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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 $\textrm{rank}\left(E\left(\mathbb{K}\right)\right)\geq r$?

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This is (unfortunately for you) well-known. Theorem 10.1 of this paper shows that over an algebraically closed field of characteristic zero the Mordell-Weil rank of any positive-dimensional abelian variety is infinite, and (as remark 1 below the theorem mentions) this implies that for any abelian variety over field of characteristic zero, there are finite extensions over which the ranks get arbitrarily large.

The result also holds for fields of positive characteristic which are not algebraic over a finite field.

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  • $\begingroup$ Ah well. Thanks for the prompt response. $\endgroup$ – MCS Feb 24 '19 at 22:03
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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.

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  • $\begingroup$ 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 $r>1$. $\endgroup$ – Wojowu Feb 24 '19 at 22:30
  • $\begingroup$ 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. $\endgroup$ – Joe Silverman Feb 24 '19 at 23:45

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