Machine-learning the Sato–Tate conjecture

He, Yang Hui; Lee, Kyu Hwan; Oliver, Thomas

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 efficiently.

Citation

He, Y. H., Lee, K. H., & Oliver, T. (2022). Machine-learning the Sato–Tate conjecture. Journal of Symbolic Computation, 111, 61-72. https://doi.org/10.1016/j.jsc.2021.11.002

• Journal Article Type: Article
• Acceptance Date: Nov 19, 2021
• Online Publication Date: Jan 10, 2022
• Publication Date: Jul 1, 2022
• Deposit Date: Feb 17, 2022
• Publicly Available Date: Jul 11, 2023
• Journal: Journal of Symbolic Computation
• Print ISSN: 0747-7171
• Electronic ISSN: 0747-7171
• Publisher: Elsevier
• Peer Reviewed: Peer Reviewed
• Volume: 111
• Pages: 61-72
• DOI: https://doi.org/10.1016/j.jsc.2021.11.002
• Keywords: Computational Mathematics; Algebra and Number Theory
• Public URL: https://nottingham-repository.worktribe.com/output/6786296
• Publisher URL: https://www.sciencedirect.com/science/article/pii/S0747717121000729#se0110
• Additional Information: This article is maintained by: Elsevier; Article Title: Machine-learning the Sato–Tate conjecture; Journal Title: Journal of Symbolic Computation; CrossRef DOI link to publisher maintained version: https://doi.org/10.1016/j.jsc.2021.11.002; Content Type: article; Copyright: © 2021 Elsevier Ltd. All rights reserved.