The Weak Form of Malle's Conjecture and Solvable Groups

TL;DR

This paper establishes upper bounds for counting solvable G-extensions of number fields with restricted local behavior, aligning with the weak form of Malle's conjecture, especially for nilpotent groups, under certain class group torsion assumptions.

Contribution

It proves upper bounds for G-extensions with restricted local behavior, confirming the weak form of Malle's conjecture for nilpotent groups and relating bounds to class group torsion.

Findings

Upper bounds match the weak form of Malle's conjecture for nilpotent groups.

Bounds depend on the size of class group torsion for non-nilpotent solvable groups.

Conditional results based on average class group torsion sizes.

Abstract

For a fixed finite solvable group $G$ and number field $K$, we prove an upper bound for the number of $G$-extensions $L/K$ with restricted local behavior (at infinitely many places) and ${\rm inv}(L/K)<X$ for a general invariant $"{\rm inv}"$. When the invariant is given by the discriminant for a transitive embedding of a nilpotent group $G\subset S_n$, this realizes the upper bound given in the weak form of Malle's conjecture. For other solvable groups, the upper bound depends on the size of torsion of the class group of number fields with fixed degree. In particular, the bounds we prove realize the upper bound given in the weak form of Malle's conjecture for the transitive embedding of a solvable group $G\subset S_n$ if we assume that for each finite abelian group $A$ the average size of class group torsion $|{\rm Hom}({\rm Cl}(L),A)|$ is smaller than $X^{\epsilon}$ as $L/K$ varies over certain families of extensions with ${\rm inv}(L/K)<X$.