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Counting Frobenius extensions over local function fields

Jürgen Klüners, Raphael Müller

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.

Counting Frobenius extensions over local function fields

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.

Paper Structure

This paper contains 12 sections, 23 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. History
  3. Structure of the paper
  4. Acknowledgements
  5. Preliminaries
  6. Tamely ramified cyclic extensions
  7. Artin-Schreier Theory
  8. Galois Groups and Discriminants
  9. Counting Business over pd Points
  10. Counting Business over p^ ell(I)d Points
  11. The case J=|I|>1
  12. General Case

Key Result

Theorem 1.1

There exist a $(p-1)pd$-periodic function $\beta_I(x)$ and a $(p^{\ell(I)}-1)pd$-periodic function $\tilde{\beta}_I(x)$ such that $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: Field diagram

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 2.1: Classification of tamely ramified extensions
  • proof
  • Theorem 2.2: Artin-Schreier theory
  • Lemma 2.3
  • Definition 2.4
  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • ...and 40 more