The comments to this answer seem to make the following claim.
Claim. Let $ K$ be the maximal abelian extension of $ \mathbb 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 $ \mathbb Q$ , and let $ K$ be the maximal abelian extension of $ \mathbb 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 $ \mathbb 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 $ \mathbb Q$ . Given that there are quite a few results about elliptic curves that we currently only know over $ \mathbb Q$ , it seems plausible that this assumption is essential.
Question. Does there exist any infinite algebraic extension $ \mathbb 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.
[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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