The density of rational points on $\mathbb{P}^1$ with three stacky points

TL;DR

This paper investigates the distribution of rational points on a specific stacky curve with three half points, confirming a conjecture by Ellenberg using a specialized height function.

Contribution

It proves Ellenberg's conjecture regarding the density of rational points on a particular stacky projective line with three half points.

Findings

Confirmed the conjecture of Ellenberg.

Established the density behavior of rational points on the stacky curve.

Utilized the Ellenberg-Satriano-Zuerick-Brown height in the analysis.

Abstract

In this paper we consider the density of rational points on the "stacky" curve $\mathcal{X}(\mathbb{P}^1;0,2;1,2;\infty,2)$ which is $\mathbb{P}^1$ with three half points, with respect to the so-called Ellenberg-Satriano-Zuerick-Brown height. In particular, we prove a conjecture of Ellenberg.