Malle's Conjecture for Galois octic fields over \mathbb Q

2025/05/29 by Arul Shankar, Shankar, Arul, Ila Varma +1 · 1 citation
Mathematics · Computer Science · #Algebraic Geometry and Number Theory #Polynomial and algebraic computation #Commutative Algebra and Its Applications

paper · pdf · doi:10.48550/arxiv.2505.23690

Abstract

We compute the asymptotic number of octic number fields whose Galois groups over \mathbb Q are isomorphic to D₄, the symmetries of a square, when ordering such fields by their absolute discriminants. In particular, we verify the strong form of Malle's conjecture for such octic D₄-fields and obtain the constant of proportionality. Our result answers the question of whether a positive proportion of Galois octic extensions of \mathbb Q have non-abelian Galois group in the negative. We further demonstrate that the constant of proportionality satisfies the Malle--Bhargava principle of being a product of local masses, despite the fact that this principle does \em not hold for discriminants of quartic D₄-fields. This is the first instance of asymptotics being recovered for a non-concentrated family of number fields of Galois group neither abelian nor symmetric. Previously, this was only known for abelian fields, degree-n S_n-fields for n=3,4,5, and degree-6 S₃-fields.

Citations

• Inductive methods for counting number fields
• Malle's conjecture and Brauer groups of stacks
• On the asymptotics of cubic fields ordered by general invariants
• The Weak Form of Malle's Conjecture and Solvable Groups
• Mass formulas for local Galois representations
• The density of discriminants of quintic rings and fields
• Counting nilpotent Galois extensions

Cited by

• Asymptotics for 6-torsion and D₆-extensions

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• Lower bounds for counting A₄-quartic fields
• A counter example to Malle's conjecture on the asymptotics of discriminants
• Malle's conjecture with multiple invariants
• Counting D₄-field extensions by multi-invariants
• Counting C₂ \wr S₄ fields with a power saving error term