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?



Curves of high genus with many rational points
Apparently it is a conjecture of Lang that there is a bound depending on the genus only. Check out [Lucia Caporaso's paper](http://www.mat.uniroma3.it/users/caporaso/CountingPoints.pdf) (I am sure there are more recent references).


