Mordell-Weil rank growth in Iwasawa tower

Question

This is more of a reference request in case anyone can direct me to the right literature. I asked originally on MathStack, but I was suggested to better post it here.

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?

On a similar flavor, which perhaps may partly answer the above, on his paper "On L functions of elliptic curves and cyclotomic towers", Rohrlich proved the following: If $E/\mathbb Q$ a CM elliptic curve and $P$ a finite set of good primes and $L$ is the compositum of $\mathbb Q(\zeta_{p^n})$ for all $n\geq 1$ and $p\in P$ then $E(L)$ is finitely generated by findng a number field $M'$ such that $E(L)=E(M')$. In particular, in the case that $P$ consists of a single prime, i.e $L=\mathbb Q(\zeta_{p^\infty})$, he describes how to explicitly construct $M'$. These results should extend to non CM elliptic curves. So my question is have these results been extended and have these constructions been made precise?

Answer 1

Thanks to the work of Kato, and his construction of Euler systems for modular forms, this can be extended to all elliptic curves over $\mathbb{Q}$. See $p$-adic Hodge theory and values of zeta functions of modular forms, Astérisque 295 (2004), Theorem 14.4: for every abelian variety $A/\mathbb{Q}$ which is a quotient of $J_1(N)$ for some $N \geq 1$, and for every $m \geq 1$, the abelian group $\bigcup_{n \geq 1} A(\mathbb{Q}(\zeta_{m^n}))$ is finitely generated.

Rohrlich's analytical result is valid for any newform of weight 2, CM or not, and you can proceed as Rohrlich to find the number field $M'$.