Counting Frobenius extensions over local function fields

TL;DR

This paper analyzes the asymptotic growth of certain Galois extensions over local function fields, providing explicit counting results for various non-abelian groups, including Frobenius groups.

Contribution

It offers a comprehensive solution to counting extensions with specified Galois groups over local function fields, extending previous results to non-abelian groups.

Findings

Determined growth rates for extensions with Galois group in AGL_1(p).

Solved counting problems for groups in cyclic extension towers.

Provided explicit counts for S_3, dihedral groups, and Frobenius groups.

Abstract

We determine the asymptotic growth of extensions of local function fields of characteristic p counted by discriminant, where the Galois group is a subgroup of the affine group AGL_1(p). More general, we solve the corresponding counting problems for all groups which arise in a tower of a cyclic extension of order p over a cyclic extension of degree d coprime to p. This in particular give answers for certain non-abelian groups including S_3, dihedral groups of order 2p, and many Frobenius groups.