Rational points of bounded height on weighted projective stacks

Abstract

A weighted projective stack is a stacky quotient $\mathcal{P}(\mathbf{a})=({\mathbf{A}}^n-\{0\})/{\mathbb{G}}_m$, where the action of ${\mathbb{G}}_m$ is with weights $\mathbf{a}\in {\mathbb{Z}}_{>0}^n$. Examples are: the compactified moduli stack of elliptic curves $\mathcal{P}(4,6)$ and the classifying stack of ${\mu }_m$-torsors $B{\mu }_m=\mathcal{P}(m)$. 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.

Author's PhD thesis, submitted