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√-3-Selmer groups, ideal class groups and large 3-Selmer ranks · arXivDesk

2502.01069Feb 3, 2025

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

Somnath Jha, Dipramit Majumdar, Pratiksha Shingavekar

Abstract

We consider the family of elliptic curves $E_{a,b}:y^2=x^3+a(x-b)^2$

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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

.

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