On the Selmer groups and Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x \pm q) $ over imaginary quadratic number fields of class number one

TL;DR

This paper explicitly determines the Selmer groups, Shafarevich-Tate groups, and Mordell-Weil groups of specific elliptic curves over imaginary quadratic fields of class number one, focusing on cases where p and q are odd primes with q-p=2.

Contribution

It provides explicit calculations of Selmer and Shafarevich-Tate groups for these elliptic curves over certain quadratic fields, extending understanding of their arithmetic properties.

Findings

Explicit descriptions of Selmer groups for the curves.

Determination of Shafarevich-Tate groups in many cases.

Insights into Mordell-Weil ranks over imaginary quadratic fields.

Abstract

Let $ p $ and $ q $ be odd prime numbers with $ q - p = 2, $ the $\varphi -$Selmer groups, Shafarevich-Tate groups ($ \varphi - $ and $ 2-$part) and their dual ones as well the Mordell-Weil groups of elliptic curves $ y^{2} = x (x \pm p) (x \pm q) $ over imaginary quadratic number fields of class number one are determined explicitly in many cases.