Machine-Learning the Sato--Tate Conjecture

TL;DR

This paper demonstrates that machine learning techniques can accurately and efficiently classify Sato-Tate groups of hyperelliptic curves using minimal data, supporting the Sato-Tate conjecture and outperforming traditional methods.

Contribution

The authors introduce a machine learning approach to classify Sato-Tate groups of hyperelliptic curves, achieving high accuracy with minimal data and rapid computation.

Findings

Bayesian classifier distinguishes Sato-Tate groups with 99-100% confidence.

Principal component analysis separates generic and non-generic groups in genus 2.

Machine learning outperforms traditional classification methods.

Abstract

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.