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CLASS GROUP STATISTICS FOR TORSION FIELDS GENERATED BY ELLIPTIC CURVES
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- Journal:
- Journal of the Australian Mathematical Society , First View
- Published online by Cambridge University Press:
- 02 December 2024, pp. 1-20
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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 all rational elliptic curves, ordered according to height. Our results are conditional, since we assume that the p-primary part of the Tate–Shafarevich group is finite. Furthermore, we assume predictions made by Delaunay for the statistical variation of the p-primary parts of Tate–Shafarevich groups. We also prove results in the case when the elliptic curve 
$E_{/\mathbb {Q}}$ is fixed and the prime p is allowed to vary.
