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 $E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. 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 nonvanishing 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 PoonenRains for the statistical variation of the $p$primary parts of TateShafarevich 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2022
 DOI:
 10.48550/arXiv.2204.09757
 arXiv:
 arXiv:2204.09757
 Bibcode:
 2022arXiv220409757R
 Keywords:

 Mathematics  Number Theory;
 11G05;
 11R29;
 11R32;
 11R34;
 11R45
 EPrint:
 Version 2: Minor corrections and expository improvements to the introduction. Paper accepted for publication in the Journal of the Australian Math Soc