Class group statistics for torsion fields generated by elliptic curves

TL;DR

This paper investigates the statistical behavior of class groups of torsion fields generated by elliptic curves over rationals, focusing on average non-vanishing of Galois-stable quotients, with results conditioned on conjectural models.

Contribution

It introduces a study of the average non-vanishing of Galois-stable quotients of class groups in torsion fields of elliptic curves, extending understanding of their distribution under conjectural frameworks.

Findings

Conditional results on non-vanishing averages for varying elliptic curves.

Results for fixed elliptic curve with varying prime p.

Insights into the structure of class groups in torsion fields.

Abstract

For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set $K=\mathbb{Q}(E[p])$ to denote the torsion field generated by $E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group $\operatorname{Cl}_K$ is a module over $\operatorname{Gal}(K/\mathbb{Q})$. Given a fixed odd prime number $p$, we study the average non-vanishing of certain Galois stable quotients of the mod-$p$ class group $\operatorname{Cl}_K/p\operatorname{Cl}_K$. Here, $E$ varies over rational elliptic curves, ordered according to \emph{height}. Our results are conditional and rely on predictions made by Delaunay and Poonen-Rains for the statistical variation of the $p$-primary parts of Tate-Shafarevich groups of elliptic curves. We also prove results in the case when the elliptic curve $E_{/\mathbb{Q}}$ is fixed and the prime $p$ is allowed to vary.