Selmer Groups in Twist Families of Elliptic Curves

TL;DR

This paper provides numerical data on Selmer groups in twist families of elliptic curves, assuming BSD, and analyzes their distribution using Waldspurger's theorem for efficient computation.

Contribution

It introduces a computational approach for Selmer groups in elliptic curve twists under BSD, with data-driven distribution analysis based on logarithmic functions.

Findings

Distribution of Selmer group orders fits a specific logarithmic function

The optimal parameter depends on the elliptic curve's conductor

Data supports conjectural models of Selmer group behavior

Abstract

The aim of this article is to give some numerical data related to the order of the Selmer groups in twist families of elliptic curves. To do this we assume the Birch and Swinnerton-Dyer conjecture is true and we use a celebrated theorem of Waldspurger to get a fast algorithm to compute $% L_{E}(1)$. Having an extensive amount of data we compare the distribution of the order of the Selmer groups by functions of type $\alpha \frac{(\log \log (X))^{1+\varepsilon}}{\log (X)}$ with $\varepsilon $ small. We discuss how the "best choice" of $\alpha $ is depending on the conductor of the chosen elliptic curves and the congruence classes of twist factors.