Qi, Y., Veenstra, T. (2025). Machine learning the vanishing order of rational \(L\)-functions. arXiv preprint arXiv:2502.10360

    work page Pith review arXiv 2025

  4. [4]

    S., Ho, W., Kaplan, N., Spicer, S., Stein, W., Weigandt, J

    Balakrishnan, J. S., Ho, W., Kaplan, N., Spicer, S., Stein, W., Weigandt, J. (2016). Databases of elliptic curves ordered by height and distributions of Selmer groups and ranks. LMS J. Comput. Math. 19, 351--370

    work page 2016

  5. [5]

    Bujanović, Z., Kazalicki, M., Novak, L. (2024). Murmurations of Mestre–Nagao sums. Int. J. Data Sci. Math. Sci. 2(1), 1--22. DOI: 10.1142/S2810939224400070 https://doi.org/10.1142/S2810939224400070

    work page doi:10.1142/s2810939224400070 2024

  6. [6]

    D., Klagsbrun, Z

    Elkies, N. D., Klagsbrun, Z. (2020). New rank records for elliptic curves having rational torsion. In ANTS XIV. Proceedings of the Fourteenth Algorithmic Number Theory Symposium, Auckland, New Zealand, virtual event, June 29--July 4, 2020, 233--250. Berkeley, CA: Mathematical Sciences Publishers (MSP)

    work page 2020

  7. [7]

    He, Y.-H., Lee, K.-H., Oliver, T. (2023). Machine learning invariants of arithmetic curves. J. Symbolic Comput. 115, 478--491. DOI: 10.1016/j.jsc.2022.11.015 https://doi.org/10.1016/j.jsc.2022.11.015

    work page doi:10.1016/j.jsc.2022.11.015 2023

  8. [8]

    He, Y.-H., Lee, K.-H., Oliver, T., Pozdnyakov, A. (2024). Murmurations of elliptic curves. Experimental Mathematics, 33(1), 1--25. DOI: 10.1080/10586458.2024.2382361 https://doi.org/10.1080/10586458.2024.2382361

    arXiv 2024

Show all 17 references
  1. [9]

    Kim, S., Murty, M. R. (2023). From the Birch and Swinnerton-Dyer conjecture to Nagao’s conjecture. Math. Comp. 92(339), 385--408. (with an appendix by A. V. Sutherland). DOI: 10.1090/mcom/3773 https://doi.org/10.1090/mcom/3773

    2023 doi

  2. [10]

    Kazalicki, M., Vlah, D. (2023). Ranks of elliptic curves and deep neural networks. Research in Number Theory, 9(3), 53. DOI: 10.1007/s40993-023-00462-w https://doi.org/10.1007/s40993-023-00462-w

    2023 doi

  3. [11]

    The L-functions and Modular Forms Database

    The LMFDB Collaboration. The L-functions and Modular Forms Database. http://www.lmfdb.org. Accessed November 2024

    2024

  4. [12]

    Lee, K.-H., Oliver, T., Pozdnyakov, A. (2025). Murmurations of Dirichlet Characters. Int. Math. Res. Not. IMRN, 2025(1), rnae277. DOI: 10.1093/imrn/rnae277 https://doi.org/10.1093/imrn/rnae277

    2025 doi

  5. [13]

    Loshchilov, I., Hutter F. (2019). Decoupled Weight Decay Regularization. arXiv preprint arXiv:1711.05101

    2019 arXiv

  6. [14]

    Paszke, A., Gross, S., Massa, F., Lerer, A., Bradbury, J., Chanan, G.\ et al. (2019). PyTorch: An imperative style, high-performance deep learning library. In H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché-Buc, E. Fox & R. Garnett (Eds.), Advances in Neural Information ...

    2019

  7. [15]

    Pozdnyakov, A. (2024). Predicting root numbers with neural networks. Int. J. Data Sci. Math. Sci. 2(1), 15--37. DOI: 10.1142/S2810939224400057 https://doi.org/10.1142/S2810939224400057

    2024 doi

  8. [16]

    N., Topin, N

    Smith, L. N., Topin, N. (2018). Super-Convergence: Very Fast Training of Neural Networks Using Large Learning Rates. arXiv preprint arXiv:1708.07120

    2018 arXiv

  9. [17]

    Zubrilina, N. (2023). Murmurations of elliptic curves. Math. Comp. 92(341), 423--445. DOI: 10.1090/mcom/3783 https://doi.org/10.1090/mcom/3783. (Also arXiv:2111.11496)

    2023 arXiv

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Reviewed August 7, 2026 · model on record in the stance chip above.