A conjecture by Malle predicts the distribution of number fields of bounded discriminant. A related problem is to count only fields satisfying certain local conditions. But given a collection of local conditions, how do we know whether any number fields satisfying them exist at all?
In this talk, I will discuss a refinement of Malle’s conjecture in which local ramification conditions are imposed at almost all places of the number fields being counted. I will show how this problem can be viewed in terms of geometric objects called stacks, and how this new perspective provides a concrete way of checking for the existence of such number fields using techniques from rational point counting. Part of this talk is based on joint work with Daniel Loughran and Katerina Santicola.