Rational Points on Curves

Nils Bruin, PIMS/SFU/UBC

We will discuss the problem of finding the set of rational solutions to a
polynomial equation in two variables. Such equations describe a plane
curve. It has been long known that the topology of the complex solution
set of such an equation largely determines the possibilities for the
rational solution set. In 1985, Faltings proved Mordell's conjecture
(1922) that many polynomial equations, corresponding to curves of
so-called general type, admit only finitely many rational solutions. His
proof does not provide a way of actually determining those solutions in
any particular case, though.