Frey and Jarden asked if any abelian variety over a number field $K$ has the infinite Mordell-Weil rank over the maximal abelian extension $K^{\operatorname{ab}}$. In this paper, we give an affirmative answer to their conjecture for the Jacobian variety of any smooth projective curve $C$ over $K$ such that $\sharp C(K^{\operatorname{ab}})=\infty$ and for any abelian variety of $\operatorname{GL}_2$-type with trivial character.