Machine learning approaches to the Shafarevich-Tate group of elliptic curves
Abstract
We train machine learning models to predict the order of the Shafarevich-Tate group Ш of an elliptic curve over $\mathbb{Q}$. Building on earlier work of He, Lee, and Oliver, we show that a feed-forward neural network classifier trained on subsets of the invariants arising in the Birch--Swinnerton-Dyer conjectural formula yields higher accuracies ($> 0.9$) than any model previously studied. In addition, we develop a regression model that may be used to predict orders of Ш not seen during training and apply this to the elliptic curve of rank 29 recently discovered by Elkies and Klagsbrun. Finally we conduct some exploratory data analyses and visualizations on our dataset. We use the elliptic curve dataset from the $L$-functions and modular forms database (LMFDB).
Fields of Study & Subjects
Advanced Numerical Analysis Techniques
Cryptography and Residue Arithmetic
Publication details
- Journal / source
- Advances in Theoretical and Mathematical Physics
- Publisher
- International Press of Boston
- Volume
- 29
- Issue
- 8
- Pages
- 2353–2379
- Publication date
- 1 January 2026
- Language
- EN