Beyond elliptic curves

A special case of an even deeper theorem of Faltings shows that if $X$ is a nonsingular³curve defined over ${\mathbb{Q}}$ of degree greater than 3, then there are only finitely many rational points on $X$. Faltings was awarded the Fields Medal (the mathematical equivalent of the Nobel Prize) for proving this theorem.

From the algorithmic point of view, however, things are still very mysterious: it is not known whether there is a method for actually listing the rational points on a given nonsingular curve of degree greater than 3.

For example, the French mathematician Jean-Pierre Serre challenged the mathematical community many years ago to prove that the eight obvious rational points on $x^4+y^4=17$ are the only ones; it took until 2001 for this to be proved.