Malle's conjecture and Brauer groups of stacks

TL;DR

This paper proposes a new conjecture for the leading constant in Malle's conjecture on number fields, inspired by stacky versions of rational points conjectures on Fano varieties, and introduces a novel framework for Brauer groups of stacks.

Contribution

It introduces a new conjecture for Malle's leading constant and develops a framework for the unramified Brauer group of algebraic stacks, connecting number theory and algebraic geometry.

Findings

Formulation of a conjecture for the leading constant in Malle's conjecture

Development of a new notion of unramified Brauer group for stacks

Establishment of connections between number fields and stacky rational points

Abstract

We put forward a conjecture for the leading constant in Malle's conjecture on number fields of bounded discriminant, guided by stacky versions of conjectures of Batyrev-Manin, Batyrev-Tschinkel, and Peyre on rational points of bounded height on Fano varieties. A new framework for Brauer groups of stacks plays a key role in our conjecture, and we define a new notion of the unramified Brauer group of an algebraic stack.