Machine Learning Approaches to the Shafarevich-Tate Group of Elliptic Curves

TL;DR

This paper applies machine learning models, including neural networks, to predict the order of the Shafarevich-Tate group of elliptic curves, achieving high accuracy and providing new insights into this complex mathematical object.

Contribution

It introduces a neural network classifier and regression model for predicting Shafarevich-Tate group orders, surpassing previous models in accuracy and enabling predictions on unseen data.

Findings

Neural network classifier achieves over 90% accuracy.

Regression model successfully predicts unseen group orders.

Applied models to a high-rank elliptic curve from recent discoveries.

Abstract

We train machine learning models to predict the order of the Shafarevich-Tate group 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 this group 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).