The Batyrev-Manin conjecture for DM stacks

TL;DR

This paper introduces a new height function for rational points on DM stacks, formulates conjectures on their distribution, and connects these to the Malle conjecture by considering orbifold invariants and thin subsets.

Contribution

It generalizes height functions and conjectures from varieties to DM stacks, including orbifold invariants and a new perspective on the Malle conjecture.

Findings

Defined a new height function on DM stacks

Formulated Batyrev-Manin type conjectures for DM stacks

Connected conjectures to the Malle conjecture and analyzed thin subsets

Abstract

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 Kl\"uners' counterexample to the Malle conjecture.