Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Malle's conjecture and Brauer groups of stacks

Daniel Loughran, Tim Santens

TL;DR

The paper develops a stack-theoretic framework for Malle's conjecture by studying the classifying stack BG of a finite group G, introducing a refined Brauer-group toolkit (including unramified and partially unramified Brauer groups) and a stacky Hensel theory to define heights and Tamagawa measures on BG. This enables a mass formula and precise leading-constant predictions, expressed as finite sums of Euler products and modulated by Brauer–Manin obstructions arising from breaking cocycles. It also integrates orbifold Picard theory, Iitaka-type fibrations, and the notion of balanced heights to explain when the Malle–Bhargava heuristics hold and how exceptional sets alter leading constants. The results apply to classical groups (e.g. S_n, D_4, A_4) and reveal intricate connections between ramification data, residuums along sectors, and arithmetic obstructions, with extensions to function fields. Overall, the work provides a coherent, computable framework linking Malle's conjecture to stack-theoretic invariants, enabling systematic leading-constant calculations and clarifying the role of Brauer obstructions in field counting problems.

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.

Malle's conjecture and Brauer groups of stacks

TL;DR

The paper develops a stack-theoretic framework for Malle's conjecture by studying the classifying stack BG of a finite group G, introducing a refined Brauer-group toolkit (including unramified and partially unramified Brauer groups) and a stacky Hensel theory to define heights and Tamagawa measures on BG. This enables a mass formula and precise leading-constant predictions, expressed as finite sums of Euler products and modulated by Brauer–Manin obstructions arising from breaking cocycles. It also integrates orbifold Picard theory, Iitaka-type fibrations, and the notion of balanced heights to explain when the Malle–Bhargava heuristics hold and how exceptional sets alter leading constants. The results apply to classical groups (e.g. S_n, D_4, A_4) and reveal intricate connections between ramification data, residuums along sectors, and arithmetic obstructions, with extensions to function fields. Overall, the work provides a coherent, computable framework linking Malle's conjecture to stack-theoretic invariants, enabling systematic leading-constant calculations and clarifying the role of Brauer obstructions in field counting problems.

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.

Paper Structure

This paper contains 109 sections, 145 theorems, 201 equations.

Table of Contents

  1. Introduction
  2. Malle's conjecture
  3. The Malle--Bhargava heuristics
  4. Malle's conjecture via stacks
  5. Main results
  6. Mass formula, Hensel's Lemma, and Tamagawa measures
  7. Brauer groups
  8. The exceptional set and the total count
  9. Structure of the paper
  10. Shortcut to the Conjecture:
  11. Notation and conventions
  12. The stack $BG$
  13. The groupoid $BG(k)$
  14. Thin sets
  15. Inner twists
  16. ...and 94 more sections

Key Result

Theorem 1.4

Let $v$ be a good place of $G$ and $w$ a weight function with height function $H$. Let $f: \mathcal{C}_G \to \mathbb{C}$ be any Galois equivariant function.

Theorems & Definitions (373)

  • Conjecture : Malle
  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Theorem 1.4: Mass formula
  • Theorem 1.5: Stacky Hensel's Lemma
  • Theorem 1.6: Convergence of global Tamagawa measure
  • Remark 1.7
  • Theorem 1.8: Purity for the unramified Brauer group
  • Theorem 1.9: Harari's formal lemma
  • ...and 363 more