The Batyrev-Manin conjecture for DM stacks II
Ratko Darda, Takehiko Yasuda
TL;DR
The paper develops a comprehensive framework for counting rational points on DM stacks in positive characteristic, emphasizing wild stacks and introducing a flexible height built from raised line bundles and sectoroids. It constructs twisted formal disks and arcs, defines augmented nef/effective cones, and formulates a Batyrev--Manin type conjecture for F-points. The work provides evidence across tame cases, classifying stacks, Fano varieties with canonical singularities, product compatibilities, and elliptic curves in characteristic 3, and demonstrates that conductor/discriminant based heights arise naturally in this setting. The approach unifies height theory for stacks in positive characteristic with classical Manin heuristics, offering tools to analyze asymptotic point counts and their dependence on ramification and orbifold structure.
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.