$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks
Somnath Jha, Dipramit Majumdar, Pratiksha Shingavekar
TL;DR
The paper investigates the $\varphi$-Selmer and $3$-Selmer groups of the family $E_{a,b}$ of elliptic curves with a rational $3$-isogeny over $K=Q(ζ_3)$. It relates the $\mathbb{F}_3$-ranks of these Selmer groups to the $3$-part of $S$-class groups of quadratic extensions $L=K(\sqrt{a})$ through explicit local data and $S$-set constructions, employing Cassels’ and Schaefer’s formulas to compare $\psi$- and $\widehat{\psi}$-Selmer groups. The authors derive sharp upper and lower bounds for $\dim_{\mathbb{F}_3}\mathrm{Sel}^{\psi_{a,b}}(E_{a,b}/K)$ and transfer them to bounds for $\mathrm{Sel}^3(E_{a,b}/K)$, enabling the construction of curves with arbitrarily large $3$-Selmer rank over $K$ (while maintaining no nontrivial $K$-rational $3$-torsion). They also show that a positive proportion of $E_{n,n}/\mathbb{Q}$ have root number $-1$ and $\mathrm{Sel}^3$-rank equal to $1$, highlighting nontrivial parity phenomena. Overall, the work links arithmetic of Selmer groups to ideal-class groups in CM settings and demonstrates unbounded growth of $3$-Selmer ranks in explicit families over a number field.
Abstract
We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$ with $a,b \in \mathbb{Z}$. These elliptic curves have a rational $3$-isogeny, say $\varphi$. We give an upper and a lower bound on the rank of the $\varphi$-Selmer group of $E_{a,b}$ over $K:=\mathbb{Q}(ζ_3)$ in terms of the $3$-part of the ideal class group of certain quadratic extension of $K$. Using our bounds on the Selmer groups, we construct infinitely many curves in this family with arbitrary large $3$-Selmer rank over $K$ and no non-trivial $K$-rational point of order $3$. We also show that for a positive proportion of natural numbers $n$, the curve $E_{n,n}/\mathbb{Q}$ has root number $-1$ and $3$-Selmer rank $=1$.