Unveiling Arithmetic Statistics of Congruent Number Elliptic Curves via Data Science and Machine Learning

This article presents a comprehensive data-scientific investigation into the arithmetic statistics of congruent number elliptic curves, leveraging a dataset of square-free integers up to $3$ million. We analyze the Mordell-Weil ranks, 2-Selmer ranks, and 3-Selmer ranks of the corresponding elliptic curves $E_D: y^2 = x^3 - D^2x$, where $D$ is a square-free number. Our study empirically examines the Heath-Brown heuristics, which predict the distribution of $2$-Selmer ranks as well as congruent numbers based on their residue modulo $8$. In particular, offering statistical insights into the proportion of numbers whose associated elliptic curves have positive rank. We provide a rigorous verification of Goldfeld's Conjecture in this context, analyzing the distribution of analytic ranks and demonstrating their alignment with the conjectured $50/50$ split for ranks $0$ and $1$. Furthermore, we explore the conjectural asymptotic distribution of $2-$ and $3$-torsion part of the Tate-Shafarevich group of these curves. Based on empirical evidence, we also suggest potential statistical distribution of $3$-Selmer and Mordell-Weil ranks. We also examine the averages of Frobenius traces and observe that they tend to zero without exhibiting any murmuration-like patterns. In addition to these number-theoretic analyses, we apply machine learning techniques to classify and predict congruent numbers, exploring the efficacy of computational methods in distinguishing congruent from non-congruent numbers based on the arithmetic properties of elliptic curves. This interdisciplinary approach blends advanced number theory with modern data science, providing empirical support for conjectures as well as discovery of new patterns.