Ranks of elliptic curves depend only on the field? Let $K/\mathbb{Q}$ be an algebraic extension, and let $E_1,E_2/\mathbb{Q}$ be elliptic curves. Is it possible that the Mordell-Weil rank of $E_1(K)$ is finite while that of $E_2(K)$ is infinite?
 A: If you're willing to replace $\mathbb Q$ by a finite (quadratic) extension $k$, then my recollection is that there are $\mathbb Z_p$ extensions $K$ of $k$ such that some elliptic curves over $k$ have finite Mordell-Weil group over $K$ and some have infinite rank. In any case, it would be worth looking up what's known about rank growth in $\mathbb Z_p$-extensions, including for $k=\mathbb Q$, starting with the work of Mazur. 
A: The answer is yes, at least if you assume that Tate-Shafarevich conjecture.
Let $E$ be an elliptic curve over a number field $k$. Under some mild hypothesis (see Corollary 1.10 of http://arxiv.org/abs/0904.3709) there exist quadratic twists of $E$ which have trivial Mordell-Weil rank. In other words, there exists quadratic extensions in which the rank of $E$ does not go up. Assuming in addition the Tate-Shafarevich conjecture there are also quadratic twists of $E$ whose Mordel-Weil rank is $1$ (see Corollary 1.12 loc. cit.), and hence quadratic extensions in which the rank of $E$ goes up. Generalizing the methods of loc. cit. along the line of http://arxiv.org/abs/1504.02343 one can show that (assuming the Tate-Shafaretich conjecture) if $E_1$ and $E_2$ are elliptic curves over $k$ satisfying certain independence conditions then one can find a quadratic extension in which the rank of $E_1$ stays the same and the rank of $E_2$ goes up. Applying this construction successively one can construct an infinite extension $K/k$ such that the rank of $E_1(K)$ is finite and the rank of $E_2(K)$ is infinite.
A: Are you thinking of the case $K=\overline{\mathbb{Q}}$? If $K$ is a finite extension, then the rank is always finite. And I think that for $K=\overline{\mathbb{Q}}$ it is always infinite. Maybe I'm missing something...
