Machine-learning the Sato-Tate conjecture

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

TitleMachine-learning the Sato-Tate conjecture
TypeJournal article
AuthorsHe, Y.-H., Lee, K.-H. and Oliver, T.
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.

JournalJournal of Symbolic Computation
Journal citation111, pp. 61-72
ISSN1095-855X
0747-7171
Year2022
PublisherElsevier
Accepted author manuscript
License
CC BY-NC-ND 4.0
File Access Level
Open (open metadata and files)
Digital Object Identifier (DOI)https://doi.org/10.1016/j.jsc.2021.11.002
Web address (URL)https://www.sciencedirect.com/science/article/abs/pii/S0747717121000729
Publication dates
PublishedJul 2022
Published online29 Nov 2021

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