Uniformity of rational points: an up-date and corrections (Q2124313)

In this paper, the authors fill in a gap in the proof of the main result of their previous paper [\textit{L. Caporaso} et al., J. Am. Math. Soc. 10, No. 1, 1--35 (1997; Zbl 0872.14017)]. Namely, the main result of [loc. cit.] asserts that assuming a strong form of Lang's conjecture, the following holds for any fixed genus \(g \geq 2\): There is a bound \(N(g)\), such that for any number field \(K\), there are only finitely many isomorphism classes of curves over \(K\) of genus \(g\) that have more than \(N(g)\) \(K-\)rational points. The proof used in [loc. cit.] establishes this only for curves defined over the algebraic closure \(\overline K\) of \(K\), and in the current paper the authors provide an argument fixing this gap.