Uniform Boundedness of Rational Points

One of the remarkable things about this theorem is the way in which it suggests that geometry informs arithmetic. The geometric genus g is a manifestly geometric condition, yet it is controlling what seems to be an arithmetic property. Why should the number of integral solutions to xn + yn = zn have anything to do with the shape of the complex solutions? You might argue that that the genus is essentially the same invariant as the degree in the the cases we discussed (plane conics and hyperelliptic curves). But smoothness, another morally geometric condition, is also crucial here. So both the “global” and “local” topological properties are reflected in the arithmetic behavior. Of course, when you see a great theorem you should ask how it can be generalized. We are going to discuss two possible and seemingly unrelated directions of generalization. The focus of the talk will then be the connection between the two.