On the essential torsion finiteness of abelian varieties over torsion fields

The classical Mordell-Weil theorem implies that an abelian variety $A$ over a number field $K$ has only finitely many $K$-rational torsion points. This finitude of torsion still holds even over the cyclotomic extension $K^{\mathrm{c}\mathrm{y}\mathrm{c}}=K{\mathbb{Q}}^{\mathrm{a}\mathrm{b}}$ by a result of Ribet. In this article, we consider the finiteness of torsion points of an abelian variety $A$ over the infinite algebraic extension $K_B$ obtained by adjoining the coordinates of all torsion points of an abelian variety $B$. Assuming the Mumford-Tate conjecture, and up to a finite extension of the base field $K$, we give a necessary and sufficient condition for the finiteness of $A(K_B{)}_{\mathrm{t}\mathrm{o}\mathrm{r}\mathrm{s}}$ in terms of Mumford–Tate groups. We give a complete answer when both abelian varieties have dimension both three, or when both have complex multiplication.