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

Abstract

Let $p$ and $q$ be odd prime numbers with $q-p=2,$ the $\phi -$Selmer groups, Shafarevich-Tate groups ($\phi -$ 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.

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