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The comments to this answer seem to make the following claim.

Claim. Let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$ of primes). Then for every elliptic curve $E$ over $K$, the group $E(K)$ is finitely generated.

However, the source cited (as well as further references found there) only prove a weaker statement:

Theorem (Kato, Ribet, Rohrlich). Let $E$ be an elliptic curve defined over $\mathbf Q$, and let $K$ be the maximal abelian extension of $\mathbf Q$ that is unramified away from $p$ (more generally, away from a finite set $S$ of primes). Then $E(K)$ is finitely generated.

See [Loz08, Thm. 1.1].

The difference between the two statements is that in the latter, $E$ needs to be defined over $\mathbf Q$, as opposed to over some finitely generated subfield $L \subseteq K$. Chasing the references given in [Loz08] does not help either: it seems that the results cited also assume that $E$ is defined over $\mathbf Q$. Given that there are quite a few results about elliptic curves that we currently only know over $\mathbf Q$, it seems plausible that this assumption is essential.

Question. Does there exist any infinite algebraic extension $\mathbf Q \subseteq K$ such that for every elliptic curve $E$ over $K$, the group $E(K)$ is finitely generated? (Or weaker: has finite rank?)

For example, is the claim above true?

I am also interested in abelian varieties, but that may be out of reach.

References.

[Loz08] Lozano-Robledo, Álvaro, Ranks of Abelian varieties over infinite extensions of the rationals, Manuscr. Math. 126.3, p. 393-407 (2008). ZBL1157.11024.

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    $\begingroup$ In the quoted reference the theorem is vaguely attributed to Kato, Ribet (?) and Rohrlich. I am not sure whether this really holds over $\mathbb{Q}$, already. My feeling, for instance, is that if $p$ is a prime of bad reduction for $E$, little can be said. Since given any $E/\mathbb{Q}$ there exists one $p$ where $E$ has bad reduction, how can you show that $E(\mathbb{Q}(\mu_{p^\infty}))$ is finitely generated? $\endgroup$
    – Filippo Alberto Edoardo
    Commented Jan 22, 2018 at 19:41
  • $\begingroup$ @FilippoAlbertoEdoardo: The Ribet attribution is probably for the finiteness of torsion; you need this on top of finite rank to deduce finite generation (although Rohrlich actually seems to prove finite generation). You might be right about the bad reduction actually; in Rohrlich's On $L$-functions of elliptic curves and cyclotomic towers, there seems to be an assumption on good reduction. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Jan 22, 2018 at 20:53
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    $\begingroup$ For the Theorem that I quoted in my paper (and that you quote above), see Greenberg's article in the 1999 PCMI volume "Arithmetic Algebraic Geometry", Theorems 1.4 and 1.5. As far as I can tell the assumption that $E$ is defined over $\mathbb{Q}$ is essential because the theorems assume that the curve is modular. $\endgroup$
    – Álvaro Lozano-Robledo
    Commented Jan 22, 2018 at 21:12
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    $\begingroup$ Ah, I see. Still, as far as I understand, Greenberg proves boundedness of the $p$-part of $\mathrm{Sel}(E(F(\mathbb{Q}_n)))$ and to deduce boundedness of the Mordell–Weil one needs finiteness of Sha (at least the $p$-part) all over the tower... $\endgroup$
    – Filippo Alberto Edoardo
    Commented Jan 22, 2018 at 23:11
  • $\begingroup$ For a nice survey in the Iwasawa theory setting, you can look at the introducion in Kim's paper in Journal of Number Theory 183 (2018) 352–387, expecially Theorem 1.2=5.17. $\endgroup$
    – Filippo Alberto Edoardo
    Commented Jan 25, 2018 at 7:35

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