City Research Online

Machine-learning the Sato–Tate conjecture

He, Y-H. ORCID: 0000-0002-0787-8380, Lee, K. H. & Oliver, T. (2021). Machine-learning the Sato–Tate conjecture. Journal of Symbolic Computation, 111, pp. 61-72. doi: 10.1016/j.jsc.2021.11.002

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.

Publication Type: Article
Additional Information: © 2021. This article has been published in Journal of Symbolic Computation by Elsevier. This manuscript version is made available under the CC-BY-NC-ND 4.0 license https://creativecommons.org/licenses/by-nc-nd/4.0/
Publisher Keywords: Sato-Tate conjecture, Machine-learning, Classifiers, L-functions, Hyper-elliptic curves, Arithmetic geometry
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA75 Electronic computers. Computer science
Departments: School of Mathematics, Computer Science & Engineering > Mathematics
[img] Text - Accepted Version
This document is not freely accessible until 29 May 2023 due to copyright restrictions.
Available under License Creative Commons Attribution Non-commercial No Derivatives.

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