This is more of a reference request in case anyone can direct me to the right literature.
If you have an elliptic curve $E/\mathbb Q$, and you consider the $\mathbb Z_p$ extension, $\mathbb Q_{\infty}$, then we know that the rank over $\mathbb Q_{\infty}$ is finite, which means that there must be a point in the tower that the rank stops growing. I wonder, are there any results that find exactly when this happens? Or can we at least find a number field in the tower above which the rank no longer grows, even if it's not the smallest one?