Uniformity of rational points: an up-date and corrections

Abstract

In [2] it is asserted that, assuming the truth of the Strong Lang Conjecture (Conjecture 1 below), a very strong form of boundedness holds: for every g ≥ 2 there is a finite bound N(g)—not depending on K!—such that for any number field K there are only finitely many isomorphism classes of curves of genus g defined over K with more than N(g) K-rational points. The issue is, in that statement do we mean finitely many isomorphism classes over K, or over the algebraic closure K? The paper asserts the statement in the stronger form—up to isomorphism over K—but the proof establishes only the weaker statement that there are finitely many curves with more than N(g) points up to isomorphism over K.