Torsors for finite group schemes of bounded height

TL;DR

This paper introduces height functions for G-torsors over global fields, formulates a conjecture on their asymptotic distribution, and proves it for commutative G, linking it to arithmetic invariants and equidistribution.

Contribution

It generalizes height functions to finite group scheme torsors, formulates a conjecture analogous to Malle's, and proves it in the commutative case, connecting to the Stacky Batyrev-Manin conjecture.

Findings

Conjecture on asymptotics of G-torsors of bounded height.

Proof of the conjecture for commutative G.

Expression of the leading constant via arithmetic invariants.

Abstract

Let $F$ be a global field. Let $G$ be a non trivial finite \'etale tame $F$-group scheme. We define height functions on the set of $G$-torsors over $F,$ which generalize the usual heights such as discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of $G$-torsors over $F$ of bounded height. This is a special case of our more general Stacky Batyrev-Manin conjecture from arXiv:2207.03645. The conjectured asymptotic is proven for the case $G$ is commutative. When $F$ is a number field, the leading constant is expressed as a product of certain arithmetic invariants of $G$ and a volume of a space attached to $G$. Moreover, an equidistribution property of $G$-torsors in the space is established.