Data-Driven Insights into the Rank of Elliptic Curves of Prime Conductors

Abstract: In this presentation, we explore the intersection of data science and elliptic curves of prime conductor. We will begin with a quick introduction to elliptic curves before introducing the celebrated Birch and Swinnerton-Dyer conjecture. We will discuss the original insight of Birch and Swinnerton-Dyer concerning the traces of Frobenius and what they know about certain mathematical data attached to elliptic curves. We will be especially interested in the rank of elliptic curves of prime conductor. All along this talk, we will present experiments performed on the largest known dataset of such elliptic curves : the Bennett-Gherga-Retchnizer dataset, and will explicitly formulate open questions based on these observations. We will discuss some tension between data and the minimalist conjecture which stipulates that the average rank should be $\frac{1}{2}$. Among the various data scientific experiments that were performed, we will describe an interesting bias that exists between the distribution of the 2-torsion coefficients and the distribution of the rank. Finally we will discuss the importance of simple machine learning models for predicting the rank based on the traces of Frobenius.

machine learningmathematical physicsalgebraic geometryalgebraic topology

Audience: researchers in the topic

DANGER3: Data, Numbers, and Geometry