Why do some equations have only finitely many integral solutions? For instance, Siegel showed that a polynomial equation in three variables with integral coefficients has only finitely many solutions if the associated complex space is hyperbolic. In this talk I will explain how Siegel's theorem fits in well with a conjecture of Serge Lang and Paul Vojta. The latter conjecture provides a framework which answers the above question by relating arithmetic properties of systems of polynomial equations to complex analysis. It naturally leads to other questions. For instance, are there only finitely many smooth projective hypersurfaces over the ring of integers? If we fix the degree and dimension and assume the Lang-Vojta conjecture, the answer to this question is positive. In this talk I will also explain how one proves the latter statement. Furthermore, I will explain how far one can get with current methods in arithmetic geometry without assuming Lang-Vojta's conjecture.
This is joint work with Daniel Loughran.