We know Lang conjecture can imply uniform boundless of rational points on higher genus (smooth projective) curves over a fixed number field by works of Mazur and others.

How is the conjecture of uniform boundness now? Are there some new results?

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We know Lang conjecture can imply uniform boundless of rational points on higher genus (smooth projective) curves over a fixed number field by works of Mazur and others.

How is the conjecture of uniform boundness now? Are there some new results?

anda uniform bound on the Mordell-Weil rank of Jacobians of curves of genus $g$, uniform boundedness of the number of rational points follows. $\endgroup$ – Michael Stoll Mar 23 at 13:12