Rational points of bounded height on weighted projective stacks

TL;DR

This paper defines height functions on weighted projective stacks, generalizes classical height concepts, and analyzes the asymptotic distribution of rational points of bounded height on these stacks.

Contribution

It introduces a framework for heights on weighted projective stacks and studies the asymptotic count of rational points, extending classical height theories.

Findings

Established height functions generalizing classical notions

Derived asymptotic formulas for rational point counts

Applied to examples like elliptic curves and torsors

Abstract

A weighted projective stack is a stacky quotient $\mathscr P(\mathbf a)=(\mathbf A^n-\{0\})/\mathbb G_m$, where the action of $\mathbb G_m$ is with weights $\mathbf a\in\mathbb Z^n_{>0}$. Examples are: the compactified moduli stack of elliptic curves $\mathscr P(4,6)$ and the classifying stack of $\mu_m$-torsors $B\mu_m=\mathscr P(m)$. We define heights on the weighted projective stacks. The heights generalize the naive height of an elliptic curve and the absolute discriminant of a torsor. We use the heights to count rational points. We find the asymptotic behaviour for the number of rational points of bounded heights.