# Machine-Learning the Sato--Tate Conjecture

arXiv: https://arxiv.org/abs/2010.01213 · HTML: https://mlsanity.com/paper/2010.01213/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 2 Oct 2020.

## Built on by (most cited first, 14 of 14)

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- [Machines Learn Number Fields, But How? The Case of Galois Groups](https://mlsanity.com/paper/2508.06670/) (2025)
- [Murmurations, Mestre--Nagao sums, and Convolutional Neural Networks for elliptic curves](https://mlsanity.com/paper/2603.17681/) (2026)
- [Machine-Learning Kronecker Coefficients](https://mlsanity.com/paper/2306.04734/) (2023)
- [Machine Learning Mutation-Acyclicity of Quivers](https://mlsanity.com/paper/2411.04209/) (2024)
- [Learning Fricke signs from Maass form Coefficients](https://mlsanity.com/paper/2501.02105/) (2025)
- [Learning to be Simple](https://mlsanity.com/paper/2312.05299/) (2023)
- [Computational Experiments in Number Theory](https://mlsanity.com/paper/2504.19451/) (2025)
- [Murmurations: a case study in AI-assisted mathematics](https://mlsanity.com/paper/2603.09680/) (2026)
- [Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves](https://mlsanity.com/paper/2607.24251/) (2026)

## Abstract (arXiv)

We apply some of the latest techniques from machine-learning to the arithmetic of hyperelliptic curves. More precisely we show that, with impressive accuracy and confidence (between 99 and 100 percent precision), and in very short time (matter of seconds on an ordinary laptop), a Bayesian classifier can distinguish between Sato-Tate groups given a small number of Euler factors for the L-function. Our observations are in keeping with the Sato-Tate conjecture for curves of low genus. For elliptic curves, this amounts to distinguishing generic curves (with Sato-Tate group SU(2)) from those with complex multiplication. In genus 2, a principal component analysis is observed to separate the generic Sato-Tate group USp(4) from the non-generic groups. Furthermore in this case, for which there are many more non-generic possibilities than in the case of elliptic curves, we demonstrate an accurate characterisation of several Sato-Tate groups with the same identity component. Throughout, our observations are verified using known results from the literature and the data available in the LMFDB. The results in this paper suggest that a machine can be trained to learn the Sato-Tate distributions and may be able to classify curves much more efficiently than the methods available in the literature.