Counting extensions of number fields with Frobenius Galois group

TL;DR

This paper establishes upper bounds on the number of algebraic extensions of number fields with a given Frobenius Galois group, aligning with Malle's conjecture under certain conjectural assumptions.

Contribution

It extends methods to bound the number of extensions with Frobenius Galois groups, including groups with abelian normal subgroups, and verifies Malle's conjecture in specific cases.

Findings

Upper bounds match Malle's conjecture under the $\, ext{l}$-torsion conjecture.

Unconditional bounds are provided for degree 6 extensions with Galois group $A_4$.

Method applies to groups with abelian normal subgroups, generalizing previous results.

Abstract

Let $G$ be a Frobenius group with an abelian Frobenius kernel $F$ and let $k$ be a finite extension of $\mathbb{Q}$. 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 $\mathcal{N}_{k/\mathbb{Q}}(d_{K/k})$ 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 $\ell$-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 $A_4$ also satisfies Malle's weak conjecture.