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 $$K(k)\colon=\lim_{\stackrel{n}{\rightarrow}}(\mathbb{Q}[{\mu}_{{p}^{\infty}},{{k}}^{{\frac{{1}}{{{p}^{n}}}}}])$$