Counting extensions of number fields with Frobenius Galois group

2019/10/31 by Harsh Mehta, Mehta, Harsh
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Coding theory and cryptography #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1911.00121

openalex publication_date 2019/10/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a Frobenius group with an abelian Frobenius kernel F and let k be a finite extension of ℚ. We obtain an upper bound for the number of degree |F| algebraic extensions K/k with Galois group G with the norm of the discriminant N_k/ℚ(d_K/k) bounded above by X. We extend this method for any group G that has an abelian normal subgroup. If G has an abelian normal subgroup, then we obtain upper bounds for the number of degree |G| extensions N/k with Galois group G with bounded norm of the discriminant. Malle made a conjecture about what the order of magnitude of this quantity should be as the degree of the extension d and underlying Galois group G vary. We show that under the ℓ-torsion conjecture, the upper bounds we achieve for certain pairs d and G agree with the prediction of Malle. Unconditionally we show that the upper bound for the number of degree 6 extensions with Galois group A₄ also satisfies Malle's weak conjecture.

Citations

• Geometry-of-numbers methods over global fields I: Prehomogeneous vector spaces

Related

• Counting G-Extensions by Discriminant
• Enumerating Galois extensions of number fields
• On the Malle conjecture and the Grunwald problem
• The number of extensions of a number field with fixed degree and bounded discriminant
• Inductive methods for counting number fields