Inductive methods for counting number fields
Brandon Alberts, Robert J. Lemke Oliver, Jiuya Wang, Melanie Matchett Wood
TL;DR
The paper develops an inductive framework to count G-extensions of number fields ordered by discriminant, by partitioning along a normal subgroup $T \lhd G$ and pushing counts to the quotient $G/T$. It leverages the Twisted Malle conjecture (via Abelts–O'Dorney for abelian $T$ and related S3-structures) to bound fibers and then sums over fibers under a convergence criterion, yielding asymptotics of the form $c X^{1/a} (\log X)^{b-1}$ for many concentrated groups. Core tools include sharp $H^1_{ur}(k,T(\pi))$ bounds tied to class-group torsion, pushforward discriminants for imprimitive extensions, and a detailed complex-analytic treatment of local Euler products to obtain uniform bounds. The results substantially broaden the families of groups for which Malle’s weak and strong conjectures (in various formulations) hold or fail, including numerous nilpotent and wreath-product cases, plus a range of new counterexamples to the strong form. The methods are scalable to other concentrated groups and connect to recent work on twisted counting, offering a roadmap for proving conjectures in broader arithmetic-statistical settings.
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.