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Machine learning the vanishing order of rational L-functions

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper claims that the order of vanishing of a rational L-function at its central point can be predicted from finitely many normalized Dirichlet coefficients, with a linear classifier reaching 95.9% accuracy and neural networks…

desk verdict Large-scale but under-benchmarked ML study of vanishing orders; transfer learning is the new part, but missing conductor baselines and scope overclaims keep it from a clean accept. read the letter →

arxiv 2502.10360 v1 pith:IPBS6IWR submitted 2025-02-14 math.NT hep-th

Joanna Bieri , Giorgi Butbaia , Edgar Costa , Alyson Deines , Kyu-Hwan Lee , David Lowry-Duda , Thomas Oliver , Yidi Qi
show 1 more author
Tamara Veenstra
This is my paper
classification math.NThep-th MSC 11G4011F6768T05
keywords rationalL-functionsvanishingorderDirichletcoefficientsmachinelearningmurmurationslineardiscriminantanalysisneuralnetworkscentralpoint
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 asks whether the order of vanishing at the central point of a rational L-function can be recovered from finitely many Dirichlet coefficients. Using a curated dataset of 176,156 primitive rational L-functions that are all of degree 4 and motivic weight 1, it represents each function by 168 normalized prime coefficients and finds that both a linear classifier and convolutional networks learn the vanishing order with high accuracy. The central result is that a linear discriminant analysis reaches 95.9% held-out accuracy, and CNNs exceed 95% accuracy on every subfamily, with transfer between the two main subfamilies above 90%. The paper also documents murmuration-like patterns in coefficient averages that visibly separate by vanishing order. If these patterns are real, finite coefficient data carries substantial information about a deep arithmetic invariant.

What carries the argument

The carrying object is the coefficient vector $v(L)\in\mathbb{R}^{168}$ formed by the normalized Dirichlet coefficients $\bar a_p = a_p/(d p^{w/2})$ for primes $p<1000$, which puts every feature in $[-1,1]$ through the Hasse bound. The paper runs three mechanisms on this point cloud: PCA diagonalizes the covariance matrix to find linear projections that separate classes; LDA finds the linear discriminant that best separates the four vanishing orders; and a 1D CNN with three convolutional layers learns a nonlinear classifier. The normalization is load-bearing: it removes the dependence on degree and weight so that functions of different origins live in the same feature space, and averaging these coefficients by vanishing order produces the murmuration-like curves that motivate the supervised approach.

What would settle it

Permute the vanishing-order labels in the training portion of the dataset and retrain the LDA with the same features and hyperparameters; if validation accuracy on the unpermuted test set does not fall to chance, the reported 95.9% stems from label leakage rather than genuine learnability.

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Extended reading notes

Core claim

The paper's central claim is that the vanishing order $r$ at the central point of a rational L-function is encoded in the normalized Dirichlet coefficient vector $v(L)=(a_p/(d p^{w/2}))_{p<1000}$, in the following sense: on the sub-dataset PRAT$^\star$ of 176,156 primitive rational L-functions (degree 4, motivic weight 1, mostly elliptic curves over number fields and genus-2 curves over $\mathbb{Q}$), LDA predicts $r\in\{0,1,2,3\}$ with 95.9% held-out accuracy and explained variance 0.982, and a CNN trained on the raw coefficient vector reaches over 95% accuracy on each subfamily (ECNF, BMF, HMF, G2Q). Additionally, PCA projections show visible clustering by $r$, and a CNN trained on only the first two principal components reaches about 91% accuracy. The paper interprets these accuracies as evidence that the order of vanishing is a learnable function of finitely many coefficients, not that the models have identified the underlying arithmetic mechanism.

Load-bearing premise

The paper assumes the vanishing-order labels in the dataset are exact and that every coefficient is correctly normalized; if any label is wrong or mis-scaled, the reported accuracies measure the noise in the labels, not learnability of the true invariant.

Editorial extensions

If this is right

  • If the result is correct, the order of vanishing is, for this class of L-functions, a function of the first 168 prime coefficients; no further analytic information is needed for accurate classification.
  • The 95.9% LDA accuracy implies the class separation is nearly linear in the normalized coefficient space, suggesting a single linear combination of coefficients carries most of the signal.
  • The transfer-learning result (train on one subfamily, test on another above 90%) implies that the coefficient–vanishing-order relationship is shared across ECNF and G2Q, not an artifact of one family.
  • Murmuration-like averages provide a quantitative counterpart to Mestre–Nagao sums: larger vanishing orders correspond to systematically smaller average coefficients across conductors.

Reading between the lines

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

  • One step the paper leaves open is whether the same protocol holds on larger conductor ranges or on the excluded degree-2 families; a failure there would mark a boundary of the learnability phenomenon.
  • If the linear discriminant is stable across random splits, it could be extracted as an explicit linear form; comparing its weights to the Mestre–Nagao logarithmic weights might reveal the number-theoretic mechanism, a step the paper leaves open.
  • Because the dataset is restricted to root analytic conductor below 4, the high accuracy may reflect a restricted range; on wider conductor ranges the signal could degrade, and the paper does not test this.
  • A practical extension is to use the classifier as a rapid pre-screen for vanishing order before exact computation, but only after a label-permutation falsifier is passed.
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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

3 major / 5 minor

Summary. The paper studies the order of vanishing at the central point for rational L-functions from a data-scientific viewpoint. Using the RAT/PRAT datasets extracted from LMFDB, the authors represent each L-function by the vector of 168 normalized Dirichlet coefficients at primes less than 1000. Restricting to the dataset PRAT* (primitive, degree 4, motivic weight 1, vanishing order <= 3), they compute PCA projections, run LDA, and train CNNs to predict the vanishing order. The main reported results are 95.9% held-out LDA accuracy on PRAT*, CNN accuracies above 95% on sub-datasets, and transfer learning between ECNF and G2Q with accuracies above 90%.

Significance. If the reported accuracies genuinely reflect information carried by the Dirichlet coefficients, the paper would provide a substantial data-driven contribution to understanding vanishing orders in a large, heterogeneous family of L-functions. The construction of the RAT/PRAT datasets, with a public DOI, is itself a useful resource, and the transfer-learning observations are interesting. However, the central claim is currently not fully established: the feature vector may encode the conductor N of the L-function, and the paper does not provide the conductor-only baseline needed to separate the coefficient signal from a conductor/type proxy. The post hoc restriction to PRAT* also makes the abstract's 'rational L-functions' claim broader than the evidence. The PCA-based experiment additionally suffers from fitting principal components on the full dataset before splitting.

major comments (3)
  1. [§3.3, Table 3.1; §3.4, Table 3.3] The claim that the normalized Dirichlet coefficient vector v(L) carries the vanishing order is not yet established, because the features may encode the conductor N. Every L-function in PRAT* has root analytic conductor < 4, and for degree 4 this gives N < 256; for each prime p dividing N, the local Euler factor is represented among the 168 feature coordinates, so the feature vector contains strong local information at exactly the primes where the conductor is supported. The manuscript reports no distribution of r by conductor, no conductor-only classifier (e.g., logistic regression on N or log N, or majority class per N), and no accuracy stratified by conductor intervals. A conductor-only baseline is essential: if such a baseline approaches 95%, the reported LDA/CNN accuracy and the ECNF-to-G2Q transfer would be explained by a shared conductor/type correlation rather than by a subtler coefficient signal, which would materially change the contribution.
  2. [§2.3, §3.1, abstract] The abstract advertises 'rational L-functions' broadly, but all supervised experiments are performed on PRAT*, a subset selected after inspecting the data: primitive, degree 4, motivic weight 1, vanishing order <= 3, and root analytic conductor < 4. This selection is reasonable, but it means the reported accuracies do not cover ECQ, CMF, DIR, or Artin L-functions; Appendix A explicitly treats those as outside the main dataset. The paper should either temper the abstract and concluding claims to the actual class studied, or provide additional experiments on the remaining part of PRAT<=3. As written, the presentation overstates the scope of the empirical evidence.
  3. [§3.4, Figure 3.2, Table 3.2] The PCA used for the principal-component CNN is computed on the full pointcloud D before the train/test split (as described at the end of §3.2 and used in §3.4), so the test set contributes to the principal axes. The 91% test accuracies in Table 3.2 may therefore be optimistically biased. The PCA should be fit on the training split only, and its weights then applied to the validation/test split, before these numbers are reported. This issue does not affect the v(L)-based results in Tables 3.1 and 3.3, which use the raw feature vector directly, but it does affect the specific PCA-based claim.
minor comments (5)
  1. [Equations (2.1) and (3.1)] The notation overloads a_p: in (2.1) the normalized coefficient is called eap, while in (3.1) the same symbol a_p is reused for the further normalized coefficient. Distinct symbols would avoid confusion.
  2. [Table 3.1] The column 'Explained Variance' is not defined for LDA; the manuscript should state whether it is the ratio of between-class to total scatter eigenvalues or another quantity, since the reported value 0.982 is used to support the accuracy claim.
  3. [References] References [Poz24a] and [Poz24b] appear to refer to the same arXiv paper; the duplicate should be removed or consolidated.
  4. [§3.4, Figure 3.5] The transfer-learning curves in Figure 3.5 lack axis labels and a precise statement of which epochs and accuracies are shown; adding these would make the transfer claim easier to verify.
  5. [§3.3, §3.4] All accuracy numbers are point estimates without confidence intervals, repeated seeds, or standard deviations; given the class imbalance (e.g., vanishing order 3 has only 2,837 examples in PRAT*), error bars would substantially strengthen the quantitative claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the vanishing-order predictions are evaluated on held-out data and do not reduce to the fitted inputs or to the labels by definition.

full rationale

The central claim is an empirical supervised-learning statement: the feature vector v(L) is built from normalized Dirichlet coefficients (Eq. 3.2), and the target r is the vanishing order supplied by the external LMFDB data described in Section 2.2. LDA and CNN accuracies (Tables 3.1 and 3.3) are reported on 20% held-out validation/test splits after an 80:20 stratified split, so the numbers are not training accuracy and are not equal by construction to a fitted parameter. No equation in the paper defines r in terms of a_p; Eq. (3.1) is merely a rescaling of the input features, and for PRAT* it reduces to the fixed divisor 4*sqrt(p), so there is no self-definitional identity between target and features. The citations to [CL25] and earlier murmuration work are data attribution and background, not load-bearing derivation: [CL25] is an externally deposited Zenodo/LMFDB-derived dataset, and the labels are independently computable invariants. Section 4 explicitly asks to what extent classifier accuracy depends on the conductor range, which flags a possible confounding variable, but a conductor-proxy concern is a robustness and validity issue, not circularity, because the reported predictions do not reduce to the conductor by construction. The absence of code is a reproducibility concern, not a circularity concern. No fitted parameter is renamed as a prediction, and no uniqueness theorem or ansatz is imported from the authors' prior work to force the choice of model. The derivation chain is therefore self-contained with respect to circularity.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the dataset's integrity and on the choice of feature truncation and normalization. No new entities are postulated. The ML model parameters are fit to data, so the accuracies are conditional on those fitted values.

free parameters (4)
  • PCA principal components (weights w_p in Eq. 3.3) = eigenvectors of covariance matrix fitted on D
    Used as CNN input in Section 3.4; computed on the full dataset including the test split, so the 91% accuracy figure is optimistically biased.
  • LDA discriminant coefficients = fitted on the 80% training split
    Table 3.1 accuracies are achieved with these learned linear boundaries; no closed form is given.
  • CNN hyperparameters = 3 conv layers (16/32/64 channels, kernel 3), 2 FC layers (128), dropout 0.5, Adam lr 0.001, batch 3000
    Chosen by hand without a reported search; accuracies in Tables 3.2 and 3.3 depend on these choices.
  • Root analytic conductor cutoff = 4
    Defines the dataset scope in Section 2.2; chosen because the dataset is relatively balanced in this range, and results may not extrapolate beyond it.
assumptions (5)
  • standard math Selberg-type axiomatisation: each L-function has an Euler product and functional equation
    Section 2.1 defines the objects of study; the dataset entries are assumed to satisfy this.
  • standard math Hasse and Weil bounds |a_p| <= d p^{w/2}
    Used in Eq. (3.1) to normalize features into [-1,1]; assumed for all L-functions in PRAT⋆.
  • domain assumption LMFDB vanishing order r is the true analytic order
    r is the supervised target in Sections 3.3 and 3.4; the paper reports no confidence or error bars on these labels.
  • domain assumption Truncation at primes below 1000 preserves the signal
    The 168-dimensional feature vector is the only input; the paper provides empirical evidence but no guarantee that longer truncations behave the same.
  • domain assumption Stratified 80:20 split is representative
    Section 3.3 states the split but gives no seeds or repeated trials, so variance across splits is unknown.

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Pith. "Pith review of Machine learning the vanishing order of rational L-functions." pith.science (2026). https://pith.science/paper/IPBS6IWR

@misc{pith2026250210360,
  author       = {Pith},
  title        = {Pith review of: Machine learning the vanishing order of rational L-functions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IPBS6IWR}},
  note         = {Machine review of arXiv:2502.10360}
}
abstract

In this paper, we study the vanishing order of rational $L$-functions from a data scientific perspective. Each $L$-function is represented in our data by finitely many Dirichlet coefficients, the normalisation of which depends on the context. We observe murmuration-like patterns in averages across our dataset, find that PCA clusters rational $L$-functions by their vanishing order, and record that LDA and neural networks may accurately predict this quantity.

Figures

Figures reproduced from arXiv: 2502.10360 by the authors.

Figure 2.1
Figure 2.1. UpSet plot for RAT. Each row corresponds to the named subset, and each column corresponds to an intersection of subsets. A black circle is used to indicate that a subset is involved in a particular intersection (the vertical lines simply indicate the orientation of the plot). The horizontal bars show the number of datapoints in each subset (row), and a vertical bar above shows the size of an exclusive intersection, … view at source ↗
Figure 2.2
Figure 2.2. Average value of eap for primitive rational L-functions with specified vanishing order, excluding the 9 L-functions with vanishing order 4. In [PITH_FULL_IMAGE:figures/full_fig_p006_2_2.png] view at source ↗
Figure 2.3
Figure 2.3. Average value of eap over L-functions with specified vanishing order in PRAT⋆ . 2 19 47 79 109 151 191 229 269 311 353 397 439 479 523 577 617 659 709 757 811 857 907 953 −6 −4 −2 0 2 Van. order 0 1 2 2 19 47 79 109 151 191 229 269 311 353 397 439 479 523 577 617 659 709 757 811 857 907 953 −6 −4 −2 0 2 Van. order 0 1 2 3 [PITH_FULL_IMAGE:figures/full_fig_p007_2_3.png] view at source ↗
Figures from the paper (8 more)
Figure 2.4
Figure 2.4. Figure 2.4: Average value of eap over L-functions with specified vanishing order in (left) ECNF and (right) G2Q excluding the 27 L-functions with vanishing order 3 for ECNF. 7 [PITH_FULL_IMAGE:figures/full_fig_p007_2_4.png]
Figure 2.5
Figure 2.5. Figure 2.5: Venn diagram for ECNF. 2 19 47 79 109 151 191 229 269 311 353 397 439 479 523 577 617 659 709 757 811 857 907 953 −6 −4 −2 0 2 Van. order 0 1 2 2 19 47 79 109 151 191 229 269 311 353 397 439 479 523 577 617 659 709 757 811 857 907 953 −6 −4 −2 0 2 Van. order 0 1 2 […
Figure 2.6
Figure 2.6. Figure 2.6: Average value of eap for (left) BMF and (right) HMF, excluding the 26 (resp. 1) L￾functions with vanishing order 3. 3. Machine learning the vanishing order of rational L-functions We present an unsupervised approach (PCA) and two supervised approaches (LDA and neural…
Figure 3.1
Figure 3.1. Figure 3.1: Two-dimensional PCA for the PRAT⋆ dataset. Each datapoint is colored by the van￾ishing order for the underlying L-function at its central point. In the left (resp. right) image we plot the points in ascending (resp. descending) order for the order of vanishing. 3.3. …
Figure 3.2
Figure 3.2. Figure 3.2: Learning the vanishing order the first and second principal components: percentage accuracy against epoch with CNN. Each colour corresponds to one category of L-functions sub-dataset test accuracy ECNF 91.22% BMF 91.48% HMF 90.54% G2Q 91.13% [PITH_FULL_IMAGE:figures…
Figure 3.3
Figure 3.3. Figure 3.3: The weights in the first and second principal components used in the training [PITH_FULL_IMAGE:figures/full_fig_p012_3_3.png]
Figure 3.4
Figure 3.4. Figure 3.4: Learning the vanishing order in PRAT⋆ from v(L) : percentage accuracy against epoch with CNN. Each colour corresponds to one category of L-functions [PITH_FULL_IMAGE:figures/full_fig_p012_3_4.png]
Figure 3.5
Figure 3.5. Figure 3.5: Train with ECNF then test with G2Q, and vice versa. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_3_5.png]

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