The seminal theorem of Faltings confirms Mordell's conjecture: that is, curves of genus at least 2 have at most finitely many rational points. The proof of Faltings' theorem is not effective, meaning there is no way to estimate the number of rational points on a given curve.

What is the conjectured truth for these curves? In some cases, a curve with a large genus actually have no non-trivial rational points (such as Fermat curves). Do there exist algebraic curves with genus at least 2 and arbitrarily many rational points?

setof rational points on a given curve (or the set of solutions in Roth's theorem). It is, however, known that an effective version of the ABC conjecture would supply such a procedure. – Vesselin Dimitrov Mar 25 at 16:49