Machine Learning Meets Number Theory: The Data Science of Birch-Swinnerton-Dyer

Empirical analysis is often the first step towards the birth of a conjecture. This is the case of the Birch-Swinnerton-Dyer (BSD) Conjecture describing the rational points on an elliptic curve, one of the most celebrated unsolved problems in mathematics. Here, we extend the original empirical approach to the analysis of the Cremona database of quantities relevant to BSD, inspecting more than 2.5 million elliptic curves by means of the latest techniques in data science, machine learning and topological data analysis. Key quantities such as rank, Weierstraβ coefficients, period, conductor, Tamagawa number, regulator and order of the Tate-Shafarevich group give rise to a high-dimensional point cloud whose statistical properties we investigate. We reveal patterns and distributions in the rank versus Weierstraβ coefficients, as well as the Beta distribution of the BSD ratio of the quantities. Via gradient-boosted trees, machine learning is applied in finding inter-correlation among the various quantities. We anticipate that our approach will spark further research on the statistical properties of large datasets in Number Theory and more in general in pure Mathematics.