Work

Counting abelian extensions by Artin–Schreier conductor

Year: 2025

Type: article

Abstract: Let G G be a finite abelian p p -group. We count étale G G -extensions of global rational function fields F q ( T ) \mathbb F_q(T) of characteristic p p by the degree of what we call their Artin–Schreier conductor. The corresponding (ordinary) generating function turns out to be rational. This gives an exact answer to the counting problem, and seems to beg for a geometric interpretation. This is in contrast with the generating functions for the ordinary conductor (from class field theory) and the discriminant, which in general have no meromorphic continuation to the entire complex plane.

Source: Proceedings of the American Mathematical Society

DOI: 10.1090/proc/17440

Author: Fabian Gundlach

Institution: Paderborn University

Language: English

Topic: Algebraic Geometry and Number Theory

Subfield: Geometry and Topology

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Deutsche Forschungsgemeinschaft, Universität Paderborn

Awards: TRR 358: Ganzzahlige Strukturen in Geometrie und Darstellungstheorie, G8945940722