Class group statistics for torsion fields generated by elliptic curves

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 $E[p]:=\ker \{E\stackrel{p}{\to }E\}$. The class group ${\mathrm{Cl}}_K$ is a module over $\mathrm{Gal}(K/\mathbb{Q})$. Given a fixed odd prime number $p$, we study the average non-vanishing of certain Galois stable quotients of the mod-$p$ class group ${\mathrm{Cl}}_K/p{\mathrm{Cl}}_K$. 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.

Version 2: Minor corrections and expository improvements to the introduction. Paper accepted for publication in the Journal of the Australian Math Soc