The local-global principle for integral points on stacky curves

TL;DR

This paper constructs a specific stacky curve of genus 1/2 over integers that has local points everywhere but no global integral point, demonstrating a failure of the local-global principle in this setting.

Contribution

It provides a counterexample of a genus 1/2 stacky curve over integers violating the local-global principle for integral points.

Findings

Constructed a genus 1/2 stacky curve with local points everywhere but no global integral point.

Proved that all stacky curves of genus less than 1/2 over rings of S-integers satisfy the local-global principle.

Abstract

We construct a stacky curve of genus $1/2$ (i.e., Euler characteristic $1$) over $\mathbb{Z}$ that has an $\mathbb{R}$-point and a $\mathbb{Z}_p$-point for every prime $p$ but no $\mathbb{Z}$-point. This is best possible: we also prove that any stacky curve of genus less than $1/2$ over a ring of $S$-integers of a global field satisfies the local-global principle for integral points.