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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?

fhn
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  • You should probably ask this question on Mathoverflow. The result you quote is a combination of a result of Rohrlich (Inventiones 1984) about the non-vanishing of twisted $L$-functions of modular forms (and hence, after Wiles et al, of elliptic curves), combined with Kato's work on BSD linking non-vanishing of $L$-functions to having algebraic rank $0$. I don't think Rohrlich's result is explicit, but one of the experts over at Mathoverflow can probably give you a better idea of the state of the art. – Mathmo123 Apr 08 '22 at 09:32
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    Now posted to MO, https://mathoverflow.net/questions/419936/mordell-weil-rank-growth-in-iwasawa-tower – Gerry Myerson Apr 09 '22 at 02:56

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