The Brauer-Manin obstruction for stacky curves

TL;DR

This paper demonstrates that the Brauer-Manin obstruction fully explains the failure of strong approximation and the integral Hasse principle for certain stacky curves over global fields, including genus 1/2 cases.

Contribution

It establishes the Brauer-Manin obstruction as the sole obstruction for all stacky curves with finite abelian fundamental groups and computes this obstruction for genus 1/2 cases.

Findings

Brauer-Manin obstruction is the only obstruction to strong approximation for these curves.

Elementary obstruction is the only obstacle to the integral Hasse principle in certain models.

Explicit computation of the Brauer-Manin obstruction for genus 1/2 stacky curves.

Abstract

We show that the Brauer-Manin obstruction is the only obstruction to strong approximation for all stacky curves over global fields with finite abelian fundamental groups. This includes all stacky curves of genus $g = \frac{1}{2}$, thus explaining a recent counterexample to the Hasse principle of Bhargava-Poonen. We will furthermore show that the elementary obstruction is the only obstruction to the integral Hasse principle for smooth proper integral models of stacky curves of genus $g < 1$. We then compute the Brauer-Manin obstruction for smooth proper integral models of stacky curves of genus $\frac{1}{2}$.