Class group statistics for torsion fields generated by elliptic curves
Anwesh Ray, Tom Weston
TL;DR
The paper investigates how often the mod-$p$ class group of the torsion field $K=\mathbb{Q}(E[p])$ contains a $G$-stable quotient isomorphic to $E[p]$, as $E$ varies or $p$ varies. It develops a framework relating $\operatorname{Cl}_K/p\operatorname{Cl}_K$ to $E[p]$ via the residual Galois representation and employs local-density and Tate–Shafarevich-group heuristics to derive conditional density bounds, including explicit lower bounds of order $p^{-1}$ for the set of curves with such quotients. For fixed $E$ and varying $p$, the paper shows that, under standard hypotheses (finite $\Sh$, non-CM, and almost-sure irreducibility), either almost all $p$ yield vanishing quotients when $\operatorname{rank}E(\mathbb{Q})\le 1$, or yield dimension bounds $\ge \operatorname{rank}E(\mathbb{Q})-1$ when $\operatorname{rank}E(\mathbb{Q})\ge 2$. The work combines Galois-module analysis, Prasad–Shekhar bounds, and Delaunay–Poonen–Rains heuristics, and supplements theory with $p=3$ computational data illuminating the interaction between $3$-Selmer groups and mod-$3$ 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.