Timeline for Artin representations appearing in Mordell-Weil groups of elliptic curves

8 events

when toggle format	what		by	license	comment
S Jul 25, 2021 at 12:07	history	bounty ended	CommunityBot
S Jul 25, 2021 at 12:07	history	notice removed	CommunityBot
Jul 17, 2021 at 10:43	comment	added	Joe Silverman		As Hindry said, one can use a height argument to get $\text{rank}\,E(\overline{\mathbb Q})=\infty$. But it's easier to just find points in lots of disjoint quadratic fields. Not that this answers your question.
S Jul 17, 2021 at 10:27	history	bounty started	R.P.
S Jul 17, 2021 at 10:27	history	notice added	R.P.		Draw attention
Sep 7, 2017 at 16:15	comment	added	Jeremy Rouse		If $K = \mathbb{Q}(\sqrt{d})$, then ${\rm rank} E(K) = {\rm rank} E(\mathbb{Q}) + {\rm rank} E_{d}(\mathbb{Q})$, where $E_{d}$ is the quadratic twist of $E$ by $d$. Hence, you can find an $E$ so that ${\rm rank} E(K) > {\rm rank} E(\mathbb{Q})$ just by taking any elliptic curve $F/\mathbb{Q}$ with positive rank and setting $E = F_{d}$.
Sep 7, 2017 at 13:34	comment	added	Chris Wuthrich		If the degree of $K$ is small, like $<8$, you should be able to pick a point $P\in\mathbb{P}^2(K)$ with well-chosen coordinates and then find a cubic $E$ that passes through $P$ (and its conjugates). That should answer it for a few representation, but won't get much further.
Sep 7, 2017 at 13:10	history	asked	François Brunault	CC BY-SA 3.0