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There are many partial results towards the following conjecture:

Every projective K3 surface over an algebraically closed field contains infinitely many integral rational curves.

My question is: is there anything known about rational curves on K3 surfaces over a fixed number field? Or at least over $\mathbb{Q}$. Any help/reference is much appreciated.

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    $\begingroup$ Welcome new contributor. What makes you believe there is any rational curve defined over $\mathbb{Q}$? A quartic Fermat hypersurface in projective $3$-space has no $\mathbb{Q}$-points, thus no copy of $\mathbb{P}^1_{\mathbb{Q}}$. Moreover, for the "generic" quartic surface, every zero-cycle on every curve has degree divisible by $4$. Since the anticanonical divisor class has degree $2$, there is no genus $0$ curve. $\endgroup$
    – Jason Starr
    Commented Mar 10, 2022 at 20:01
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    $\begingroup$ If $X$ is a K3 surface over a number field $K$ with $Aut(X)$ infinite (i.e., there is an automorphism $\sigma$ of infinite order), then there is a number field $L/K$ such that $X_L$ contains infinitely many pairwise distinct (split) rational curves. (That is, there are infinitely many pairwise distinct morphisms $\mathbb{P}^1_L\to X_L$.) This was proven by Bogomolov-Tschinkel. $\endgroup$
    – Ariyan Javanpeykar
    Commented Mar 11, 2022 at 8:18

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