Machine learning invariants of arithmetic curves. (March 2023)

• Record Type: Journal Article
• Authors: He, Yang-Hui; Lee, Kyu-Hwan; Oliver, Thomas
• Is Part Of: Journal of symbolic computation. Volume 115(2023)
• Page Start: 478
• Page End: 491
• Publication Date: 2023-03
• Subjects: Machine-learning -- Arithmetic geometry -- Elliptic curves -- Hyper-elliptic curves -- Birch-Swinnerton-Dyer conjecture; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.08.017
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We show that standard machine learning algorithms may be trained to predict certain invariants of low genus arithmetic curves. Using datasets of size around 10 5, 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.