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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] Lozano-Robledo, Álvaro, Ranks of Abelian varieties over infinite extensions of the rationals, Manuscr. Math. 126.3, p. 393-407 (2008). ZBL1157.11024. [1]: https://mathoverflow.net/a/213174

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.

The comments to [this answer][1] 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.

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. [1]: https://mathoverflow.net/a/213174

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] Lozano-Robledo, Álvaro, Ranks of Abelian varieties over infinite extensions of the rationals, Manuscr. Math. 126.3, p. 393-407 (2008). ZBL1157.11024. [1]: https://mathoverflow.net/a/213174

The comments to [this answer][1] 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.

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. [1]: https://mathoverflow.net/a/213174

The comments to [this answer][1] 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.

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. [1]: https://mathoverflow.net/a/213174