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Malle's conjecture and Brauer groups of stacks

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arxiv 2412.04196 v3 pith:AXH3P3AC submitted 2024-12-05 math.NT math.AG

Daniel Loughran , Tim Santens This is my paper · ORCID
classification math.NTmath.AG
keywords conjecturebrauerboundedgroupsmallestacksalgebraicbatyrev-manin
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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.

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Cited by 7 Pith papers

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  1. Inductive methods for counting number fields

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    Introduces a fiber-summation inductive method that proves new cases of Malle's conjecture and gives counterexamples to Malle's predicted exponents for wreath products.

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    Every smooth del Pezzo surface of degree at least 3 over an algebraically closed field in characteristic 0 is defined over its field of moduli, but degree-1 and degree-2 counterexamples exist over C/R.

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  4. A refined Malle conjecture for Heisenberg groups

    math.NT 2026-07 conditional novelty 7.0 of 10

    The leading constant for Malle's conjecture for Heis_4-extensions of Q decomposes as a sum of two Euler products due to a transcendental Brauer–Manin obstruction, yielding the first discriminant-ordering failure of lo...

  5. Counting abelian number fields with restricted ramification type

    math.NT 2025-07 conditional novelty 7.0 of 10

    For finite abelian G, G-extensions of bounded height with restricted tame ramification type satisfy an explicit Malle-type asymptotic whose constant is governed by a partially unramified Brauer group.

  6. Counterexamples for T\"urkelli's Modification on Malle's Conjecture

    math.NT 2025-02 conditional novelty 7.0 of 10

    Türkelli's modification to Malle's conjecture fails for infinite families of wreath-product groups, and a refined Malle conjecture with a corrected b-constant is proposed.

  7. Counting $D_4$-field extensions by multi-invariants

    math.NT 2025-07 conditional novelty 6.0 of 10

    For D4-Galois extensions of Q, the number with all four Gundlach multi-invariants bounded by X1, X2, X3, X4 is asymptotic to (27/8) times the Euler product over odd primes of (1-1/p)^4(1+4/p) times X1X2X3X4, provided ...

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