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arXiv · 2502.01069

$\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks

Somnath Jha·Dipramit Majumdar·Pratiksha Shingavekar
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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 $K:=\mathbb{Q}(\zeta_3)$ 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 $=1$.

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Somnath Jha, Dipramit Majumdar, Pratiksha Shingavekar. 2025-02-03. $\sqrt{-3}$-Selmer groups, ideal class groups and large $3$-Selmer ranks. https://arxiv.org/abs/2502.01069

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