The Batyrev-Manin conjecture for DM stacks II

TL;DR

This paper extends the Batyrev-Manin conjecture to Deligne-Mumford stacks in positive characteristic by introducing a new height function suitable for wild stacks, supported by several evidences.

Contribution

It proposes a new height function for wild DM stacks in positive characteristic and generalizes the Batyrev-Manin conjecture accordingly.

Findings

Introduced a flexible height function for wild stacks

Formulated a generalized Batyrev-Manin conjecture

Provided supporting evidence for the conjecture

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.