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For $E$ a fixed elliptic curve (or more generally, abelian variety) over $\mathbb Q$, the condition "$E(K)$ is a finitely generated abelian group" is an answer to your question. It is satisfied by any finite extension $K$ of $\mathbb Q$, by the theorem of Mordell-Weil. If $K \subset L$ and the condition is satisfied by $L$, then it is also satisfied by $K$ since $E(K)$ is a subgroup of $E(L)$ and a subgroup of a finitely generated abelian group is still finitely generated.

The condition is definitely not satisfied for $K=\overline{ \mathbb Q}$. But a theorem of Ribet (combined with results of Kato, Röhrlich and Greenberg) states that this condition is satisfied for $K$ the maximal abelian extension of $\mathbb Q$ unramified outside a fixed set of primes. For other infinite extensions of $\mathbb Q$, the situation is very complex (certainly not fully understood even conjecturally), but there are a good deal of deep conjectures, and a good lot of them proved, concerning extensions of number fields whose Galois group is an analytic pro-$p$-group: this is the so-called "Iwasawa theory for elliptic curve". Of course, the abelian Iwasawa theory, where we deal with abelian such extension, that is of Galois group $\mathbb Z_p^n$, is more advanced, but by no means complete. For a thorough discussion of that theory, see Greenberg's Introduction to Iwasawa Theory for Elliptic Curves.

For $E$ a fixed elliptic curve (or more generally, abelian variety) over $\mathbb Q$, the condition "$E(K)$ is a finitely generated abelian group" is an answer to your question. It is satisfied by any finite extension $K$ of $\mathbb Q$, by the theorem of Mordell-Weil. If $K \subset L$ and the condition is satisfied by $L$, then it is also satisfied by $K$ since $E(K)$ is a subgroup of $E(L)$ and a subgroup of a finitely generated abelian group is still finitely generated.

The condition is definitely not satisfied for $K=\overline{ \mathbb Q}$. But a theorem of Ribet (combined with results of Kato, Rohrlich and Greenberg) states that this condition is satisfied for $K$ the maximal abelian extension of $\mathbb Q$ unramified outside a fixed set of primes. For other infinite extensions of $\mathbb Q$, the situation is very complex (certainly not fully understood even conjecturally), but there are a good deal of deep conjectures, and a good lot of them proved, concerning extensions of number fields whose Galois group is an analytic pro-$p$-group: this is the so-called "Iwasawa theory for elliptic curve". Of course, the abelian Iwasawa theory, where we deal with abelian such extension, that is of Galois group $\mathbb Z_p^n$, is more advanced, but by no means complete. For a thorough discussion of that theory, see Greenberg's Introduction to Iwasawa Theory for Elliptic Curves.

For $E$ a fixed elliptic curve (or more generally, abelian variety) over $\mathbb Q$, the condition "$E(K)$ is a finitely generated abelian group." It is satisfied by any finite extension $K$ of $\mathbb Q$, by the theorem of Mordell-Weil. If $K \subset L$ and the condition is satisfied by $L$, then it is also satisfied by $K$ since $E(K)$ is a subgroup of $E(L)$ and a subgroup of a finitely generated abelian group is still finitely generated.

The condition is definitely not satisfied for $K=\overline{ \mathbb Q}$. But a theorem of Ribet (combined with results of Kato, Röhrlich and Greenberg) states that this condition is satisfied for $K$ the maximal abelian extension of $\mathbb Q$ unramified outside a fixed set of primes. For other infinite extensions of $\mathbb Q$, the situation is very complex (certainly not fully understood even conjecturally), but there are a good deal of deep conjectures, and a good lot of them proved, concerning extensions of number fields whose Galois group is a pro-$p$-group (especially of the form $\mathbb Z_p^n$). For examples of such conjectures and theorem, see Greenberg's Introduction to Iwasawa Theory for Elliptic Curves.

For $E$ a fixed elliptic curve (or more generally, abelian variety) over $\mathbb Q$, the condition "$E(K)$ is a finitely generated abelian group" is an answer to your question. It is satisfied by any finite extension $K$ of $\mathbb Q$, by the theorem of Mordell-Weil. If $K \subset L$ and the condition is satisfied by $L$, then it is also satisfied by $K$ since $E(K)$ is a subgroup of $E(L)$ and a subgroup of a finitely generated abelian group is still finitely generated.

The condition is definitely not satisfied for $K=\overline{ \mathbb Q}$. But a theorem of Ribet (combined with results of Kato, Röhrlich and Greenberg) states that this condition is satisfied for $K$ the maximal abelian extension of $\mathbb Q$ unramified outside a fixed set of primes. For other infinite extensions of $\mathbb Q$, the situation is very complex (certainly not fully understood even conjecturally), but there are a good deal of deep conjectures, and a good lot of them proved, concerning extensions of number fields whose Galois group is an analytic pro-$p$-group: this is the so-called "Iwasawa theory for elliptic curve". Of course, the abelian Iwasawa theory, where we deal with abelian such extension, that is of Galois group $\mathbb Z_p^n$, is more advanced, but by no means complete. For a thorough discussion of that theory, see Greenberg's Introduction to Iwasawa Theory for Elliptic Curves.

For $E$ a fixed elliptic curve (or more generally, abelian variety) over $\mathbb Q$, the condition "$E(K)$ is a finitely generated abelian group." It is satisfied by any finite extension $K$ of $\mathbb Q$, by the theorem of Mordell-Weil. If $K \subset L$ and the condition is satisfied by $L$, then it is also satisfied by $K$ since $E(K)$ is a subgroup of $E(L)$ and a subgroup of a finitely generated abelian group is still finitely generated.

The condition is definitely not satisfied for $K=\overline{ \mathbb Q}$. But a theorem of Ribet states that this condition is satisfied for $K$ the maximal abelian extension of $\mathbb Q$ unramified outside a fixed set of primes. For other infinite extensions of $\mathbb Q$, the situation is very complex (certainly not fully understood even conjecturally), but there are a good deal of deep conjectures, and a good lot of them proved, concerning extensions of number fields whose Galois group is a pro-$p$-group (especially of the form $\mathbb Z_p^n$). For examples of such conjectures and theorem, see Greenberg's Introduction to Iwasawa Theory for Elliptic Curves.

For $E$ a fixed elliptic curve (or more generally, abelian variety) over $\mathbb Q$, the condition "$E(K)$ is a finitely generated abelian group." It is satisfied by any finite extension $K$ of $\mathbb Q$, by the theorem of Mordell-Weil. If $K \subset L$ and the condition is satisfied by $L$, then it is also satisfied by $K$ since $E(K)$ is a subgroup of $E(L)$ and a subgroup of a finitely generated abelian group is still finitely generated.

The condition is definitely not satisfied for $K=\overline{ \mathbb Q}$. But a theorem of Ribet (combined with results of Kato, Röhrlich and Greenberg) states that this condition is satisfied for $K$ the maximal abelian extension of $\mathbb Q$ unramified outside a fixed set of primes. For other infinite extensions of $\mathbb Q$, the situation is very complex (certainly not fully understood even conjecturally), but there are a good deal of deep conjectures, and a good lot of them proved, concerning extensions of number fields whose Galois group is a pro-$p$-group (especially of the form $\mathbb Z_p^n$). For examples of such conjectures and theorem, see Greenberg's Introduction to Iwasawa Theory for Elliptic Curves.