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?