The torsion subgroup of the elliptic curve $Y^2 = X^3 + AX$ over the maximal abelian extension of $\mathbb{Q}$

Abstract

We determine explicitly the structure of the torsion group over the maximal abelian extension of $\mathbb{Q}$ and over the maximal $p$-cyclotomic extensions of $\mathbb{Q}$ for the family of rational elliptic curves given by $Y^2 = X^3 + AX$, where $A$ is an integer.