Counting extensions of number fields with Frobenius Galois group

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.