Counting abelian extensions by Artin-Schreier conductor

TL;DR

This paper provides an exact count of abelian G-extensions of rational function fields in characteristic p using Artin-Schreier conductors, revealing a rational generating function and suggesting a geometric interpretation.

Contribution

It introduces a novel counting method for G-extensions via Artin-Schreier conductors and establishes the rationality of the associated generating function.

Findings

The generating function for counting is rational.

Provides an exact counting formula for G-extensions.

Suggests a potential geometric interpretation.

Abstract

Let $G$ be a finite abelian $p$-group. We count \'etale $G$-extensions of global rational function fields $\mathbb F_q(T)$ of characteristic $p$ 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.