Let $K$ be a global field, does there always exist an elliptic curve $(E,e)$ over $K$, such that $E(K)=\{e\}$? (Is there an explicit way to find such a curve?)
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2$\begingroup$ If one could generalize the results of this paper arxiv.org/abs/1312.7859 to an arbitary global field that should be sufficient as 100% of elliptic curves have no torsion points. I don't know any obstruction to doing this but Arul Shankar, if no one else, should know. $\endgroup$– Will SawinCommented Dec 5, 2019 at 3:30
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2$\begingroup$ The answer is yes, if $K$ is a number field, see theorem 1.1 here (and if $K=\Bbb Q$, see also here). $\endgroup$– WatsonCommented Dec 5, 2019 at 17:41
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$\begingroup$ Possibly related: mathoverflow.net/questions/333406 $\endgroup$– WatsonCommented Jan 10, 2020 at 16:57
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