Uniformity of rational points an up-date and corrections

Abstract

The purpose of this note is to correct, and enlarge on, an argument in a paper we published a quarter century ago (J. Amer. Math. Soc. 10:1 (1997), 1–35). The question raised is a simple one to state: given that a curve $C$ of genus $g\ge 2$ defined over a number field $K$ has only finitely many rational points, we ask if the number of points is bounded as $C$ varies.