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REVIEW 3 major objections 6 minor 3 cited by

Learning Euler Factors of Elliptic Curves

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read An encoder-only transformer trained on Frobenius traces $a_q$ for $q<100$, $q\neq p$ predicts $a_p \bmod 2$ with around $0.94$ accuracy and MCC around $0.84$ for most primes $p<100$, without access to L-functions or functional equations.

desk verdict A promising ML-plus-number-theory proof-of-concept whose headline accuracies are compromised by an unmeasured train/test overlap on repeated input tuples. read the letter →

arxiv 2502.10357 v1 pith:6HXGPTDN submitted 2025-02-14 math.NT cs.LG

Angelica Babei , François Charton , Edgar Costa , Xiaoyu Huang , Kyu-Hwan Lee , David Lowry-Duda , Ashvni Narayanan , Alexey Pozdnyakov This is my paper · ORCID
classification math.NTcs.LG MSC 11G0511G4068T07
keywords FrobeniustracesellipticcurvesEulerfactorstransformersmodulararithmeticL-functionsisogenyclassesmachinelearning
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 a machine-learning model can determine one Euler factor of an elliptic curve from the others. Given the sequence of Frobenius traces $a_q$ for $q<100$, $q\neq p$, an encoder-only transformer trained on isogeny classes (curves related by isogeny) with conductors below $10^6$ predicts $a_p$ with accuracy far above chance, and its prediction of $a_p \bmod 2$ reaches about $0.94$ accuracy and a Matthews correlation coefficient near $0.84$ for most $p<100$. Notably, the models receive no L-functions, functional equations, or explicit Galois-representation input, so the authors interpret the performance as evidence that the models learn the congruence structure of Frobenius traces. They support this with confusion matrices and PCA of the learned embeddings, which separate by parity and modulo-4 classes. The result matters because no efficient classical algorithm is known for completing a finite trace sequence, despite a theorem of Serre guaranteeing the sequence determines the isogeny class.

What carries the argument

The load-bearing object is the Frobenius trace $a_p(E) = p + 1 - |E(\mathbb{F}_p)|$, the coefficient in the local Euler factor $1/(1 - a_p p^{-s} + p^{1 - 2s})$. The argument is carried by an encoder-only transformer whose vocabulary tokenizes signed integers; it reads the sequence $(a_q)_{q\neq p,\ q<100}$ and is trained, in the parity experiments, on targets reduced modulo $2$. Two auxiliary mechanisms are essential: the dataset of isogeny classes with conductors below $10^7$, and the modulo-2 duplicate-removal and balancing procedure that separates the training and test sets. The interpretability step uses PCA of token embeddings and of decoder hidden states to show that the model organizes its representations by residues modulo $2$ and $4$.

What would settle it

Build the test set for the $a_p \bmod 2$ task by discarding every input tuple $(a_q)_{q\neq p,\ q<100}$ that also appears in training, then retrain; if accuracy on that tuple-disjoint test set is near $0.5$, the claimed parity learning is an artifact of overlap.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is that a transformer can learn the arithmetic of Frobenius traces from data alone. For the exact-value task, $a_{97}$ is predicted with test MCC $0.4711$ against a most-common-class baseline of $0.082$, and the model captures $|a_{97}|$ better than the sign (sign-agnostic MCC $0.6266$); $a_2$ and $a_3$ show the same pattern. When the target is reduced modulo $2$, the encoder-only transformer reaches accuracies close to $0.94$ and MCCs around $0.84$ for most primes $p<100$ (accuracy $0.9472$ at $p=83$, MCC $0.8703$ at $p=5$), with the lowest results at $p=2$. The authors read this as evidence that the model implicitly computes $a_p \bmod 2$ as an intermediate step, and they corroborate it by showing confusion matrices consistent with parity classes and PCA projections of embeddings and decoder hidden states that cluster by residues modulo $2$, $4$, and in one case $6$.

Load-bearing premise

The empirical claim rests on the assumption that test accuracy measures learning of new input sequences rather than recall of sequences already seen during training.

Editorial extensions

If this is right

  • If the central claim is correct, a neural network can recover the parity of a missing Euler factor from its neighbors in a regime where no efficient deterministic completion algorithm is known.
  • The parity signal is the model's main strength, and exact-value predictions inherit it: models trained on exact $a_p$ already predict $a_p \bmod 2$ with MCC near the models trained directly on the parity task.
  • The sign of $a_p$ is the principal obstacle; the sign-agnostic MCC exceeds the full MCC in every exact-value experiment.
  • The pattern is stable across the dataset range: results on a test set with conductors up to $10^7$ remain well above chance, though the highest conductor band $10^6$--$10^7$ is the least accurate.

Reading between the lines

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

  • Read together with the paper's own Section 4.2, which reports that reducing inputs modulo 2 makes over 80% of test tuples appear in training, the Section 3 exact-value results motivate a tuple-disjoint retest before concluding the model generalizes to genuinely unseen sequences.
  • If the parity result survives the tuple-disjoint retest, the PCA-visible parity structure could be used to generate explicit conjectural congruences among Frobenius traces for verification by traditional Galois-representation methods.
  • The parity prediction task may be equivalent, in effect, to learning the mod-2 Galois representation attached to the curve; the paper's embedding evidence suggests the model finds this representation before it finds the full trace.
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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 / 6 minor

Summary. The paper trains encoder-only transformers and feedforward neural networks on datasets of isogeny classes of elliptic curves over Q (conductors up to 10^7) to predict Frobenius traces a_p, or their residues mod 2, from traces at other small primes q < 100. The main empirical claims are that exact a_p prediction for p = 97 reaches accuracy 0.4923 against a baseline of 0.082, that a_p mod 2 prediction from a_q mod 2 reaches accuracies near 0.94 and MCC near 0.84, and that PCA and saliency analyses suggest the models implicitly learn mod-2 and mod-4 structure. The paper interprets these results as evidence that neural networks can learn relationships among Euler factors without explicit number-theoretic tools such as functional equations.

Significance. If the quantitative claims are robust, this is a useful experimental data point for the machine-learning/number-theory interface: it would show that standard transformer architectures can pick up congruence structure in Frobenius traces and expose it in learned embeddings. The paper is also explicit about some limitations, such as the overlap problem for mod-2 inputs. However, the strength of the claims depends critically on whether test accuracy measures generalization to genuinely new input tuples, and the current manuscript does not establish this for the exact-trace experiments. The PCA and saliency sections are suggestive but informal; they do not by themselves carry the central quantitative claim.

major comments (3)
  1. [§3.1, §3.2, §4.1, §4.2] The train/test split is by isogeny class, not by the input tuple (a_q)_{q<100, q≠p}. Section 4.2 demonstrates that this distinction matters: after reducing inputs modulo 2, more than 80% of the test data appears in the training set, and the authors had to remove duplicates before reporting results. For the exact-trace experiments in Sections 3 and 4.1, no duplicate removal or overlap statistic is reported. If identical input tuples recur in both training and test, then the reported accuracies and MCCs (e.g., a_97 accuracy 0.4923 in Table 1, parity accuracies near 0.94 in Figure 5) may reflect memorization of tuple-to-label mappings rather than generalization to new trace sequences. The authors should quantify, for each experiment, the fraction of test tuples that also occur in the training set, and should report results after deduplication or after explicitly removing overlapping tuples from the test set.
  2. [Table 1, Table 3, Figure 5, Figure 6] The paper reports 'maximum' test accuracy and 'maximum' test MCC obtained during training, but it does not specify whether the checkpoint used for evaluation was selected on a validation set or on the test set. Selecting the best epoch by test performance inflates the reported numbers and makes the results non-reproducible without additional protocol details. The authors should report early stopping on a validation split, or else provide the full training curve and state explicitly that the maximum is over validation epochs. In addition, no repeated-seed experiments or confidence intervals are reported, so it is unclear how stable the numbers in Table 1 and Figure 5 are.
  3. [§4.2] The handling of indeterminate cases after duplicate removal is ambiguous. The text says that the dataset is balanced and that between 5% and 11.5% of the training set consists of indeterminate cases (same input tuple with different a_p mod 2 labels). It is not stated whether the test set likewise contains indeterminate cases, nor how predictions are scored when the input admits two valid labels. If indeterminate cases are present in the test set, the achievable accuracy is bounded by the majority fraction for those tuples, and reporting accuracy without separating deterministic and indeterminate inputs makes the parity results difficult to interpret. The authors should specify the evaluation rule and report results separately for deterministic and indeterminate test cases.
minor comments (6)
  1. [§2.4.2] There is a typo: 'the the Mathews correlation coefficient' should be 'the Matthews correlation coefficient', and the attribution should be to Matthews.
  2. [Figure 5 caption] The caption contains a stray '5,' in 'The highest MCC, 5, 0.8703'; this should read 'The highest MCC, 0.8703'.
  3. [§1 and §3.1] The introduction says models predict a_97 from a_q with q < 97, whereas Section 3.1 specifies the input as (a_q)_{q≠p, q≤100}; these descriptions should be reconciled.
  4. [§5] The convention 'a ≡ b (mod 0) ⇔ a = b' is nonstandard and should be explained more clearly in the main text rather than in a footnote-like parenthetical.
  5. [Table 3] The caption 'Primes excluded' is unclear: it should state explicitly that the model's input excludes a_q for the listed primes while predicting a_97 from the remaining a_q values.
  6. [References] The reference [AHL+] is listed as 'In preparation'; if it is used as context, the authors should provide a stable reference or remove it.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: reported predictions are measured on held-out isogeny classes and no derived quantity is reused as an input.

full rationale

The paper's central claim is empirical: transformers and feedforward networks predict Frobenius traces and their parities from neighboring traces, with accuracies reported on data reserved as isogeny classes distinct from training (Section 3.1 and Section 4.1). This is a direct measurement rather than a construction that reuses the target as an input. The mathematical motivation cites Serre's theorem and related results, but the ML predictions do not assume those theorems; they are evaluated against Sutherland's ECQ6/ECQ7 databases. Self-citations to Int2Int and prior ML-number-theory papers are tooling or related-work references and are not load-bearing. The paper itself flags a genuine limitation in Section 4.2: after reducing inputs modulo 2, more than 80% of test tuples occur in training, and it removes duplicates for those experiments. For full-trace experiments, no duplicate removal is reported, so the magnitudes of the reported accuracies could in principle be affected by tuple overlap; this is an experimental-validity caveat, not a circular derivation. The claimed ability to learn Euler-factor structure is therefore not equivalent by construction to the training inputs, and no circular step can be identified from the text.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claims are empirical and rest on standard bounds, cited conditional theorems, the fidelity of the Sutherland dataset, small indeterminacy in the mod-2 task, and the adequacy of the q<100 input window. No new entities are introduced. The model weights and hyperparameters are fitted quantities rather than derived constants.

free parameters (2)
  • Neural network parameters = 4,223,805 trainable parameters (encoder-only transformer); FFN sizes vary, up to 1,709,353
    Every reported accuracy is a property of a fitted model. The learned function is not extracted into a closed-form rule, so the paper's empirical claims are inseparable from these fitted values.
  • Training hyperparameters = learning rates 3e-5 and 5e-5; 100 to 1,000 epochs; embedding dim 256; 8 heads; 4 layers; batch sizes 100,000…
    Chosen by hand without sensitivity analysis. Different hyperparameters could change the reported accuracies, and the paper provides no measure of that variation.
assumptions (5)
  • standard math Hasse bound |a_p| <= 2*sqrt(p) gives finite label sets for a_2, a_3, a_97 used as classification targets.
    Invoked implicitly in Section 3.1 to define 5, 7, and 39 classes for a_2, a_3, a_97. This is standard.
  • domain assumption Serre's theorem that, under the Riemann hypothesis for Artin L-functions, finitely many Frobenius traces determine the isogeny class of a Q-nonisogenous, non-CM elliptic curve.
    Used in Section 1 as the motivation that the prediction task is well-posed. The theorem is conditional on RH for Artin L-functions and does not give an efficient algorithm, so the paper leans on it as background.
  • domain assumption The Sutherland ECQ6/ECQ7 datasets faithfully represent isogeny classes of elliptic curves over Q with conductor less than 10^6 and 10^7, with correctly computed traces.
    Section 2.5 states the datasets are extracted from Sutherland's extension of Stein-Watkins. The entire experiment rests on these traces being correct and the dataset being complete in the stated ranges.
  • domain assumption In the mod-2 experiments, the tuple (a_q mod 2)_{q<100, q!=p} almost always determines a_p mod 2; the small fraction of indeterminate cases is kept rather than excluded.
    Section 4.2 reports 1 to 2.5 percent indeterminates for conductor below 10^6 and 5 to 11.5 percent after balancing on ECQ7. The paper assumes this residual ambiguity is small enough that high MCC still reflects learning, but it does not compare against the maximum attainable MCC given the indeterminacy.
  • ad hoc to paper The 24-prime window q<100, q!=p is sufficient input for the models to identify the relevant mod-2 Galois representation for most curves in the dataset.
    This is implicit in the task design in Sections 3 and 4. The paper does not prove window sufficiency; the high empirical accuracy is the only evidence. The choice of q<100 is a modeling decision, not a derived quantity.

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Cite this review

Pith. "Pith review of Learning Euler Factors of Elliptic Curves." pith.science (2026). https://pith.science/paper/6HXGPTDN

@misc{pith2026250210357,
  author       = {Pith},
  title        = {Pith review of: Learning Euler Factors of Elliptic Curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6HXGPTDN}},
  note         = {Machine review of arXiv:2502.10357}
}
abstract

We apply transformer models and feedforward neural networks to predict Frobenius traces $a_p$ from elliptic curves given other traces $a_q$. We train further models to predict $a_p \bmod 2$ from $a_q \bmod 2$, and cross-analysis such as $a_p \bmod 2$ from $a_q$. Our experiments reveal that these models achieve high accuracy, even in the absence of explicit number-theoretic tools like functional equations of $L$-functions. We also present partial interpretability findings.

Figures

Figures reproduced from arXiv: 2502.10357 by the authors.

Figure 1
Figure 1. We implement an encoder only transformer architecture using Int2Int. 2.4. Performance metrics. 2.4.1. Accuracy. In the case of binary classification on balanced datasets, the most instructive performance metric is accuracy, which measures the proportion of correct predictions. We use this metric in the experiments in Sections 4 and 5. 4 [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Predicting a97. Proportion of cases where the model predicts a given value as a function of its true value [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Confusion matrix heat map for predicting a2 (left) and a3 (right). the majority class ratio (where ap ≡ 0 mod 2) ranges from 52.45% for p = 2 to 72.62% for p = 97 in the combined training and test datasets, as shown in [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The ratios of curves with ap ≡ 0 (mod 2) for various p < 100 when curves have good reduction at p. There is a consistent bias of ap ≡ 0 mod 2 when p is not “small”. (a) Accuracies (b) MCCs [PITH_FULL_IMAGE:figures/full_fig_p008_4.png]
Figure 5
Figure 5. Figure 5: The results of predicting ap mod 2 using {aq}q̸=p,q<100 for curves with good reduction at p show that, except for p = 2, 3, 43, the model achieves accuracies close to 0.94. The highest accuracy, 0.9472, occurs at p = 83, while the lowest, 0.8935, is observed at p = 2. …
Figure 6
Figure 6. Figure 6: shows the maximum attained MCC across the three conductor ranges for each prime. As expected, the performance is reduced for curves of the highest conductor range 106 ≤ N(E) < 107 [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 7
Figure 7. Figure 7: shows saliency plots for the experiments in [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: Saliency plots for important features. The top left is for the 24 primes governing a97 (mod 2), the top right shows the importance of primes when using the first 100 primes. The bottom left shows the same plot with conductor and root number added as well, while the bot…
Figure 9
Figure 9. Figure 9: PCA plots of the initial embeddings for predicting a2, a3, and a97 are shown: a2 (upper left), a3 (upper right), and a97 (bottom). Red, green, blue, and purple dots represent aq ≡ 0, 1, 2, 3 (mod 4), respectively. All show clear separation between odd and even integers…
Figure 10
Figure 10. Figure 10: PCA plots of the initial embeddings for predicting a2 (mod 2) (upper left), a3 (mod 2) (upper right), and a97 (mod 2) (bottom). The distinction between modular–4 classes resembles that in [PITH_FULL_IMAGE:figures/full_fig_p014_10.png]
Figure 11
Figure 11. Figure 11: We also implement an encoder-decoder Transformer for further insights into a2 predictions using Int2Int. As a result, we obtain a clear structure of the hidden states in the decoder compared to the encoder [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: The 2D PCA of the model’s hidden states at the decoder transformer layer for predicting a2 with conductor up to 104 . The dots, representing the hidden states of curves, are separated by their residue classes mod-2 and mod-4 from left to right [PITH_FULL_IMAGE:figure…
Figure 13
Figure 13. Figure 13: PCA plots of the initial embeddings in the encoder (left) and decoder (right) for predicting a2 with an encoder-decoder model. The encoder PCA shows a clear separation between even and odd integers, as well as integers divisible by 3. As a result, we observe clusterin…

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Reference graph

Works this paper leans on

27 extracted references · 22 canonical work pages · cited by 3 Pith papers

  1. [1]

    Amir, Y.-H

    M. Amir, Y.-H. He, K.-H. Lee, T. Oliver, and E. Sultanow. On the machine-learning of the class number of real quadratic fields. In preparation

    work page

  2. [2]

    Pca plotting for int2int

    Angelica Babei. Pca plotting for int2int. https://github.com/ababei/Int2Int

    work page

  3. [3]

    Machine learning approaches to the shafarevich-tate group of elliptic curves

    Angelica Babei, Barinder S Banwait, AJ Fong, Xiaoyu Huang, and Deependra Singh. Machine learning approaches to the shafarevich-tate group of elliptic curves. arXiv preprint arXiv:2412.18576 , 2024

    arXiv 2024

  4. [4]

    An application of the effective sato-tate conjecture

    Alina Bucur and Kiran S Kedlaya. An application of the effective sato-tate conjecture. Frobenius distributions: Lang-Trotter and Sato-Tate conjectures , 663:45--56, 2016

    work page 2016

  5. [5]

    A toy model of universality: Reverse engineering how networks learn group operations

    Bilal Chughtai, Lawrence Chan, and Neel Nanda. A toy model of universality: Reverse engineering how networks learn group operations. In International Conference on Machine Learning , pages 6243--6267. PMLR, 2023

    work page 2023

  6. [6]

    Fran c ois Charton. Int2int. https://github.com/f-charton/Int2Int

    work page

  7. [7]

    Can transformers learn the greatest common divisor? arXiv preprint arXiv:2308.15594 , 2023

    Fran c ois Charton. Can transformers learn the greatest common divisor? arXiv preprint arXiv:2308.15594 , 2023

    arXiv 2023

  8. [8]

    A mathematical framework for transformer circuits

    Nelson Elhage, Neel Nanda, Catherine Olsson, Tom Henighan, Nicholas Joseph, Ben Mann, Amanda Askell, Yuntao Bai, Anna Chen, Tom Conerly, Nova DasSarma, Dawn Drain, Deep Ganguli, Zac Hatfield-Dodds, Danny Hernandez, Andy Jones, Jackson Kernion, Liane Lovitt, Kamal Ndousse, Dario Amodei, Tom Brown, Jack Clark, Jared Kaplan, Sam McCandlish, and Chris Olah. A...

    work page 2021

Show all 27 references
  1. [9]

    Comparing two k-category assignments by a k-category correlation coefficient

    Jan Gorodkin. Comparing two k-category assignments by a k-category correlation coefficient. Computational biology and chemistry , 28(5-6):367--374, 2004

    2004

  2. [10]

    Machine-learning number fields

    Yang-Hui He, Kyu-Hwan Lee, and Thomas Oliver. Machine-learning number fields. Math. Comput. Geom. Data , 2(1):49--66, 2022

    2022

  3. [11]

    Machine-learning the S ato- T ate conjecture

    Yang-Hui He, Kyu-Hwan Lee, and Thomas Oliver. Machine-learning the S ato- T ate conjecture. J. Symbolic Comput. , 111:61--72, 2022

    2022

  4. [12]

    Machine learning invariants of arithmetic curves

    Yang-Hui He, Kyu-Hwan Lee, and Thomas Oliver. Machine learning invariants of arithmetic curves. J. Symbolic Comput. , 115:478--491, 2023

    2023

  5. [13]

    Murmurations of elliptic curves

    Yang-Hui He, Kyu-Hwan Lee, Thomas Oliver, and Alexey Pozdnyakov. Murmurations of elliptic curves. Experimental Mathematics , pages 1--13, 2024

    2024

  6. [14]

    How does gpt-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language model

    Michael Hanna, Ollie Liu, and Alexandre Variengien. How does gpt-2 compute greater-than?: Interpreting mathematical abilities in a pre-trained language model. Advances in Neural Information Processing Systems , 36, 2024

    2024

  7. [15]

    Ranks of elliptic curves and deep neural networks

    Matija Kazalicki and Domagoj Vlah. Ranks of elliptic curves and deep neural networks. Res. Number Theory , 9(3):Paper No. 53, 21, 2023

    2023

  8. [16]

    Towards understanding grokking: An effective theory of representation learning

    Ziming Liu, Ouail Kitouni, Niklas S Nolte, Eric Michaud, Max Tegmark, and Mike Williams. Towards understanding grokking: An effective theory of representation learning. Advances in Neural Information Processing Systems , 35:34651--34663, 2022

    2022

  9. [17]

    The L -functions and modular forms database

    The LMFDB Collaboration . The L -functions and modular forms database. https://www.lmfdb.org, 2024. [Online; accessed 29 December 2024]

    2024

  10. [18]

    Comparison of the predicted and observed secondary structure of t4 phage lysozyme

    Brian W Matthews. Comparison of the predicted and observed secondary structure of t4 phage lysozyme. Biochimica et Biophysica Acta (BBA)-Protein Structure , 405(2):442--451, 1975

    1975

  11. [19]

    Progress measures for grokking via mechanistic interpretability

    Neel Nanda, Lawrence Chan, Tom Lieberum, Jess Smith, and Jacob Steinhardt. Progress measures for grokking via mechanistic interpretability. arXiv preprint arXiv:2301.05217 , 2023

    2023 arXiv

  12. [20]

    Predicting root numbers with neural networks

    Alexey Pozdnyakov. Predicting root numbers with neural networks. arXiv , 2024. arXiv:math.NT:2403.14631v1

    2024 arXiv

  13. [21]

    Understanding addition in transformers

    Philip Quirke and Fazl Barez. Understanding addition in transformers. arXiv preprint arXiv:2310.13121 , 2023

    2023 arXiv

  14. [22]

    Teaching transformers modular arithmetic at scale

    Eshika Saxena, Alberto Alfarano, Emily Wenger, and Kristin Lauter. Teaching transformers modular arithmetic at scale. arXiv preprint arXiv:2410.03569 , 2024

    2024 arXiv

  15. [23]

    Quelques applications du th \'e oreme de densit \'e de chebotarev

    Jean-Pierre Serre. Quelques applications du th \'e oreme de densit \'e de chebotarev. Publications Math \'e matiques de l'IH \'E S , 54:123--201, 1981

    1981

  16. [24]

    Sutherland

    Andrew V. Sutherland. A set of isogeny classes of elliptic curves of conductor up to 10^7 . https://math.mit.edu/ edgarc/data/ECQ7.txt

  17. [25]

    Stein and Mark Watkins

    William A. Stein and Mark Watkins. A database of elliptic curves---first report. In Algorithmic number theory ( S ydney, 2002) , volume 2369 of Lecture Notes in Comput. Sci. , pages 267--275. Springer, Berlin, 2002

    2002

  18. [26]

    Attention is all you need

    Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in Neural Information Processing Systems , volume 30, pages 5998--6008. Curran Associates, Inc., 2017

    2017

  19. [27]

    The clock and the pizza: Two stories in mechanistic explanation of neural networks

    Ziqian Zhong, Ziming Liu, Max Tegmark, and Jacob Andreas. The clock and the pizza: Two stories in mechanistic explanation of neural networks. Advances in Neural Information Processing Systems , 36, 2024

    2024

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