The Batyrev-Manin conjecture for stacks

This talk is based on my joint works with Ratko Darda. The Batyrev-Manin conjecture predicts the growth of the number of rational points of bounded height. Malle’s conjecture is a conjecture of similar kind, but it deals with extensions of a fixed number field, rather than rational points of an algebraic variety. I will talk about how these two conjectures are unified by considering rational points of stacks. We may try to formulate a similar conjecture for function fields in place of number fields. However, the case where stabilizers of the stack have order divisible by the characteristic of the base field is the so-called wild case and known to be very difficult to analyze. Our recent work has succeeded in extending predictions even to the wild case. This stacky approach would provide geometric insights to the problem of counting field extensions as well as enable us to address the problem of counting various objects which can be interpreted as rational points of a moduli stack.