# Machine-Learning Arithmetic Curves

arXiv: https://arxiv.org/abs/2012.04084 · HTML: https://mlsanity.com/paper/2012.04084/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 Dec 2020.

## Built on by (most cited first, 10 of 10)

- [Murmurations of elliptic curves](https://mlsanity.com/paper/2204.10140/) (2022)
- [From Black Box to Bijection: Interpreting Machine Learning to Build a Zeta Map Algorithm](https://mlsanity.com/paper/2511.12421/) (2025)
- [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)
- [Murmurations: a case study in AI-assisted mathematics](https://mlsanity.com/paper/2603.09680/) (2026)
- [Transformers Learn the Mestre-Nagao Heuristic](https://mlsanity.com/paper/2606.15036/) (2026)
- [Decision trees, Frobenius traces, and Weierstrass coefficients of elliptic curves](https://mlsanity.com/paper/2607.24251/) (2026)

## Abstract (arXiv)

We show that standard machine-learning algorithms may be trained to predict certain invariants of low genus arithmetic curves. Using datasets of size around one hundred thousand, we demonstrate the utility of machine-learning in classification problems pertaining to the BSD invariants of an elliptic curve (including its rank and torsion subgroup), and the analogous invariants of a genus 2 curve. Our results show that a trained machine can efficiently classify curves according to these invariants with high accuracies (>0.97). For problems such as distinguishing between torsion orders, and the recognition of integral points, the accuracies can reach 0.998.