Finite extensions preserve finiteness of Elliptic curve points

Question

Suppose $E/K$ is an elliptic curve over a field $\mathbb Q\subset K \subset \overline{\mathbb Q}$ such that $E(K)$ is finite. Let $[L:K]$ be a finite abelian extension. Is it necessarily the case that $E(L)$ is finite too?

Answer 1

As your question was already answered in the comments, I'll just add a couple of things.

Firstly how might you find an example if nobody else knew one already? You can use the LMFDB to do this! It's not the quickest method ever, but I went to http://www.lmfdb.org/EllipticCurve/2.2.5.1/?field=2.2.5.1&include_base_change=only to find elliptic curves over $\mathbf Q(\sqrt 5)$ which are base changes from $\mathbf Q$, clicked on a few till I found some of rank 1, such as http://www.lmfdb.org/EllipticCurve/2.2.5.1/729.1/b/3 then scrolled to the bottom of the page and found that this particular curve is a base change of http://www.lmfdb.org/EllipticCurve/Q/27/a/4 which has rank 0. It took a couple of minutes of clicking around to find this one. So we have another simple example of a curve $/\mathbf Q$ with finitely many rational points which grows to a rank 1 group under a small quadratic extension.

As for the generalities, it is a very interesting question of which abelian extensions can the rank grow under in general, and it is natural to reduce to cyclic extensions of degree $p$ a prime, as others can be built out of these. In this case there are number of conjectures floating around about how the rank can grow, and one conjecture is that for $p > 5$ there are only finitely many cyclic extensions $L/K$ of degree $p$ for which $E/K$ can have $\operatorname{rank} E(L) > \operatorname{rank} E(K)$.

For more on this you might enjoy Barry Mazur's talk http://www.math.harvard.edu/~mazur/papers/heuristics.Toronto.12.pdf see video at http://www.fields.utoronto.ca/video-archive/2016/11/1570-16199 or https://www.youtube.com/watch?v=BmnPn7mssvk