Multi-height distribution of rational points of split toric stacks

TL;DR

This paper investigates the distribution of rational points on split toric stacks over $ extbf{Q}$ by lifting the counting problem to an extended universal torsor, enabling a new approach to height and Tamagawa number calculations.

Contribution

It introduces an integral parametrization of rational points on toric stacks and defines the Tamagawa number as an Euler product, linking heights to point counting over finite fields.

Findings

Established the existence of an integral parametrization for rational points.

Defined the Tamagawa number of a toric stack as an Euler product.

Interpreted the $p$-adic factor via a mass formula for $ extbf{F}_p$-points.

Abstract

We study the distribution of rational points of split toric stacks with all heights bounded over $\mathbf{Q}$ by lifting the counting problem to an extended universal torsor under the torus associated with the orbifold Picard group. To achieve this, we prove the existence of an integral parametrization of rational points on toric stacks, which allows us to define a lift of the stacky height to this extended universal torsor. This allows us to define the Tamagawa number of a toric stack $X$ as an Euler product and, for a prime number $p$, to interpret the $p$-adic factor via a mass formula counting $\mathbf{F}_p$-points of the sectors of $X$.