Lower bounds for counting A₄-quartic fields

2025/10/06 by Daniel Loughran, Loughran, Daniel, Ross Paterson +1
Mathematics · #14D23 #Advanced Topology and Set Theory #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Number Theory (math.NT) #Primary 11N45 #Secondary 11R20

paper · pdf · doi:10.48550/arxiv.2510.05248

openalex publication_date 2025/10/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A conjecture of Malle predicts the quantity of number fields with bounded discriminant of given Galois group. We present a lower bound matching this in the case of quartic fields with Galois group A₄.

Citations

• Counting quadratic points on Fano varieties
• Local solubility of generalised Fermat equations
• Malle's conjecture and Brauer groups of stacks
• The Batyrev-Manin conjecture for DM stacks
• Diagonal quartic surfaces with a Brauer-Manin obstruction
• Heights on stacks and a generalized Batyrev-Manin-Malle conjecture
• Uniform bounds for lattice point counting and partial sums of zeta functions
• Bounds on 2-torsion in class groups of number fields and integral points on elliptic curves
• Dirichlet series associated to cubic fields with given quadratic resolvent
• The density of discriminants of quintic rings and fields
• On character sums and class numbers

Related

• An improved error term for counting D₄-quartic fields
• Progress Towards Counting D₅ Quintic Fields
• Malle's Conjecture for Galois octic fields over \mathbb Q
• The number of quartic D₄-fields ordered by conductor
• Upper Bounds for the Number of Number Fields with Alternating Galois\n Group