The Batyrev-Manin conjecture for DM stacks II

Abstract

In this paper, we propose a new framework for studying the distribution of rational points on DM stacks of positive characteristic. Our primary focus is on wild stacks, which existing frameworks do not address. There was not even a satisfactory notion of heights for such stacks. First, we introduce a new kind of height function that extends the authors' idea from their preceding paper on characteristic-zero stacks. This new height function is more general and flexible than the previous one. Examples of the new height function include discriminants of torsors, minimal discriminants, and conductors of elliptic curves in characteristic three. Next, we formulate a generalization of the Batyrev-Manin conjecture for rational points of DM stacks in positive characteristic relative to this new type of height function. We provide several pieces of evidence for this generalization.