  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.
