# Growth of Mordell-Weil Rank of Elliptic Curves over Field Extensions

I'm a graduate student just checking to make sure that what he's researching isn't already known.

Let $$\mathbb{F}$$ be a number field, and let $$E$$ be an elliptic curve defined over $$\mathbb{F}$$. Is it already known that, for any integer $$r\geq1$$, there exists a finite-degree extension $$\mathbb{K}$$ of $$\mathbb{F}$$ so that $$\textrm{rank}\left(E\left(\mathbb{K}\right)\right)\geq r$$?

For elliptic curves you can be more specific. For example, it's not hard to show that there are elements $$a_1,\ldots,a_r\in\mathbb{F}$$ so that for $$\mathbb{K}=\mathbb{F}(\sqrt{a_1},\ldots,\sqrt{a_r})$$ one has $$\operatorname{rank}E(\mathbb{K})\ge r$$. In particular, $$E$$ has infinite rank over the maximal abelian extension of $$\mathbb F$$ of exponent 2. However, if you ask for the minimal degree of an extension $$\mathbb K/\mathbb F$$ such that $$E(\mathbb K)$$ has rank $$r$$, then I think that you get an interesting question, although that, too, has certainly been studied.
• Could you outline or provide a reference regarding the first claim? I am aware of how to do that for $r=1$, but I'm not so sure how to do that for $r>1$. – Wojowu Feb 24 '19 at 22:30
• Outline: $E:y^2=f(x)$. Find values $x_1,x_2,\ldots,x_r$ so that the fields $\mathbb{F}(\sqrt{f(x_i)})$ are distinct. To do this, choose the $x_i$ successively so that each $f(x_i)$ is divisible by a prime (to an odd power) that doesn't divide any of the others. There's a similar, but rather more elaborate, argument in my paper with Rosen: On the independence of Heegner points associated to distinct quadratic imaginary fields. J. Number Theory 127 (2007), no. 1, 10–36. – Joe Silverman Feb 24 '19 at 23:45