Inductive methods for counting number fields

TL;DR

This paper introduces a novel inductive approach for counting number field extensions by discriminant, leading to new results and counterexamples related to Malle's Conjecture, applicable to various Galois groups.

Contribution

The paper presents a new inductive method for counting number fields that allows proving new cases and counterexamples to Malle's Conjecture, expanding the understanding of extension distributions.

Findings

Proved many new cases of Malle's Conjecture.

Constructed counterexamples to Malle's Conjecture.

Developed a general framework applicable to various Galois groups.

Abstract

We give a new method for counting extensions of a number field asymptotically by discriminant, which we employ to prove many new cases of Malle's Conjecture and counterexamples to Malle's Conjecture. We consider families of extensions whose Galois closure is a fixed permutation group $G$. Our method relies on having asymptotic counts for $T$-extensions for some normal subgroup $T$ of $G$, uniform bounds for the number of such $T$-extensions, and possibly weak bounds on the asymptotic number of $G/T$-extensions. However, we do not require that most $T$-extensions of a $G/T$-extension are $G$-extensions. Our new results use $T$ either abelian or $S_3^m$, though our framework is general.