It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$. Much less is known if $K$ is infinite-dimensional over $\mathbb{Q}$.

The rank of $E$ over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},...)$ is infinite. It would however suffice for my purposes if I knew that for every prime $p$ there is an infinite family $Q$ of primes so that:

(1) The rank of $E(\mathbb{Q}(\sqrt{-q}))$ equals that of $E(\mathbb{Q})$ for all $q\in Q$.

(2) $p$ has the same prescribed splitting behavior (i.e. inert, split or ramified) in $\mathbb{Q}(\sqrt{-q})/\mathbb{Q}$ for all $q\in Q$.

(3) $p\notin Q$.

Is it known that such a family exists?

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$. Much less is known if $K$ is infinite-dimensional over $\mathbb{Q}$.

The rank of $E$ over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},...)$ is infinite. It would however suffice for my purposes if I knew that for every prime $p$ there is an infinite family $Q$ of primes, $p\notin Q$, so that:

(1) The rank of $E(\mathbb{Q}(\sqrt{-q}))$ equals that of $E(\mathbb{Q})$ for all $q\in Q$.

(2) $p$ has the same prescribed splitting behavior (i.e. inert, split or ramified) in $\mathbb{Q}(\sqrt{-q})/\mathbb{Q}$ for all $q\in Q$.

Is it known that such a family exists?

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$. Much less is known if $K$ is infinite-dimensional over $\mathbb{Q}$.

What is known about the rank of $E$ over $\mathbb{Q}(\sqrt{-q_{1}},\sqrt{-q_{2}},...)$?

The rank of $E$ over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},...)$ is infinite. It would however suffice for my purposes if I could show that for every prime $p$ there is an infinite family $Q$ of primes so that:

(1) The rank of $E(\mathbb{Q}(\sqrt{-q}))$ equals that of $E(\mathbb{Q})$ for all $q\in Q$.

(2) $p$ has the same prescribed splitting behavior (i.e. inert, split or ramified) in $\mathbb{Q}(\sqrt{-q})/\mathbb{Q}$ for all $q\in Q$.

It is known that the group of $K$-rational points of an elliptic curve $E$ is finitely generated if $K$ is a number field of finite degree over $\mathbb{Q}$. Much less is known if $K$ is infinite-dimensional over $\mathbb{Q}$.

The rank of $E$ over $\mathbb{Q}(\sqrt{-1},\sqrt{2},\sqrt{3},\sqrt{5},...)$ is infinite. It would however suffice for my purposes if I knew that for every prime $p$ there is an infinite family $Q$ of primes so that:

(1) The rank of $E(\mathbb{Q}(\sqrt{-q}))$ equals that of $E(\mathbb{Q})$ for all $q\in Q$.

(2) $p$ has the same prescribed splitting behavior (i.e. inert, split or ramified) in $\mathbb{Q}(\sqrt{-q})/\mathbb{Q}$ for all $q\in Q$.

(3) $p\notin Q$.

Is it known that such a family exists?