Counting abelian extensions by Artin–Schreier conductor

Let G G be a finite abelian p p -group. We count étale G G -extensions of global rational function fields F q ( T ) 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.