The comments to [this answer][1] 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] <cite authors="Lozano-Robledo, Álvaro">Lozano-Robledo, Álvaro, [*Ranks of Abelian varieties over infinite extensions of the rationals*](http://dx.doi.org/10.1007/s00229-008-0189-4), Manuscr. Math. 126.**3**, p. 393-407 (2008). [ZBL1157.11024](https://zbmath.org/?q=an:1157.11024).</cite>
  [1]: https://mathoverflow.net/a/213174