Torsion of elliptic curves over imaginary quadratic fields of class number 1

Abstract

Filip Najman examined the possibilities for the group of torsion points on elliptic curves over the number fields $\mathbb {Q}(\sqrt {-1})$ and $\mathbb {Q}(\sqrt {-3})$ in Najman (2011, 2010). In this article, we study the possible torsion structures of elliptic curves over the remaining imaginary quadratic fields of class numbers $1$, i.e., over the fields $\mathbb {Q}(\sqrt {-2})$, $\mathbb {Q}(\sqrt {-7})$, $\mathbb {Q}(\sqrt {-11})$, $\mathbb {Q}(\sqrt {-19})$, $\mathbb {Q}(\sqrt {-43})$, $\mathbb {Q}(\sqrt {-67})$ and $\mathbb {Q}(\sqrt {-163})$.