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I am interested what is known about the following statement:

For every number field $K$, there exists an elliptic curve $E$ defined over $K$ with algebraic rank equal to $1$.

Is this statement known to be true unconditionally? If not, has this been conjectured anywhere in the literature?

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    $\begingroup$ See this related question. The references in the question prove this result conditionally on some standard conjectures. $\endgroup$
    – Wojowu
    Commented Jul 22 at 17:23
  • $\begingroup$ Thanks a lot for linking this question! I am aware of the Mazur--Rubin reference, and would be specifically interested in what (if anything) is known unconditionally. $\endgroup$
    – P. Koymans
    Commented Jul 22 at 17:27
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    $\begingroup$ @P.Koymans There's a preprint of Bhargava Shankar Wang "Geometry-of-numbers methods over global fields II" which calculates the average 3 Selmer group over a general number field. Then assuming that TS groups are finite and a version of the p-parity conjecture over $K$ one should be able to deduce that a positive density of elliptic curves have rank 1, generalizing the argument of Theorem 41 of "Ternary cubic forms having bounded invariants..."-Bhargava Shankar. One can also look into the paper of Bhargava and Skinner in the JRMS but I think that will also require some version of p-parity. $\endgroup$
    – Anwesh Ray
    Commented Jul 23 at 2:50

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