Work

Inductive methods for counting number fields

Year: 2025

Type: article

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 G/T -extensions. However, we do not require that most 𝑇-extensions of a G / T G/T -extension are 𝐺-extensions. Our new results use 𝑇 either abelian or S 3 m S_{3}^{m} , though our framework is general.

Source: Journal für die reine und angewandte Mathematik (Crelles Journal)

DOI: 10.1515/crelle-2025-0074

Authors: Brandon Alberts, Robert J. Lemke Oliver, Jiuya Wang, Melanie Matchett Wood

Institutions: Eastern Michigan University, University of Wisconsin–Madison, University of Georgia, Harvard University

Language: English

Topic: Advanced Algebra and Geometry

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $2,712

Funders: National Science Foundation, Harvard University, Radcliffe Institute for Advanced Study, Harvard University