Let F be a global field. Let G be a non trivial finite étale 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 behaviour of the number of G-torsors over F of bounded height. The conjectured asymptotic is proven for the case when G is commutative. As an application of our theory, we prove that the inverse Galois problem has an affirmative answer for what we call semicommutative group schemes.

We define a new height function on rational points of a DM (Deligne-Mumford) stack over a number field. This generalizes a generalized discriminant of Ellenberg-Venkatesh, the height function recently introduced by Ellenberg-Satriano-Zureick-Brown (as far as DM stacks over number fields are concerned), and the quasi-toric height function on weighted projective stacks by Darda. Generalizing the Manin conjecture and the more general Batyrev-Manin conjecture, we formulate a few conjectures on the asymptotic behavior of the number of rational points of a DM stack with bounded height. To formulate the Batyrev-Manin conjecture for DM stacks, we introduce the orbifold versions of the so-called a-and b-invariants. When applied to the classifying stack of a finite group, these conjectures specialize to the Malle conjecture, except that we remove certain thin subsets from counting. More precisely, we remove breaking thin subsets, which have been studied in the case of varieties by people including Hassett, Tschinkel, Tanimoto, Lehmann and Sengupta, and can be generalized to DM stack thanks to our generalization of a-and b-invariants. The breaking thin subset enables us to reinterpret Klüners' counterexample to the Malle conjecture. 9.3. Fano stacks revisited 39 9.4. The Malle conjecture revisited 40 9.5. Klüners' counterexample revisited 44 9.6. Comprehensiveness 46 References 47

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A semicommutative finite group scheme is a finite group scheme which can be obtained from commutative finite group schemes by iterated performing semidirect products with commutative kernels and taking quotients by normal subgroups. In this article, for an étale tame semicommutative finite group scheme G, we give a lower bound on the number of connected G-torsors of bounded height (such as discriminant).Definition 1.1.2. Denote by BG(F ) the set of G-torsors over F and for a place v of F , denote by BG(F v ) the set of G-torsors over the completion F v . We say that