Torsors for finite group schemes of bounded height

Abstract

Let be a nontrivial finite étale tame group scheme over a global field . We define height functions on the set of ‐torsors over , which generalize the usual heights such as the discriminant. As an analogue of the Malle conjecture for group schemes, we formulate a conjecture on the asymptotic behavior of the number of ‐torsors over of bounded height. This is a special case of our more general “stacky Batyrev–Manin” conjecture [18]. The conjectured asymptotic is proven for the case when is commutative. When is a number field, the leading constant is expressed as a product of certain arithmetic invariants of and the volume of a space attached to . Moreover, an equidistribution property of ‐torsors in the space is established.