Machine-Learning Arithmetic Curves
This work provides a new computational tool for number theorists to classify arithmetic curves based on their invariants, potentially accelerating research in this domain.
This paper demonstrates that machine learning algorithms can predict invariants of low genus arithmetic curves. Using datasets of approximately 100,000 curves, the authors achieved high accuracies (over 0.97) in classifying elliptic and genus 2 curves based on their BSD invariants, including rank and torsion subgroup, with some specific tasks reaching 0.998 accuracy.
We show that standard machine-learning algorithms may be trained to predict certain invariants of low genus arithmetic curves. Using datasets of size around one hundred thousand, we demonstrate the utility of machine-learning in classification problems pertaining to the BSD invariants of an elliptic curve (including its rank and torsion subgroup), and the analogous invariants of a genus 2 curve. Our results show that a trained machine can efficiently classify curves according to these invariants with high accuracies (>0.97). For problems such as distinguishing between torsion orders, and the recognition of integral points, the accuracies can reach 0.998.