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.
Keywords
Cite
@article{arxiv.2502.07157,
title = {The Batyrev-Manin conjecture for DM stacks II},
author = {Ratko Darda and Takehiko Yasuda},
journal= {arXiv preprint arXiv:2502.07157},
year = {2025}
}
Comments
77 pages. v3: The abstract and the introduction were rewritten. "Main Speculation" in the previous version is now called a conjecture. Proposition 4.5 has been added. The nefness of raised line bundle was defined and added in the assumptions of the conjecture