What is known about the structure of $J\_{0}(N)$ over $\mathbb{Q}[\mu\_{p^{\infty}}]$?


More generally, what do we know about $J\_{0}(N)$ over 
$\mathbb{Q}[\mu\_{p^{\infty}},k^{1/p^{n}}]$, where $k\in\mathbb{Z}$?


Does it have infinite rank? Does it have finite torsion?


I am especially interested in the case where $N$ is the conductor of an elliptic curve with additive reduction over $\mathbb{Q}$ at $p$. (This curve acquires, of course, a Néron model with semistable reduction st $p$ over $\mathbb{Q}[\mu\_{p^{\infty}}]$.)


EDIT:
In view of the previous comments, I would like to ask whether the $rank({J}\_{0}(N))$ is still finite over
$$K(k)\colon=\lim\_{\stackrel{\rightarrow}{n}}(\mathbb{Q}[{\mu}\_{{p}^{\infty}},{{k}}^{{\frac{{1}}{{{p}^{n}}}}}])$$



What do we know about the structure of $J_{0}(N)$ over $\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{\frac{{1}}{{p}^{n}}}])$?
Theorem 14.4 of Kato's paper in Asterisque 295 (2004), on page 236, says:



> 
> Let $A$ be an abelian variety over $\mathbf{Q}$ such that there is a surjective homomorphism $J\_1(N) \to A$ for some integer $N$. Then for any $m \ge 1$, $\bigcup\_n A(\mathbf{Q}(\mu\_{m^n}))$ is a finitely-generated abelian group.
> 
> 
> 


(I'll leave the case of Kummer extensions to others, but this answers the question in the title.)


